--- a/src/HOL/Import/HOLLight/HOLLight.thy Wed Jul 13 00:23:24 2011 +0900
+++ b/src/HOL/Import/HOLLight/HOLLight.thy Wed Jul 13 00:29:33 2011 +0900
@@ -1,6 +1,6 @@
(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
-theory HOLLight = "../HOLLightCompat" + "../HOL4Syntax":
+theory HOLLight imports "../HOLLightCompat" "../HOL4Syntax" begin
;setup_theory hollight
@@ -13,235 +13,290 @@
lemma DEF__FALSITY_: "_FALSITY_ = False"
by (import hollight DEF__FALSITY_)
-lemma CONJ_ACI: "((p::bool) & (q::bool)) = (q & p) &
-((p & q) & (r::bool)) = (p & q & r) &
+lemma CONJ_ACI: "(p & q) = (q & p) &
+((p & q) & r) = (p & q & r) &
(p & q & r) = (q & p & r) & (p & p) = p & (p & p & q) = (p & q)"
by (import hollight CONJ_ACI)
-lemma DISJ_ACI: "((p::bool) | (q::bool)) = (q | p) &
-((p | q) | (r::bool)) = (p | q | r) &
+lemma DISJ_ACI: "(p | q) = (q | p) &
+((p | q) | r) = (p | q | r) &
(p | q | r) = (q | p | r) & (p | p) = p & (p | p | q) = (p | q)"
by (import hollight DISJ_ACI)
-lemma EQ_CLAUSES: "ALL t::bool.
- (True = t) = t &
- (t = True) = t & (False = t) = (~ t) & (t = False) = (~ t)"
+lemma IMP_CONJ_ALT: "(p & q --> r) = (q --> p --> r)"
+ by (import hollight IMP_CONJ_ALT)
+
+lemma EQ_CLAUSES: "(True = t) = t & (t = True) = t & (False = t) = (~ t) & (t = False) = (~ t)"
by (import hollight EQ_CLAUSES)
lemma NOT_CLAUSES_WEAK: "(~ True) = False & (~ False) = True"
by (import hollight NOT_CLAUSES_WEAK)
-lemma AND_CLAUSES: "ALL t::bool.
- (True & t) = t &
- (t & True) = t & (False & t) = False & (t & False) = False & (t & t) = t"
+lemma AND_CLAUSES: "(True & t) = t &
+(t & True) = t & (False & t) = False & (t & False) = False & (t & t) = t"
by (import hollight AND_CLAUSES)
-lemma OR_CLAUSES: "ALL t::bool.
- (True | t) = True &
- (t | True) = True & (False | t) = t & (t | False) = t & (t | t) = t"
+lemma OR_CLAUSES: "(True | t) = True &
+(t | True) = True & (False | t) = t & (t | False) = t & (t | t) = t"
by (import hollight OR_CLAUSES)
-lemma IMP_CLAUSES: "ALL t::bool.
- (True --> t) = t &
- (t --> True) = True &
- (False --> t) = True & (t --> t) = True & (t --> False) = (~ t)"
+lemma IMP_CLAUSES: "(True --> t) = t &
+(t --> True) = True &
+(False --> t) = True & (t --> t) = True & (t --> False) = (~ t)"
by (import hollight IMP_CLAUSES)
-lemma IMP_EQ_CLAUSE: "((x::'q_864::type) = x --> (p::bool)) = p"
+lemma IMP_EQ_CLAUSE: "((x::'q_851) = x --> (p::bool)) = p"
by (import hollight IMP_EQ_CLAUSE)
-lemma SWAP_FORALL_THM: "ALL P::'A::type => 'B::type => bool.
- (ALL x::'A::type. All (P x)) = (ALL (y::'B::type) x::'A::type. P x y)"
- by (import hollight SWAP_FORALL_THM)
-
-lemma SWAP_EXISTS_THM: "ALL P::'A::type => 'B::type => bool.
- (EX x::'A::type. Ex (P x)) = (EX (x::'B::type) xa::'A::type. P xa x)"
- by (import hollight SWAP_EXISTS_THM)
-
-lemma TRIV_EXISTS_AND_THM: "ALL (P::bool) Q::bool.
- (EX x::'A::type. P & Q) = ((EX x::'A::type. P) & (EX x::'A::type. Q))"
+lemma TRIV_EXISTS_AND_THM: "(EX x::'A. (P::bool) & (Q::bool)) = ((EX x::'A. P) & (EX x::'A. Q))"
by (import hollight TRIV_EXISTS_AND_THM)
-lemma TRIV_AND_EXISTS_THM: "ALL (P::bool) Q::bool.
- ((EX x::'A::type. P) & (EX x::'A::type. Q)) = (EX x::'A::type. P & Q)"
+lemma TRIV_AND_EXISTS_THM: "((EX x::'A. (P::bool)) & (EX x::'A. (Q::bool))) = (EX x::'A. P & Q)"
by (import hollight TRIV_AND_EXISTS_THM)
-lemma TRIV_FORALL_OR_THM: "ALL (P::bool) Q::bool.
- (ALL x::'A::type. P | Q) = ((ALL x::'A::type. P) | (ALL x::'A::type. Q))"
+lemma TRIV_FORALL_OR_THM: "(ALL x::'A. (P::bool) | (Q::bool)) = ((ALL x::'A. P) | (ALL x::'A. Q))"
by (import hollight TRIV_FORALL_OR_THM)
-lemma TRIV_OR_FORALL_THM: "ALL (P::bool) Q::bool.
- ((ALL x::'A::type. P) | (ALL x::'A::type. Q)) = (ALL x::'A::type. P | Q)"
+lemma TRIV_OR_FORALL_THM: "((ALL x::'A. (P::bool)) | (ALL x::'A. (Q::bool))) = (ALL x::'A. P | Q)"
by (import hollight TRIV_OR_FORALL_THM)
-lemma TRIV_FORALL_IMP_THM: "ALL (P::bool) Q::bool.
- (ALL x::'A::type. P --> Q) =
- ((EX x::'A::type. P) --> (ALL x::'A::type. Q))"
+lemma TRIV_FORALL_IMP_THM: "(ALL x::'A. (P::bool) --> (Q::bool)) = ((EX x::'A. P) --> (ALL x::'A. Q))"
by (import hollight TRIV_FORALL_IMP_THM)
-lemma TRIV_EXISTS_IMP_THM: "ALL (P::bool) Q::bool.
- (EX x::'A::type. P --> Q) =
- ((ALL x::'A::type. P) --> (EX x::'A::type. Q))"
+lemma TRIV_EXISTS_IMP_THM: "(EX x::'A. (P::bool) --> (Q::bool)) = ((ALL x::'A. P) --> (EX x::'A. Q))"
by (import hollight TRIV_EXISTS_IMP_THM)
-lemma EXISTS_UNIQUE_ALT: "ALL P::'A::type => bool.
- Ex1 P = (EX x::'A::type. ALL y::'A::type. P y = (x = y))"
+lemma EXISTS_UNIQUE_ALT: "Ex1 (P::'A => bool) = (EX x::'A. ALL y::'A. P y = (x = y))"
by (import hollight EXISTS_UNIQUE_ALT)
-lemma SELECT_UNIQUE: "ALL (P::'A::type => bool) x::'A::type.
- (ALL y::'A::type. P y = (y = x)) --> Eps P = x"
+lemma SELECT_UNIQUE: "(!!y::'A. (P::'A => bool) y = (y = (x::'A))) ==> Eps P = x"
by (import hollight SELECT_UNIQUE)
-lemma EXCLUDED_MIDDLE: "ALL t::bool. t | ~ t"
+lemma EXCLUDED_MIDDLE: "t | ~ t"
by (import hollight EXCLUDED_MIDDLE)
-definition COND :: "bool => 'A => 'A => 'A" where
- "COND ==
-%(t::bool) (t1::'A::type) t2::'A::type.
- SOME x::'A::type. (t = True --> x = t1) & (t = False --> x = t2)"
-
-lemma DEF_COND: "COND =
-(%(t::bool) (t1::'A::type) t2::'A::type.
- SOME x::'A::type. (t = True --> x = t1) & (t = False --> x = t2))"
- by (import hollight DEF_COND)
-
-lemma COND_CLAUSES: "ALL (x::'A::type) xa::'A::type. COND True x xa = x & COND False x xa = xa"
+lemma COND_CLAUSES: "(if True then x::'A else (xa::'A)) = x & (if False then x else xa) = xa"
by (import hollight COND_CLAUSES)
-lemma COND_EXPAND: "ALL (b::bool) (t1::bool) t2::bool. COND b t1 t2 = ((~ b | t1) & (b | t2))"
+lemma COND_EXPAND: "(if b then t1 else t2) = ((~ b | t1) & (b | t2))"
by (import hollight COND_EXPAND)
-lemma COND_ID: "ALL (b::bool) t::'A::type. COND b t t = t"
- by (import hollight COND_ID)
-
-lemma COND_RAND: "ALL (b::bool) (f::'A::type => 'B::type) (x::'A::type) y::'A::type.
- f (COND b x y) = COND b (f x) (f y)"
- by (import hollight COND_RAND)
-
-lemma COND_RATOR: "ALL (b::bool) (f::'A::type => 'B::type) (g::'A::type => 'B::type)
- x::'A::type. COND b f g x = COND b (f x) (g x)"
+lemma COND_RATOR: "(if b::bool then f::'A => 'B else (g::'A => 'B)) (x::'A) =
+(if b then f x else g x)"
by (import hollight COND_RATOR)
-lemma COND_ABS: "ALL (b::bool) (f::'A::type => 'B::type) g::'A::type => 'B::type.
- (%x::'A::type. COND b (f x) (g x)) = COND b f g"
+lemma COND_ABS: "(%x::'A. if b::bool then (f::'A => 'B) x else (g::'A => 'B) x) =
+(if b then f else g)"
by (import hollight COND_ABS)
-lemma MONO_COND: "((A::bool) --> (B::bool)) & ((C::bool) --> (D::bool)) -->
-COND (b::bool) A C --> COND b B D"
+lemma MONO_COND: "[| (A --> B) & (C --> D); if b then A else C |] ==> if b then B else D"
by (import hollight MONO_COND)
-lemma COND_ELIM_THM: "(P::'A::type => bool) (COND (c::bool) (x::'A::type) (y::'A::type)) =
-((c --> P x) & (~ c --> P y))"
- by (import hollight COND_ELIM_THM)
-
-lemma SKOLEM_THM: "ALL P::'A::type => 'B::type => bool.
- (ALL x::'A::type. Ex (P x)) =
- (EX x::'A::type => 'B::type. ALL xa::'A::type. P xa (x xa))"
+lemma SKOLEM_THM: "(ALL x::'A. Ex ((P::'A => 'B => bool) x)) =
+(EX x::'A => 'B. ALL xa::'A. P xa (x xa))"
by (import hollight SKOLEM_THM)
-lemma UNIQUE_SKOLEM_ALT: "ALL P::'A::type => 'B::type => bool.
- (ALL x::'A::type. Ex1 (P x)) =
- (EX f::'A::type => 'B::type.
- ALL (x::'A::type) y::'B::type. P x y = (f x = y))"
+lemma UNIQUE_SKOLEM_ALT: "(ALL x::'A. Ex1 ((P::'A => 'B => bool) x)) =
+(EX f::'A => 'B. ALL (x::'A) y::'B. P x y = (f x = y))"
by (import hollight UNIQUE_SKOLEM_ALT)
-lemma COND_EQ_CLAUSE: "COND ((x::'q_3000::type) = x) (y::'q_2993::type) (z::'q_2993::type) = y"
+lemma COND_EQ_CLAUSE: "(if (x::'q_2963) = x then y::'q_2956 else (z::'q_2956)) = y"
by (import hollight COND_EQ_CLAUSE)
-lemma o_ASSOC: "ALL (f::'C::type => 'D::type) (g::'B::type => 'C::type)
- h::'A::type => 'B::type. f o (g o h) = f o g o h"
+lemma bool_RECURSION: "EX x::bool => 'A. x False = (a::'A) & x True = (b::'A)"
+ by (import hollight bool_RECURSION)
+
+lemma o_ASSOC: "(f::'C => 'D) o ((g::'B => 'C) o (h::'A => 'B)) = f o g o h"
by (import hollight o_ASSOC)
-lemma I_O_ID: "ALL f::'A::type => 'B::type. id o f = f & f o id = f"
+lemma I_O_ID: "id o (f::'A => 'B) = f & f o id = f"
by (import hollight I_O_ID)
-lemma EXISTS_ONE_REP: "EX x::bool. x"
+lemma EXISTS_ONE_REP: "EX x. x"
by (import hollight EXISTS_ONE_REP)
-lemma one_axiom: "ALL f::'A::type => unit. All (op = f)"
+lemma one_axiom: "(f::'A => unit) = (x::'A => unit)"
by (import hollight one_axiom)
-lemma one_RECURSION: "ALL e::'A::type. EX x::unit => 'A::type. x () = e"
+lemma one_RECURSION: "EX x::unit => 'A. x () = (e::'A)"
by (import hollight one_RECURSION)
-lemma one_Axiom: "ALL e::'A::type. EX! fn::unit => 'A::type. fn () = e"
+lemma one_Axiom: "EX! fn::unit => 'A. fn () = (e::'A)"
by (import hollight one_Axiom)
-lemma th_cond: "(P::'A::type => bool => bool) (COND (b::bool) (x::'A::type) (y::'A::type))
- b =
-(b & P x True | ~ b & P y False)"
+lemma th_cond: "(b = False --> x = x0) & (b = True --> x = x1) ==> x = (b & x1 | ~ b & x0)"
by (import hollight th_cond)
-definition LET_END :: "'A => 'A" where
- "LET_END == %t::'A::type. t"
-
-lemma DEF_LET_END: "LET_END = (%t::'A::type. t)"
+definition
+ LET_END :: "'A => 'A" where
+ "LET_END == %t::'A. t"
+
+lemma DEF_LET_END: "LET_END = (%t::'A. t)"
by (import hollight DEF_LET_END)
-definition GABS :: "('A => bool) => 'A" where
- "(op ==::(('A::type => bool) => 'A::type)
- => (('A::type => bool) => 'A::type) => prop)
- (GABS::('A::type => bool) => 'A::type)
- (Eps::('A::type => bool) => 'A::type)"
-
-lemma DEF_GABS: "(op =::(('A::type => bool) => 'A::type)
- => (('A::type => bool) => 'A::type) => bool)
- (GABS::('A::type => bool) => 'A::type)
- (Eps::('A::type => bool) => 'A::type)"
- by (import hollight DEF_GABS)
-
-definition GEQ :: "'A => 'A => bool" where
- "(op ==::('A::type => 'A::type => bool)
- => ('A::type => 'A::type => bool) => prop)
- (GEQ::'A::type => 'A::type => bool) (op =::'A::type => 'A::type => bool)"
-
-lemma DEF_GEQ: "(op =::('A::type => 'A::type => bool)
- => ('A::type => 'A::type => bool) => bool)
- (GEQ::'A::type => 'A::type => bool) (op =::'A::type => 'A::type => bool)"
- by (import hollight DEF_GEQ)
-
-lemma PAIR_EXISTS_THM: "EX (x::'A::type => 'B::type => bool) (a::'A::type) b::'B::type.
- x = Pair_Rep a b"
+consts
+ "_SEQPATTERN" :: "('q_4007 => 'q_4004 => bool)
+=> ('q_4007 => 'q_4004 => bool) => 'q_4007 => 'q_4004 => bool" ("'_SEQPATTERN")
+
+defs
+ "_SEQPATTERN_def": "_SEQPATTERN ==
+%(r::'q_4007 => 'q_4004 => bool) (s::'q_4007 => 'q_4004 => bool) x::'q_4007.
+ if Ex (r x) then r x else s x"
+
+lemma DEF__SEQPATTERN: "_SEQPATTERN =
+(%(r::'q_4007 => 'q_4004 => bool) (s::'q_4007 => 'q_4004 => bool)
+ x::'q_4007. if Ex (r x) then r x else s x)"
+ by (import hollight DEF__SEQPATTERN)
+
+consts
+ "_UNGUARDED_PATTERN" :: "bool => bool => bool" ("'_UNGUARDED'_PATTERN")
+
+defs
+ "_UNGUARDED_PATTERN_def": "_UNGUARDED_PATTERN == op &"
+
+lemma DEF__UNGUARDED_PATTERN: "_UNGUARDED_PATTERN = op &"
+ by (import hollight DEF__UNGUARDED_PATTERN)
+
+consts
+ "_GUARDED_PATTERN" :: "bool => bool => bool => bool" ("'_GUARDED'_PATTERN")
+
+defs
+ "_GUARDED_PATTERN_def": "_GUARDED_PATTERN == %p g r. p & g & r"
+
+lemma DEF__GUARDED_PATTERN: "_GUARDED_PATTERN = (%p g r. p & g & r)"
+ by (import hollight DEF__GUARDED_PATTERN)
+
+consts
+ "_MATCH" :: "'q_4049 => ('q_4049 => 'q_4053 => bool) => 'q_4053" ("'_MATCH")
+
+defs
+ "_MATCH_def": "_MATCH ==
+%(e::'q_4049) r::'q_4049 => 'q_4053 => bool.
+ if Ex1 (r e) then Eps (r e) else SOME z::'q_4053. False"
+
+lemma DEF__MATCH: "_MATCH =
+(%(e::'q_4049) r::'q_4049 => 'q_4053 => bool.
+ if Ex1 (r e) then Eps (r e) else SOME z::'q_4053. False)"
+ by (import hollight DEF__MATCH)
+
+consts
+ "_FUNCTION" :: "('q_4071 => 'q_4075 => bool) => 'q_4071 => 'q_4075" ("'_FUNCTION")
+
+defs
+ "_FUNCTION_def": "_FUNCTION ==
+%(r::'q_4071 => 'q_4075 => bool) x::'q_4071.
+ if Ex1 (r x) then Eps (r x) else SOME z::'q_4075. False"
+
+lemma DEF__FUNCTION: "_FUNCTION =
+(%(r::'q_4071 => 'q_4075 => bool) x::'q_4071.
+ if Ex1 (r x) then Eps (r x) else SOME z::'q_4075. False)"
+ by (import hollight DEF__FUNCTION)
+
+lemma PAIR_EXISTS_THM: "EX (x::'A => 'B => bool) (a::'A) b::'B. x = Pair_Rep a b"
by (import hollight PAIR_EXISTS_THM)
-definition CURRY :: "('A * 'B => 'C) => 'A => 'B => 'C" where
- "CURRY ==
-%(u::'A::type * 'B::type => 'C::type) (ua::'A::type) ub::'B::type.
- u (ua, ub)"
-
-lemma DEF_CURRY: "CURRY =
-(%(u::'A::type * 'B::type => 'C::type) (ua::'A::type) ub::'B::type.
- u (ua, ub))"
- by (import hollight DEF_CURRY)
-
-definition UNCURRY :: "('A => 'B => 'C) => 'A * 'B => 'C" where
- "UNCURRY ==
-%(u::'A::type => 'B::type => 'C::type) ua::'A::type * 'B::type.
- u (fst ua) (snd ua)"
-
-lemma DEF_UNCURRY: "UNCURRY =
-(%(u::'A::type => 'B::type => 'C::type) ua::'A::type * 'B::type.
- u (fst ua) (snd ua))"
+lemma pair_RECURSION: "EX x::'A * 'B => 'C.
+ ALL (a0::'A) a1::'B. x (a0, a1) = (PAIR'::'A => 'B => 'C) a0 a1"
+ by (import hollight pair_RECURSION)
+
+definition
+ UNCURRY :: "('A => 'B => 'C) => 'A * 'B => 'C" where
+ "UNCURRY == %(u::'A => 'B => 'C) ua::'A * 'B. u (fst ua) (snd ua)"
+
+lemma DEF_UNCURRY: "UNCURRY = (%(u::'A => 'B => 'C) ua::'A * 'B. u (fst ua) (snd ua))"
by (import hollight DEF_UNCURRY)
-definition PASSOC :: "(('A * 'B) * 'C => 'D) => 'A * 'B * 'C => 'D" where
+definition
+ PASSOC :: "(('A * 'B) * 'C => 'D) => 'A * 'B * 'C => 'D" where
"PASSOC ==
-%(u::('A::type * 'B::type) * 'C::type => 'D::type)
- ua::'A::type * 'B::type * 'C::type.
+%(u::('A * 'B) * 'C => 'D) ua::'A * 'B * 'C.
u ((fst ua, fst (snd ua)), snd (snd ua))"
lemma DEF_PASSOC: "PASSOC =
-(%(u::('A::type * 'B::type) * 'C::type => 'D::type)
- ua::'A::type * 'B::type * 'C::type.
+(%(u::('A * 'B) * 'C => 'D) ua::'A * 'B * 'C.
u ((fst ua, fst (snd ua)), snd (snd ua)))"
by (import hollight DEF_PASSOC)
-lemma num_Axiom: "ALL (e::'A::type) f::'A::type => nat => 'A::type.
- EX! fn::nat => 'A::type. fn 0 = e & (ALL n::nat. fn (Suc n) = f (fn n) n)"
- by (import hollight num_Axiom)
-
-lemma ADD_CLAUSES: "(ALL x::nat. 0 + x = x) &
-(ALL x::nat. x + 0 = x) &
+lemma LAMBDA_PAIR_THM: "(x::'q_4547 * 'q_4546 => 'q_4539) =
+(SOME f::'q_4547 * 'q_4546 => 'q_4539.
+ ALL (xa::'q_4547) y::'q_4546. f (xa, y) = x (xa, y))"
+ by (import hollight LAMBDA_PAIR_THM)
+
+lemma FORALL_PAIRED_THM: "All (SOME f::'q_4576 * 'q_4575 => bool.
+ ALL (x::'q_4576) y::'q_4575.
+ f (x, y) = (P::'q_4576 => 'q_4575 => bool) x y) =
+(ALL x::'q_4576. All (P x))"
+ by (import hollight FORALL_PAIRED_THM)
+
+lemma EXISTS_PAIRED_THM: "Ex (SOME f::'q_4612 * 'q_4611 => bool.
+ ALL (x::'q_4612) y::'q_4611.
+ f (x, y) = (P::'q_4612 => 'q_4611 => bool) x y) =
+(EX x::'q_4612. Ex (P x))"
+ by (import hollight EXISTS_PAIRED_THM)
+
+lemma FORALL_TRIPLED_THM: "All (SOME f::'q_4649 * 'q_4648 * 'q_4647 => bool.
+ ALL (x::'q_4649) (y::'q_4648) z::'q_4647.
+ f (x, y, z) = (P::'q_4649 => 'q_4648 => 'q_4647 => bool) x y z) =
+(ALL (x::'q_4649) y::'q_4648. All (P x y))"
+ by (import hollight FORALL_TRIPLED_THM)
+
+lemma EXISTS_TRIPLED_THM: "Ex (SOME f::'q_4695 * 'q_4694 * 'q_4693 => bool.
+ ALL (x::'q_4695) (y::'q_4694) z::'q_4693.
+ f (x, y, z) = (P::'q_4695 => 'q_4694 => 'q_4693 => bool) x y z) =
+(EX (x::'q_4695) y::'q_4694. Ex (P x y))"
+ by (import hollight EXISTS_TRIPLED_THM)
+
+lemma IND_SUC_0_EXISTS: "EX (x::ind => ind) z::ind.
+ (ALL (x1::ind) x2::ind. (x x1 = x x2) = (x1 = x2)) &
+ (ALL xa::ind. x xa ~= z)"
+ by (import hollight IND_SUC_0_EXISTS)
+
+definition
+ IND_SUC :: "ind => ind" where
+ "IND_SUC ==
+SOME f. EX z. (ALL x1 x2. (f x1 = f x2) = (x1 = x2)) & (ALL x. f x ~= z)"
+
+lemma DEF_IND_SUC: "IND_SUC =
+(SOME f. EX z. (ALL x1 x2. (f x1 = f x2) = (x1 = x2)) & (ALL x. f x ~= z))"
+ by (import hollight DEF_IND_SUC)
+
+definition
+ IND_0 :: "ind" where
+ "IND_0 ==
+SOME z.
+ (ALL x1 x2. (IND_SUC x1 = IND_SUC x2) = (x1 = x2)) &
+ (ALL x. IND_SUC x ~= z)"
+
+lemma DEF_IND_0: "IND_0 =
+(SOME z.
+ (ALL x1 x2. (IND_SUC x1 = IND_SUC x2) = (x1 = x2)) &
+ (ALL x. IND_SUC x ~= z))"
+ by (import hollight DEF_IND_0)
+
+definition
+ NUM_REP :: "ind => bool" where
+ "NUM_REP ==
+%a. ALL NUM_REP'.
+ (ALL a.
+ a = IND_0 | (EX i. a = IND_SUC i & NUM_REP' i) -->
+ NUM_REP' a) -->
+ NUM_REP' a"
+
+lemma DEF_NUM_REP: "NUM_REP =
+(%a. ALL NUM_REP'.
+ (ALL a.
+ a = IND_0 | (EX i. a = IND_SUC i & NUM_REP' i) -->
+ NUM_REP' a) -->
+ NUM_REP' a)"
+ by (import hollight DEF_NUM_REP)
+
+lemma num_RECURSION_STD: "EX fn::nat => 'Z.
+ fn (0::nat) = (e::'Z) &
+ (ALL n::nat. fn (Suc n) = (f::nat => 'Z => 'Z) n (fn n))"
+ by (import hollight num_RECURSION_STD)
+
+lemma ADD_CLAUSES: "(ALL x::nat. (0::nat) + x = x) &
+(ALL x::nat. x + (0::nat) = x) &
(ALL (x::nat) xa::nat. Suc x + xa = Suc (x + xa)) &
(ALL (x::nat) xa::nat. x + Suc xa = Suc (x + xa))"
by (import hollight ADD_CLAUSES)
@@ -250,25 +305,25 @@
m + n + (p::nat) = m + (n + p) & m + (n + p) = n + (m + p)"
by (import hollight ADD_AC)
-lemma EQ_ADD_LCANCEL_0: "ALL (m::nat) n::nat. (m + n = m) = (n = 0)"
+lemma EQ_ADD_LCANCEL_0: "((m::nat) + (n::nat) = m) = (n = (0::nat))"
by (import hollight EQ_ADD_LCANCEL_0)
-lemma EQ_ADD_RCANCEL_0: "ALL (x::nat) xa::nat. (x + xa = xa) = (x = 0)"
+lemma EQ_ADD_RCANCEL_0: "((x::nat) + (xa::nat) = xa) = (x = (0::nat))"
by (import hollight EQ_ADD_RCANCEL_0)
-lemma ONE: "NUMERAL_BIT1 0 = Suc 0"
- by (import hollight ONE)
-
-lemma TWO: "NUMERAL_BIT0 (NUMERAL_BIT1 0) = Suc (NUMERAL_BIT1 0)"
+lemma BIT1: "2 * x + 1 = Suc (x + x)"
+ by (import hollight BIT1)
+
+lemma BIT1_THM: "2 * x + 1 = Suc (x + x)"
+ by (import hollight BIT1_THM)
+
+lemma TWO: "2 = Suc 1"
by (import hollight TWO)
-lemma ADD1: "ALL x::nat. Suc x = x + NUMERAL_BIT1 0"
- by (import hollight ADD1)
-
-lemma MULT_CLAUSES: "(ALL x::nat. 0 * x = 0) &
-(ALL x::nat. x * 0 = 0) &
-(ALL x::nat. NUMERAL_BIT1 0 * x = x) &
-(ALL x::nat. x * NUMERAL_BIT1 0 = x) &
+lemma MULT_CLAUSES: "(ALL x::nat. (0::nat) * x = (0::nat)) &
+(ALL x::nat. x * (0::nat) = (0::nat)) &
+(ALL x::nat. (1::nat) * x = x) &
+(ALL x::nat. x * (1::nat) = x) &
(ALL (x::nat) xa::nat. Suc x * xa = x * xa + xa) &
(ALL (x::nat) xa::nat. x * Suc xa = x + x * xa)"
by (import hollight MULT_CLAUSES)
@@ -277,1496 +332,831 @@
m * n * (p::nat) = m * (n * p) & m * (n * p) = n * (m * p)"
by (import hollight MULT_AC)
-lemma MULT_2: "ALL n::nat. NUMERAL_BIT0 (NUMERAL_BIT1 0) * n = n + n"
- by (import hollight MULT_2)
-
-lemma MULT_EQ_1: "ALL (m::nat) n::nat.
- (m * n = NUMERAL_BIT1 0) = (m = NUMERAL_BIT1 0 & n = NUMERAL_BIT1 0)"
- by (import hollight MULT_EQ_1)
-
-definition EXP :: "nat => nat => nat" where
- "EXP ==
-SOME EXP::nat => nat => nat.
- (ALL m::nat. EXP m 0 = NUMERAL_BIT1 0) &
- (ALL (m::nat) n::nat. EXP m (Suc n) = m * EXP m n)"
-
-lemma DEF_EXP: "EXP =
-(SOME EXP::nat => nat => nat.
- (ALL m::nat. EXP m 0 = NUMERAL_BIT1 0) &
- (ALL (m::nat) n::nat. EXP m (Suc n) = m * EXP m n))"
- by (import hollight DEF_EXP)
-
-lemma EXP_EQ_0: "ALL (m::nat) n::nat. (EXP m n = 0) = (m = 0 & n ~= 0)"
- by (import hollight EXP_EQ_0)
-
-lemma EXP_ADD: "ALL (m::nat) (n::nat) p::nat. EXP m (n + p) = EXP m n * EXP m p"
- by (import hollight EXP_ADD)
-
-lemma EXP_ONE: "ALL n::nat. EXP (NUMERAL_BIT1 0) n = NUMERAL_BIT1 0"
- by (import hollight EXP_ONE)
-
-lemma EXP_1: "ALL x::nat. EXP x (NUMERAL_BIT1 0) = x"
- by (import hollight EXP_1)
-
-lemma EXP_2: "ALL x::nat. EXP x (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = x * x"
- by (import hollight EXP_2)
-
-lemma MULT_EXP: "ALL (p::nat) (m::nat) n::nat. EXP (m * n) p = EXP m p * EXP n p"
- by (import hollight MULT_EXP)
-
-lemma EXP_MULT: "ALL (m::nat) (n::nat) p::nat. EXP m (n * p) = EXP (EXP m n) p"
- by (import hollight EXP_MULT)
-
-consts
- "<=" :: "nat => nat => bool" ("<=")
-
-defs
- "<=_def": "<= ==
-SOME u::nat => nat => bool.
- (ALL m::nat. u m 0 = (m = 0)) &
- (ALL (m::nat) n::nat. u m (Suc n) = (m = Suc n | u m n))"
-
-lemma DEF__lessthan__equal_: "<= =
-(SOME u::nat => nat => bool.
- (ALL m::nat. u m 0 = (m = 0)) &
- (ALL (m::nat) n::nat. u m (Suc n) = (m = Suc n | u m n)))"
- by (import hollight DEF__lessthan__equal_)
-
-consts
- "<" :: "nat => nat => bool" ("<")
-
-defs
- "<_def": "< ==
-SOME u::nat => nat => bool.
- (ALL m::nat. u m 0 = False) &
- (ALL (m::nat) n::nat. u m (Suc n) = (m = n | u m n))"
-
-lemma DEF__lessthan_: "< =
-(SOME u::nat => nat => bool.
- (ALL m::nat. u m 0 = False) &
- (ALL (m::nat) n::nat. u m (Suc n) = (m = n | u m n)))"
- by (import hollight DEF__lessthan_)
-
-consts
- ">=" :: "nat => nat => bool" (">=")
-
-defs
- ">=_def": ">= == %(u::nat) ua::nat. <= ua u"
-
-lemma DEF__greaterthan__equal_: ">= = (%(u::nat) ua::nat. <= ua u)"
- by (import hollight DEF__greaterthan__equal_)
-
-consts
- ">" :: "nat => nat => bool" (">")
-
-defs
- ">_def": "> == %(u::nat) ua::nat. < ua u"
-
-lemma DEF__greaterthan_: "> = (%(u::nat) ua::nat. < ua u)"
- by (import hollight DEF__greaterthan_)
-
-lemma LE_SUC_LT: "ALL (m::nat) n::nat. <= (Suc m) n = < m n"
- by (import hollight LE_SUC_LT)
-
-lemma LT_SUC_LE: "ALL (m::nat) n::nat. < m (Suc n) = <= m n"
- by (import hollight LT_SUC_LE)
-
-lemma LE_SUC: "ALL (x::nat) xa::nat. <= (Suc x) (Suc xa) = <= x xa"
- by (import hollight LE_SUC)
-
-lemma LT_SUC: "ALL (x::nat) xa::nat. < (Suc x) (Suc xa) = < x xa"
- by (import hollight LT_SUC)
-
-lemma LE_0: "All (<= 0)"
- by (import hollight LE_0)
-
-lemma LT_0: "ALL x::nat. < 0 (Suc x)"
- by (import hollight LT_0)
-
-lemma LE_REFL: "ALL n::nat. <= n n"
- by (import hollight LE_REFL)
-
-lemma LT_REFL: "ALL n::nat. ~ < n n"
- by (import hollight LT_REFL)
-
-lemma LE_ANTISYM: "ALL (m::nat) n::nat. (<= m n & <= n m) = (m = n)"
- by (import hollight LE_ANTISYM)
-
-lemma LT_ANTISYM: "ALL (m::nat) n::nat. ~ (< m n & < n m)"
+lemma EXP_EQ_1: "((x::nat) ^ (n::nat) = (1::nat)) = (x = (1::nat) | n = (0::nat))"
+ by (import hollight EXP_EQ_1)
+
+lemma LT_ANTISYM: "~ ((m::nat) < (n::nat) & n < m)"
by (import hollight LT_ANTISYM)
-lemma LET_ANTISYM: "ALL (m::nat) n::nat. ~ (<= m n & < n m)"
+lemma LET_ANTISYM: "~ ((m::nat) <= (n::nat) & n < m)"
by (import hollight LET_ANTISYM)
-lemma LTE_ANTISYM: "ALL (x::nat) xa::nat. ~ (< x xa & <= xa x)"
+lemma LTE_ANTISYM: "~ ((x::nat) < (xa::nat) & xa <= x)"
by (import hollight LTE_ANTISYM)
-lemma LE_TRANS: "ALL (m::nat) (n::nat) p::nat. <= m n & <= n p --> <= m p"
- by (import hollight LE_TRANS)
-
-lemma LT_TRANS: "ALL (m::nat) (n::nat) p::nat. < m n & < n p --> < m p"
- by (import hollight LT_TRANS)
-
-lemma LET_TRANS: "ALL (m::nat) (n::nat) p::nat. <= m n & < n p --> < m p"
- by (import hollight LET_TRANS)
-
-lemma LTE_TRANS: "ALL (m::nat) (n::nat) p::nat. < m n & <= n p --> < m p"
- by (import hollight LTE_TRANS)
-
-lemma LE_CASES: "ALL (m::nat) n::nat. <= m n | <= n m"
- by (import hollight LE_CASES)
-
-lemma LT_CASES: "ALL (m::nat) n::nat. < m n | < n m | m = n"
+lemma LT_CASES: "(m::nat) < (n::nat) | n < m | m = n"
by (import hollight LT_CASES)
-lemma LET_CASES: "ALL (m::nat) n::nat. <= m n | < n m"
- by (import hollight LET_CASES)
-
-lemma LTE_CASES: "ALL (x::nat) xa::nat. < x xa | <= xa x"
+lemma LTE_CASES: "(x::nat) < (xa::nat) | xa <= x"
by (import hollight LTE_CASES)
-lemma LT_NZ: "ALL n::nat. < 0 n = (n ~= 0)"
- by (import hollight LT_NZ)
-
-lemma LE_LT: "ALL (m::nat) n::nat. <= m n = (< m n | m = n)"
- by (import hollight LE_LT)
-
-lemma LT_LE: "ALL (x::nat) xa::nat. < x xa = (<= x xa & x ~= xa)"
- by (import hollight LT_LE)
-
-lemma NOT_LE: "ALL (m::nat) n::nat. (~ <= m n) = < n m"
- by (import hollight NOT_LE)
-
-lemma NOT_LT: "ALL (m::nat) n::nat. (~ < m n) = <= n m"
- by (import hollight NOT_LT)
-
-lemma LT_IMP_LE: "ALL (x::nat) xa::nat. < x xa --> <= x xa"
- by (import hollight LT_IMP_LE)
-
-lemma EQ_IMP_LE: "ALL (m::nat) n::nat. m = n --> <= m n"
- by (import hollight EQ_IMP_LE)
-
-lemma LE_EXISTS: "ALL (m::nat) n::nat. <= m n = (EX d::nat. n = m + d)"
- by (import hollight LE_EXISTS)
-
-lemma LT_EXISTS: "ALL (m::nat) n::nat. < m n = (EX d::nat. n = m + Suc d)"
+lemma LE_1: "(ALL x::nat. x ~= (0::nat) --> (0::nat) < x) &
+(ALL x::nat. x ~= (0::nat) --> (1::nat) <= x) &
+(ALL x>0::nat. x ~= (0::nat)) &
+(ALL x>0::nat. (1::nat) <= x) &
+(ALL x>=1::nat. (0::nat) < x) & (ALL x>=1::nat. x ~= (0::nat))"
+ by (import hollight LE_1)
+
+lemma LT_EXISTS: "(m < n) = (EX d. n = m + Suc d)"
by (import hollight LT_EXISTS)
-lemma LE_ADD: "ALL (m::nat) n::nat. <= m (m + n)"
- by (import hollight LE_ADD)
-
-lemma LE_ADDR: "ALL (x::nat) xa::nat. <= xa (x + xa)"
- by (import hollight LE_ADDR)
-
-lemma LT_ADD: "ALL (m::nat) n::nat. < m (m + n) = < 0 n"
+lemma LT_ADD: "((m::nat) < m + (n::nat)) = ((0::nat) < n)"
by (import hollight LT_ADD)
-lemma LT_ADDR: "ALL (x::nat) xa::nat. < xa (x + xa) = < 0 x"
+lemma LT_ADDR: "((xa::nat) < (x::nat) + xa) = ((0::nat) < x)"
by (import hollight LT_ADDR)
-lemma LE_ADD_LCANCEL: "ALL (x::nat) (xa::nat) xb::nat. <= (x + xa) (x + xb) = <= xa xb"
- by (import hollight LE_ADD_LCANCEL)
-
-lemma LE_ADD_RCANCEL: "ALL (x::nat) (xa::nat) xb::nat. <= (x + xb) (xa + xb) = <= x xa"
- by (import hollight LE_ADD_RCANCEL)
-
-lemma LT_ADD_LCANCEL: "ALL (x::nat) (xa::nat) xb::nat. < (x + xa) (x + xb) = < xa xb"
- by (import hollight LT_ADD_LCANCEL)
-
-lemma LT_ADD_RCANCEL: "ALL (x::nat) (xa::nat) xb::nat. < (x + xb) (xa + xb) = < x xa"
- by (import hollight LT_ADD_RCANCEL)
-
-lemma LE_ADD2: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- <= m p & <= n q --> <= (m + n) (p + q)"
- by (import hollight LE_ADD2)
-
-lemma LET_ADD2: "ALL (m::nat) (n::nat) (p::nat) q::nat. <= m p & < n q --> < (m + n) (p + q)"
- by (import hollight LET_ADD2)
-
-lemma LTE_ADD2: "ALL (x::nat) (xa::nat) (xb::nat) xc::nat.
- < x xb & <= xa xc --> < (x + xa) (xb + xc)"
- by (import hollight LTE_ADD2)
-
-lemma LT_ADD2: "ALL (m::nat) (n::nat) (p::nat) q::nat. < m p & < n q --> < (m + n) (p + q)"
- by (import hollight LT_ADD2)
-
-lemma LT_MULT: "ALL (m::nat) n::nat. < 0 (m * n) = (< 0 m & < 0 n)"
- by (import hollight LT_MULT)
-
-lemma LE_MULT2: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- <= m n & <= p q --> <= (m * p) (n * q)"
- by (import hollight LE_MULT2)
-
-lemma LT_LMULT: "ALL (m::nat) (n::nat) p::nat. m ~= 0 & < n p --> < (m * n) (m * p)"
+lemma LT_LMULT: "(m::nat) ~= (0::nat) & (n::nat) < (p::nat) ==> m * n < m * p"
by (import hollight LT_LMULT)
-lemma LE_MULT_LCANCEL: "ALL (m::nat) (n::nat) p::nat. <= (m * n) (m * p) = (m = 0 | <= n p)"
+lemma LE_MULT_LCANCEL: "((m::nat) * (n::nat) <= m * (p::nat)) = (m = (0::nat) | n <= p)"
by (import hollight LE_MULT_LCANCEL)
-lemma LE_MULT_RCANCEL: "ALL (x::nat) (xa::nat) xb::nat. <= (x * xb) (xa * xb) = (<= x xa | xb = 0)"
+lemma LE_MULT_RCANCEL: "((x::nat) * (xb::nat) <= (xa::nat) * xb) = (x <= xa | xb = (0::nat))"
by (import hollight LE_MULT_RCANCEL)
-lemma LT_MULT_LCANCEL: "ALL (m::nat) (n::nat) p::nat. < (m * n) (m * p) = (m ~= 0 & < n p)"
+lemma LT_MULT_LCANCEL: "((m::nat) * (n::nat) < m * (p::nat)) = (m ~= (0::nat) & n < p)"
by (import hollight LT_MULT_LCANCEL)
-lemma LT_MULT_RCANCEL: "ALL (x::nat) (xa::nat) xb::nat. < (x * xb) (xa * xb) = (< x xa & xb ~= 0)"
+lemma LT_MULT_RCANCEL: "((x::nat) * (xb::nat) < (xa::nat) * xb) = (x < xa & xb ~= (0::nat))"
by (import hollight LT_MULT_RCANCEL)
-lemma LT_MULT2: "ALL (m::nat) (n::nat) (p::nat) q::nat. < m n & < p q --> < (m * p) (n * q)"
+lemma LT_MULT2: "(m::nat) < (n::nat) & (p::nat) < (q::nat) ==> m * p < n * q"
by (import hollight LT_MULT2)
-lemma LE_SQUARE_REFL: "ALL n::nat. <= n (n * n)"
- by (import hollight LE_SQUARE_REFL)
-
lemma WLOG_LE: "(ALL (m::nat) n::nat. (P::nat => nat => bool) m n = P n m) &
-(ALL (m::nat) n::nat. <= m n --> P m n) -->
-(ALL m::nat. All (P m))"
+(ALL (m::nat) n::nat. m <= n --> P m n)
+==> P (m::nat) (x::nat)"
by (import hollight WLOG_LE)
lemma WLOG_LT: "(ALL m::nat. (P::nat => nat => bool) m m) &
(ALL (m::nat) n::nat. P m n = P n m) &
-(ALL (m::nat) n::nat. < m n --> P m n) -->
-(ALL m::nat. All (P m))"
+(ALL (m::nat) n::nat. m < n --> P m n)
+==> P (m::nat) (x::nat)"
by (import hollight WLOG_LT)
-lemma num_WF: "ALL P::nat => bool.
- (ALL n::nat. (ALL m::nat. < m n --> P m) --> P n) --> All P"
- by (import hollight num_WF)
-
-lemma num_WOP: "ALL P::nat => bool. Ex P = (EX n::nat. P n & (ALL m::nat. < m n --> ~ P m))"
+lemma num_WOP: "Ex (P::nat => bool) = (EX n::nat. P n & (ALL m<n. ~ P m))"
by (import hollight num_WOP)
-lemma num_MAX: "ALL P::nat => bool.
- (Ex P & (EX M::nat. ALL x::nat. P x --> <= x M)) =
- (EX m::nat. P m & (ALL x::nat. P x --> <= x m))"
+lemma num_MAX: "(Ex (P::nat => bool) & (EX M::nat. ALL x::nat. P x --> x <= M)) =
+(EX m::nat. P m & (ALL x::nat. P x --> x <= m))"
by (import hollight num_MAX)
-definition EVEN :: "nat => bool" where
- "EVEN ==
-SOME EVEN::nat => bool.
- EVEN 0 = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
-
-lemma DEF_EVEN: "EVEN =
-(SOME EVEN::nat => bool.
- EVEN 0 = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n)))"
- by (import hollight DEF_EVEN)
-
-definition ODD :: "nat => bool" where
- "ODD ==
-SOME ODD::nat => bool. ODD 0 = False & (ALL n::nat. ODD (Suc n) = (~ ODD n))"
-
-lemma DEF_ODD: "ODD =
-(SOME ODD::nat => bool.
- ODD 0 = False & (ALL n::nat. ODD (Suc n) = (~ ODD n)))"
- by (import hollight DEF_ODD)
-
-lemma NOT_EVEN: "ALL n::nat. (~ EVEN n) = ODD n"
+lemma NOT_EVEN: "odd (n::nat) = odd n"
by (import hollight NOT_EVEN)
-lemma NOT_ODD: "ALL n::nat. (~ ODD n) = EVEN n"
+lemma NOT_ODD: "(~ odd (n::nat)) = even n"
by (import hollight NOT_ODD)
-lemma EVEN_OR_ODD: "ALL n::nat. EVEN n | ODD n"
+lemma EVEN_OR_ODD: "even (n::nat) | odd n"
by (import hollight EVEN_OR_ODD)
-lemma EVEN_AND_ODD: "ALL x::nat. ~ (EVEN x & ODD x)"
+lemma EVEN_AND_ODD: "~ (even (x::nat) & odd x)"
by (import hollight EVEN_AND_ODD)
-lemma EVEN_ADD: "ALL (m::nat) n::nat. EVEN (m + n) = (EVEN m = EVEN n)"
- by (import hollight EVEN_ADD)
-
-lemma EVEN_MULT: "ALL (m::nat) n::nat. EVEN (m * n) = (EVEN m | EVEN n)"
- by (import hollight EVEN_MULT)
-
-lemma EVEN_EXP: "ALL (m::nat) n::nat. EVEN (EXP m n) = (EVEN m & n ~= 0)"
+lemma EVEN_EXP: "even ((m::nat) ^ (n::nat)) = (even m & n ~= (0::nat))"
by (import hollight EVEN_EXP)
-lemma ODD_ADD: "ALL (m::nat) n::nat. ODD (m + n) = (ODD m ~= ODD n)"
- by (import hollight ODD_ADD)
-
-lemma ODD_MULT: "ALL (m::nat) n::nat. ODD (m * n) = (ODD m & ODD n)"
+lemma ODD_MULT: "odd ((m::nat) * (n::nat)) = (odd m & odd n)"
by (import hollight ODD_MULT)
-lemma ODD_EXP: "ALL (m::nat) n::nat. ODD (EXP m n) = (ODD m | n = 0)"
+lemma ODD_EXP: "odd ((m::nat) ^ (n::nat)) = (odd m | n = (0::nat))"
by (import hollight ODD_EXP)
-lemma EVEN_DOUBLE: "ALL n::nat. EVEN (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n)"
+lemma EVEN_DOUBLE: "even ((2::nat) * (n::nat))"
by (import hollight EVEN_DOUBLE)
-lemma ODD_DOUBLE: "ALL x::nat. ODD (Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * x))"
+lemma ODD_DOUBLE: "odd (Suc (2 * x))"
by (import hollight ODD_DOUBLE)
-lemma EVEN_EXISTS_LEMMA: "ALL n::nat.
- (EVEN n --> (EX m::nat. n = NUMERAL_BIT0 (NUMERAL_BIT1 0) * m)) &
- (~ EVEN n --> (EX m::nat. n = Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * m)))"
+lemma EVEN_EXISTS_LEMMA: "(even n --> (EX m. n = 2 * m)) & (odd n --> (EX m. n = Suc (2 * m)))"
by (import hollight EVEN_EXISTS_LEMMA)
-lemma EVEN_EXISTS: "ALL n::nat. EVEN n = (EX m::nat. n = NUMERAL_BIT0 (NUMERAL_BIT1 0) * m)"
- by (import hollight EVEN_EXISTS)
-
-lemma ODD_EXISTS: "ALL n::nat. ODD n = (EX m::nat. n = Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * m))"
- by (import hollight ODD_EXISTS)
-
-lemma EVEN_ODD_DECOMPOSITION: "ALL n::nat.
- (EX (k::nat) m::nat.
- ODD m & n = EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) k * m) =
- (n ~= 0)"
+lemma EVEN_ODD_DECOMPOSITION: "(EX (k::nat) m::nat. odd m & (n::nat) = (2::nat) ^ k * m) = (n ~= (0::nat))"
by (import hollight EVEN_ODD_DECOMPOSITION)
-lemma SUB_0: "ALL x::nat. 0 - x = 0 & x - 0 = x"
+lemma SUB_0: "(0::nat) - (x::nat) = (0::nat) & x - (0::nat) = x"
by (import hollight SUB_0)
-lemma SUB_PRESUC: "ALL (m::nat) n::nat. Pred (Suc m - n) = m - n"
+lemma SUB_PRESUC: "Suc m - n - Suc 0 = m - n"
by (import hollight SUB_PRESUC)
-lemma SUB_EQ_0: "ALL (m::nat) n::nat. (m - n = 0) = <= m n"
- by (import hollight SUB_EQ_0)
-
-lemma ADD_SUBR: "ALL (x::nat) xa::nat. xa - (x + xa) = 0"
+lemma ADD_SUBR: "(xa::nat) - ((x::nat) + xa) = (0::nat)"
by (import hollight ADD_SUBR)
-lemma SUB_ADD: "ALL (x::nat) xa::nat. <= xa x --> x - xa + xa = x"
- by (import hollight SUB_ADD)
-
-lemma SUC_SUB1: "ALL x::nat. Suc x - NUMERAL_BIT1 0 = x"
- by (import hollight SUC_SUB1)
-
-definition FACT :: "nat => nat" where
- "FACT ==
-SOME FACT::nat => nat.
- FACT 0 = NUMERAL_BIT1 0 & (ALL n::nat. FACT (Suc n) = Suc n * FACT n)"
-
-lemma DEF_FACT: "FACT =
-(SOME FACT::nat => nat.
- FACT 0 = NUMERAL_BIT1 0 & (ALL n::nat. FACT (Suc n) = Suc n * FACT n))"
- by (import hollight DEF_FACT)
-
-lemma FACT_LT: "ALL n::nat. < 0 (FACT n)"
- by (import hollight FACT_LT)
-
-lemma FACT_LE: "ALL x::nat. <= (NUMERAL_BIT1 0) (FACT x)"
- by (import hollight FACT_LE)
-
-lemma FACT_MONO: "ALL (m::nat) n::nat. <= m n --> <= (FACT m) (FACT n)"
- by (import hollight FACT_MONO)
-
-lemma DIVMOD_EXIST: "ALL (m::nat) n::nat. n ~= 0 --> (EX (q::nat) r::nat. m = q * n + r & < r n)"
+lemma EVEN_SUB: "even ((m::nat) - (n::nat)) = (m <= n | even m = even n)"
+ by (import hollight EVEN_SUB)
+
+lemma ODD_SUB: "odd ((x::nat) - (xa::nat)) = (xa < x & odd x ~= odd xa)"
+ by (import hollight ODD_SUB)
+
+lemma EXP_LT_0: "((0::nat) < (xa::nat) ^ (x::nat)) = (xa ~= (0::nat) | x = (0::nat))"
+ by (import hollight EXP_LT_0)
+
+lemma LT_EXP: "((x::nat) ^ (m::nat) < x ^ (n::nat)) =
+((2::nat) <= x & m < n | x = (0::nat) & m ~= (0::nat) & n = (0::nat))"
+ by (import hollight LT_EXP)
+
+lemma LE_EXP: "((x::nat) ^ (m::nat) <= x ^ (n::nat)) =
+(if x = (0::nat) then m = (0::nat) --> n = (0::nat)
+ else x = (1::nat) | m <= n)"
+ by (import hollight LE_EXP)
+
+lemma EQ_EXP: "((x::nat) ^ (m::nat) = x ^ (n::nat)) =
+(if x = (0::nat) then (m = (0::nat)) = (n = (0::nat))
+ else x = (1::nat) | m = n)"
+ by (import hollight EQ_EXP)
+
+lemma EXP_MONO_LE_IMP: "(x::nat) <= (xa::nat) ==> x ^ (xb::nat) <= xa ^ xb"
+ by (import hollight EXP_MONO_LE_IMP)
+
+lemma EXP_MONO_LT_IMP: "(x::nat) < (y::nat) & (n::nat) ~= (0::nat) ==> x ^ n < y ^ n"
+ by (import hollight EXP_MONO_LT_IMP)
+
+lemma EXP_MONO_LE: "((x::nat) ^ (n::nat) <= (y::nat) ^ n) = (x <= y | n = (0::nat))"
+ by (import hollight EXP_MONO_LE)
+
+lemma EXP_MONO_LT: "((x::nat) ^ (xb::nat) < (xa::nat) ^ xb) = (x < xa & xb ~= (0::nat))"
+ by (import hollight EXP_MONO_LT)
+
+lemma EXP_MONO_EQ: "((x::nat) ^ (xb::nat) = (xa::nat) ^ xb) = (x = xa | xb = (0::nat))"
+ by (import hollight EXP_MONO_EQ)
+
+lemma DIVMOD_EXIST: "(n::nat) ~= (0::nat) ==> EX (q::nat) r::nat. (m::nat) = q * n + r & r < n"
by (import hollight DIVMOD_EXIST)
-lemma DIVMOD_EXIST_0: "ALL (m::nat) n::nat.
- EX (x::nat) xa::nat.
- COND (n = 0) (x = 0 & xa = 0) (m = x * n + xa & < xa n)"
+lemma DIVMOD_EXIST_0: "EX (x::nat) xa::nat.
+ if (n::nat) = (0::nat) then x = (0::nat) & xa = (m::nat)
+ else m = x * n + xa & xa < n"
by (import hollight DIVMOD_EXIST_0)
-definition DIV :: "nat => nat => nat" where
- "DIV ==
-SOME q::nat => nat => nat.
- EX r::nat => nat => nat.
- ALL (m::nat) n::nat.
- COND (n = 0) (q m n = 0 & r m n = 0)
- (m = q m n * n + r m n & < (r m n) n)"
-
-lemma DEF_DIV: "DIV =
-(SOME q::nat => nat => nat.
- EX r::nat => nat => nat.
- ALL (m::nat) n::nat.
- COND (n = 0) (q m n = 0 & r m n = 0)
- (m = q m n * n + r m n & < (r m n) n))"
- by (import hollight DEF_DIV)
-
-definition MOD :: "nat => nat => nat" where
- "MOD ==
-SOME r::nat => nat => nat.
- ALL (m::nat) n::nat.
- COND (n = 0) (DIV m n = 0 & r m n = 0)
- (m = DIV m n * n + r m n & < (r m n) n)"
-
-lemma DEF_MOD: "MOD =
-(SOME r::nat => nat => nat.
- ALL (m::nat) n::nat.
- COND (n = 0) (DIV m n = 0 & r m n = 0)
- (m = DIV m n * n + r m n & < (r m n) n))"
- by (import hollight DEF_MOD)
-
-lemma DIVISION: "ALL (m::nat) n::nat. n ~= 0 --> m = DIV m n * n + MOD m n & < (MOD m n) n"
+lemma DIVISION: "(n::nat) ~= (0::nat) ==> (m::nat) = m div n * n + m mod n & m mod n < n"
by (import hollight DIVISION)
-lemma DIVMOD_UNIQ_LEMMA: "ALL (m::nat) (n::nat) (q1::nat) (r1::nat) (q2::nat) r2::nat.
- (m = q1 * n + r1 & < r1 n) & m = q2 * n + r2 & < r2 n -->
- q1 = q2 & r1 = r2"
+lemma DIVMOD_UNIQ_LEMMA: "((m::nat) = (q1::nat) * (n::nat) + (r1::nat) & r1 < n) &
+m = (q2::nat) * n + (r2::nat) & r2 < n
+==> q1 = q2 & r1 = r2"
by (import hollight DIVMOD_UNIQ_LEMMA)
-lemma DIVMOD_UNIQ: "ALL (m::nat) (n::nat) (q::nat) r::nat.
- m = q * n + r & < r n --> DIV m n = q & MOD m n = r"
+lemma DIVMOD_UNIQ: "(m::nat) = (q::nat) * (n::nat) + (r::nat) & r < n
+==> m div n = q & m mod n = r"
by (import hollight DIVMOD_UNIQ)
-lemma MOD_UNIQ: "ALL (m::nat) (n::nat) (q::nat) r::nat. m = q * n + r & < r n --> MOD m n = r"
+lemma MOD_UNIQ: "(m::nat) = (q::nat) * (n::nat) + (r::nat) & r < n ==> m mod n = r"
by (import hollight MOD_UNIQ)
-lemma DIV_UNIQ: "ALL (m::nat) (n::nat) (q::nat) r::nat. m = q * n + r & < r n --> DIV m n = q"
+lemma DIV_UNIQ: "(m::nat) = (q::nat) * (n::nat) + (r::nat) & r < n ==> m div n = q"
by (import hollight DIV_UNIQ)
-lemma MOD_MULT: "ALL (x::nat) xa::nat. x ~= 0 --> MOD (x * xa) x = 0"
- by (import hollight MOD_MULT)
-
-lemma DIV_MULT: "ALL (x::nat) xa::nat. x ~= 0 --> DIV (x * xa) x = xa"
- by (import hollight DIV_MULT)
-
-lemma DIV_DIV: "ALL (m::nat) (n::nat) p::nat. n * p ~= 0 --> DIV (DIV m n) p = DIV m (n * p)"
- by (import hollight DIV_DIV)
-
-lemma MOD_LT: "ALL (m::nat) n::nat. < m n --> MOD m n = m"
- by (import hollight MOD_LT)
-
-lemma MOD_EQ: "ALL (m::nat) (n::nat) (p::nat) q::nat. m = n + q * p --> MOD m p = MOD n p"
+lemma MOD_EQ: "(m::nat) = (n::nat) + (q::nat) * (p::nat) ==> m mod p = n mod p"
by (import hollight MOD_EQ)
-lemma DIV_MOD: "ALL (m::nat) (n::nat) p::nat.
- n * p ~= 0 --> MOD (DIV m n) p = DIV (MOD m (n * p)) n"
- by (import hollight DIV_MOD)
-
-lemma DIV_1: "ALL n::nat. DIV n (NUMERAL_BIT1 0) = n"
- by (import hollight DIV_1)
-
-lemma EXP_LT_0: "ALL (x::nat) xa::nat. < 0 (EXP xa x) = (xa ~= 0 | x = 0)"
- by (import hollight EXP_LT_0)
-
-lemma DIV_LE: "ALL (m::nat) n::nat. n ~= 0 --> <= (DIV m n) m"
+lemma DIV_LE: "(n::nat) ~= (0::nat) ==> (m::nat) div n <= m"
by (import hollight DIV_LE)
-lemma DIV_MUL_LE: "ALL (m::nat) n::nat. <= (n * DIV m n) m"
+lemma DIV_MUL_LE: "(n::nat) * ((m::nat) div n) <= m"
by (import hollight DIV_MUL_LE)
-lemma DIV_0: "ALL n::nat. n ~= 0 --> DIV 0 n = 0"
- by (import hollight DIV_0)
-
-lemma MOD_0: "ALL n::nat. n ~= 0 --> MOD 0 n = 0"
- by (import hollight MOD_0)
-
-lemma DIV_LT: "ALL (m::nat) n::nat. < m n --> DIV m n = 0"
- by (import hollight DIV_LT)
-
-lemma MOD_MOD: "ALL (m::nat) (n::nat) p::nat. n * p ~= 0 --> MOD (MOD m (n * p)) n = MOD m n"
+lemma MOD_MOD: "(n::nat) * (p::nat) ~= (0::nat) ==> (m::nat) mod (n * p) mod n = m mod n"
by (import hollight MOD_MOD)
-lemma MOD_MOD_REFL: "ALL (m::nat) n::nat. n ~= 0 --> MOD (MOD m n) n = MOD m n"
+lemma MOD_MOD_REFL: "(n::nat) ~= (0::nat) ==> (m::nat) mod n mod n = m mod n"
by (import hollight MOD_MOD_REFL)
-lemma DIV_MULT2: "ALL (x::nat) (xa::nat) xb::nat.
- x * xb ~= 0 --> DIV (x * xa) (x * xb) = DIV xa xb"
+lemma DIV_MULT2: "(x::nat) * (xb::nat) ~= (0::nat) ==> x * (xa::nat) div (x * xb) = xa div xb"
by (import hollight DIV_MULT2)
-lemma MOD_MULT2: "ALL (x::nat) (xa::nat) xb::nat.
- x * xb ~= 0 --> MOD (x * xa) (x * xb) = x * MOD xa xb"
+lemma MOD_MULT2: "(x::nat) * (xb::nat) ~= (0::nat)
+==> x * (xa::nat) mod (x * xb) = x * (xa mod xb)"
by (import hollight MOD_MULT2)
-lemma MOD_1: "ALL n::nat. MOD n (NUMERAL_BIT1 0) = 0"
- by (import hollight MOD_1)
-
-lemma MOD_EXISTS: "ALL (m::nat) n::nat.
- (EX q::nat. m = n * q) = COND (n = 0) (m = 0) (MOD m n = 0)"
+lemma MOD_EXISTS: "(EX q::nat. (m::nat) = (n::nat) * q) =
+(if n = (0::nat) then m = (0::nat) else m mod n = (0::nat))"
by (import hollight MOD_EXISTS)
-lemma LT_EXP: "ALL (x::nat) (m::nat) n::nat.
- < (EXP x m) (EXP x n) =
- (<= (NUMERAL_BIT0 (NUMERAL_BIT1 0)) x & < m n | x = 0 & m ~= 0 & n = 0)"
- by (import hollight LT_EXP)
-
-lemma LE_EXP: "ALL (x::nat) (m::nat) n::nat.
- <= (EXP x m) (EXP x n) =
- COND (x = 0) (m = 0 --> n = 0) (x = NUMERAL_BIT1 0 | <= m n)"
- by (import hollight LE_EXP)
-
-lemma DIV_MONO: "ALL (m::nat) (n::nat) p::nat. p ~= 0 & <= m n --> <= (DIV m p) (DIV n p)"
+lemma LE_RDIV_EQ: "(a::nat) ~= (0::nat) ==> ((n::nat) <= (b::nat) div a) = (a * n <= b)"
+ by (import hollight LE_RDIV_EQ)
+
+lemma LE_LDIV_EQ: "(a::nat) ~= (0::nat)
+==> ((b::nat) div a <= (n::nat)) = (b < a * (n + (1::nat)))"
+ by (import hollight LE_LDIV_EQ)
+
+lemma LE_LDIV: "(x::nat) ~= (0::nat) & (xa::nat) <= x * (xb::nat) ==> xa div x <= xb"
+ by (import hollight LE_LDIV)
+
+lemma DIV_MONO: "(p::nat) ~= (0::nat) & (m::nat) <= (n::nat) ==> m div p <= n div p"
by (import hollight DIV_MONO)
-lemma DIV_MONO_LT: "ALL (m::nat) (n::nat) p::nat.
- p ~= 0 & <= (m + p) n --> < (DIV m p) (DIV n p)"
+lemma DIV_MONO_LT: "(p::nat) ~= (0::nat) & (m::nat) + p <= (n::nat) ==> m div p < n div p"
by (import hollight DIV_MONO_LT)
-lemma LE_LDIV: "ALL (a::nat) (b::nat) n::nat. a ~= 0 & <= b (a * n) --> <= (DIV b a) n"
- by (import hollight LE_LDIV)
-
-lemma LE_RDIV_EQ: "ALL (a::nat) (b::nat) n::nat. a ~= 0 --> <= n (DIV b a) = <= (a * n) b"
- by (import hollight LE_RDIV_EQ)
-
-lemma LE_LDIV_EQ: "ALL (a::nat) (b::nat) n::nat.
- a ~= 0 --> <= (DIV b a) n = < b (a * (n + NUMERAL_BIT1 0))"
- by (import hollight LE_LDIV_EQ)
-
-lemma DIV_EQ_0: "ALL (m::nat) n::nat. n ~= 0 --> (DIV m n = 0) = < m n"
+lemma DIV_EQ_0: "(n::nat) ~= (0::nat) ==> ((m::nat) div n = (0::nat)) = (m < n)"
by (import hollight DIV_EQ_0)
-lemma MOD_EQ_0: "ALL (m::nat) n::nat. n ~= 0 --> (MOD m n = 0) = (EX q::nat. m = q * n)"
+lemma MOD_EQ_0: "(n::nat) ~= (0::nat)
+==> ((m::nat) mod n = (0::nat)) = (EX q::nat. m = q * n)"
by (import hollight MOD_EQ_0)
-lemma EVEN_MOD: "ALL n::nat. EVEN n = (MOD n (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = 0)"
+lemma EVEN_MOD: "even (n::nat) = (n mod (2::nat) = (0::nat))"
by (import hollight EVEN_MOD)
-lemma ODD_MOD: "ALL n::nat. ODD n = (MOD n (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = NUMERAL_BIT1 0)"
+lemma ODD_MOD: "odd (n::nat) = (n mod (2::nat) = (1::nat))"
by (import hollight ODD_MOD)
-lemma MOD_MULT_RMOD: "ALL (m::nat) (n::nat) p::nat. n ~= 0 --> MOD (m * MOD p n) n = MOD (m * p) n"
+lemma MOD_MULT_RMOD: "(n::nat) ~= (0::nat) ==> (m::nat) * ((p::nat) mod n) mod n = m * p mod n"
by (import hollight MOD_MULT_RMOD)
-lemma MOD_MULT_LMOD: "ALL (x::nat) (xa::nat) xb::nat.
- xa ~= 0 --> MOD (MOD x xa * xb) xa = MOD (x * xb) xa"
+lemma MOD_MULT_LMOD: "(xa::nat) ~= (0::nat) ==> (x::nat) mod xa * (xb::nat) mod xa = x * xb mod xa"
by (import hollight MOD_MULT_LMOD)
-lemma MOD_MULT_MOD2: "ALL (x::nat) (xa::nat) xb::nat.
- xa ~= 0 --> MOD (MOD x xa * MOD xb xa) xa = MOD (x * xb) xa"
+lemma MOD_MULT_MOD2: "(xa::nat) ~= (0::nat)
+==> (x::nat) mod xa * ((xb::nat) mod xa) mod xa = x * xb mod xa"
by (import hollight MOD_MULT_MOD2)
-lemma MOD_EXP_MOD: "ALL (m::nat) (n::nat) p::nat.
- n ~= 0 --> MOD (EXP (MOD m n) p) n = MOD (EXP m p) n"
+lemma MOD_EXP_MOD: "(n::nat) ~= (0::nat) ==> ((m::nat) mod n) ^ (p::nat) mod n = m ^ p mod n"
by (import hollight MOD_EXP_MOD)
-lemma MOD_MULT_ADD: "ALL (m::nat) (n::nat) p::nat. MOD (m * n + p) n = MOD p n"
- by (import hollight MOD_MULT_ADD)
-
-lemma MOD_ADD_MOD: "ALL (a::nat) (b::nat) n::nat.
- n ~= 0 --> MOD (MOD a n + MOD b n) n = MOD (a + b) n"
+lemma MOD_ADD_MOD: "(n::nat) ~= (0::nat)
+==> ((a::nat) mod n + (b::nat) mod n) mod n = (a + b) mod n"
by (import hollight MOD_ADD_MOD)
-lemma DIV_ADD_MOD: "ALL (a::nat) (b::nat) n::nat.
- n ~= 0 -->
- (MOD (a + b) n = MOD a n + MOD b n) = (DIV (a + b) n = DIV a n + DIV b n)"
+lemma DIV_ADD_MOD: "(n::nat) ~= (0::nat)
+==> (((a::nat) + (b::nat)) mod n = a mod n + b mod n) =
+ ((a + b) div n = a div n + b div n)"
by (import hollight DIV_ADD_MOD)
-lemma DIV_REFL: "ALL n::nat. n ~= 0 --> DIV n n = NUMERAL_BIT1 0"
- by (import hollight DIV_REFL)
-
-lemma MOD_LE: "ALL (m::nat) n::nat. n ~= 0 --> <= (MOD m n) m"
+lemma MOD_LE: "(n::nat) ~= (0::nat) ==> (m::nat) mod n <= m"
by (import hollight MOD_LE)
-lemma DIV_MONO2: "ALL (m::nat) (n::nat) p::nat. p ~= 0 & <= p m --> <= (DIV n m) (DIV n p)"
+lemma DIV_MONO2: "(p::nat) ~= (0::nat) & p <= (m::nat) ==> (n::nat) div m <= n div p"
by (import hollight DIV_MONO2)
-lemma DIV_LE_EXCLUSION: "ALL (a::nat) (b::nat) (c::nat) d::nat.
- b ~= 0 & < (b * c) ((a + NUMERAL_BIT1 0) * d) --> <= (DIV c d) (DIV a b)"
+lemma DIV_LE_EXCLUSION: "(b::nat) ~= (0::nat) & b * (c::nat) < ((a::nat) + (1::nat)) * (d::nat)
+==> c div d <= a div b"
by (import hollight DIV_LE_EXCLUSION)
-lemma DIV_EQ_EXCLUSION: "< ((b::nat) * (c::nat)) (((a::nat) + NUMERAL_BIT1 0) * (d::nat)) &
-< (a * d) ((c + NUMERAL_BIT1 0) * b) -->
-DIV a b = DIV c d"
+lemma DIV_EQ_EXCLUSION: "(b::nat) * (c::nat) < ((a::nat) + (1::nat)) * (d::nat) &
+a * d < (c + (1::nat)) * b
+==> a div b = c div d"
by (import hollight DIV_EQ_EXCLUSION)
+lemma MULT_DIV_LE: "(p::nat) ~= (0::nat) ==> (m::nat) * ((n::nat) div p) <= m * n div p"
+ by (import hollight MULT_DIV_LE)
+
+lemma DIV_DIV: "(xa::nat) * (xb::nat) ~= (0::nat)
+==> (x::nat) div xa div xb = x div (xa * xb)"
+ by (import hollight DIV_DIV)
+
+lemma DIV_MOD: "(xa::nat) * (xb::nat) ~= (0::nat)
+==> (x::nat) div xa mod xb = x mod (xa * xb) div xa"
+ by (import hollight DIV_MOD)
+
+lemma PRE_ELIM_THM: "P (n - Suc 0) = (ALL m. n = Suc m | m = 0 & n = 0 --> P m)"
+ by (import hollight PRE_ELIM_THM)
+
lemma SUB_ELIM_THM: "(P::nat => bool) ((a::nat) - (b::nat)) =
-(ALL x::nat. (b = a + x --> P 0) & (a = b + x --> P x))"
+(ALL d::nat. a = b + d | a < b & d = (0::nat) --> P d)"
by (import hollight SUB_ELIM_THM)
-lemma PRE_ELIM_THM: "(P::nat => bool) (Pred (n::nat)) =
-(ALL m::nat. (n = 0 --> P 0) & (n = Suc m --> P m))"
- by (import hollight PRE_ELIM_THM)
-
-lemma DIVMOD_ELIM_THM: "(P::nat => nat => bool) (DIV (m::nat) (n::nat)) (MOD m n) =
-(n = 0 & P 0 0 |
- n ~= 0 & (ALL (q::nat) r::nat. m = q * n + r & < r n --> P q r))"
+lemma DIVMOD_ELIM_THM: "(P::nat => nat => bool) ((m::nat) div (n::nat)) (m mod n) =
+(ALL (x::nat) xa::nat.
+ n = (0::nat) & x = (0::nat) & xa = m | m = x * n + xa & xa < n -->
+ P x xa)"
by (import hollight DIVMOD_ELIM_THM)
-definition eqeq :: "'q_9910 => 'q_9909 => ('q_9910 => 'q_9909 => bool) => bool" where
- "eqeq ==
-%(u::'q_9910::type) (ua::'q_9909::type)
- ub::'q_9910::type => 'q_9909::type => bool. ub u ua"
-
-lemma DEF__equal__equal_: "eqeq =
-(%(u::'q_9910::type) (ua::'q_9909::type)
- ub::'q_9910::type => 'q_9909::type => bool. ub u ua)"
- by (import hollight DEF__equal__equal_)
-
-definition mod_nat :: "nat => nat => nat => bool" where
- "mod_nat ==
-%(u::nat) (ua::nat) ub::nat. EX (q1::nat) q2::nat. ua + u * q1 = ub + u * q2"
-
-lemma DEF_mod_nat: "mod_nat =
-(%(u::nat) (ua::nat) ub::nat.
- EX (q1::nat) q2::nat. ua + u * q1 = ub + u * q2)"
- by (import hollight DEF_mod_nat)
-
-definition minimal :: "(nat => bool) => nat" where
- "minimal == %u::nat => bool. SOME n::nat. u n & (ALL m::nat. < m n --> ~ u m)"
-
-lemma DEF_minimal: "minimal =
-(%u::nat => bool. SOME n::nat. u n & (ALL m::nat. < m n --> ~ u m))"
+definition
+ minimal :: "(nat => bool) => nat" where
+ "minimal == %u. SOME n. u n & (ALL m<n. ~ u m)"
+
+lemma DEF_minimal: "minimal = (%u. SOME n. u n & (ALL m<n. ~ u m))"
by (import hollight DEF_minimal)
-lemma MINIMAL: "ALL P::nat => bool.
- Ex P = (P (minimal P) & (ALL x::nat. < x (minimal P) --> ~ P x))"
+lemma MINIMAL: "Ex P = (P (minimal P) & (ALL x<minimal P. ~ P x))"
by (import hollight MINIMAL)
-definition WF :: "('A => 'A => bool) => bool" where
- "WF ==
-%u::'A::type => 'A::type => bool.
- ALL P::'A::type => bool.
- Ex P --> (EX x::'A::type. P x & (ALL y::'A::type. u y x --> ~ P y))"
-
-lemma DEF_WF: "WF =
-(%u::'A::type => 'A::type => bool.
- ALL P::'A::type => bool.
- Ex P --> (EX x::'A::type. P x & (ALL y::'A::type. u y x --> ~ P y)))"
- by (import hollight DEF_WF)
-
-lemma WF_EQ: "WF (u_353::'A::type => 'A::type => bool) =
-(ALL P::'A::type => bool.
- Ex P = (EX x::'A::type. P x & (ALL y::'A::type. u_353 y x --> ~ P y)))"
+lemma TRANSITIVE_STEPWISE_LT_EQ: "(!!x y z. R x y & R y z ==> R x z)
+==> (ALL m n. m < n --> R m n) = (ALL n. R n (Suc n))"
+ by (import hollight TRANSITIVE_STEPWISE_LT_EQ)
+
+lemma TRANSITIVE_STEPWISE_LT: "[| (ALL x y z. R x y & R y z --> R x z) & (ALL n. R n (Suc n)); m < n |]
+==> R m n"
+ by (import hollight TRANSITIVE_STEPWISE_LT)
+
+lemma TRANSITIVE_STEPWISE_LE_EQ: "(ALL x. R x x) & (ALL x y z. R x y & R y z --> R x z)
+==> (ALL m n. m <= n --> R m n) = (ALL n. R n (Suc n))"
+ by (import hollight TRANSITIVE_STEPWISE_LE_EQ)
+
+lemma TRANSITIVE_STEPWISE_LE: "[| (ALL x. R x x) &
+ (ALL x y z. R x y & R y z --> R x z) & (ALL n. R n (Suc n));
+ m <= n |]
+==> R m n"
+ by (import hollight TRANSITIVE_STEPWISE_LE)
+
+lemma WF_EQ: "wfP (u_556::'A => 'A => bool) =
+(ALL P::'A => bool.
+ Ex P = (EX x::'A. P x & (ALL y::'A. u_556 y x --> ~ P y)))"
by (import hollight WF_EQ)
-lemma WF_IND: "WF (u_353::'A::type => 'A::type => bool) =
-(ALL P::'A::type => bool.
- (ALL x::'A::type. (ALL y::'A::type. u_353 y x --> P y) --> P x) -->
- All P)"
+lemma WF_IND: "wfP (u_556::'A => 'A => bool) =
+(ALL P::'A => bool.
+ (ALL x::'A. (ALL y::'A. u_556 y x --> P y) --> P x) --> All P)"
by (import hollight WF_IND)
-lemma WF_DCHAIN: "WF (u_353::'A::type => 'A::type => bool) =
-(~ (EX s::nat => 'A::type. ALL n::nat. u_353 (s (Suc n)) (s n)))"
+lemma WF_DCHAIN: "wfP (u_556::'A => 'A => bool) =
+(~ (EX s::nat => 'A. ALL n::nat. u_556 (s (Suc n)) (s n)))"
by (import hollight WF_DCHAIN)
-lemma WF_UREC: "WF (u_353::'A::type => 'A::type => bool) -->
-(ALL H::('A::type => 'B::type) => 'A::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z) --> H f x = H g x) -->
- (ALL (f::'A::type => 'B::type) g::'A::type => 'B::type.
- (ALL x::'A::type. f x = H f x) & (ALL x::'A::type. g x = H g x) -->
- f = g))"
+lemma WF_UREC: "[| wfP (u_556::'A => 'A => bool);
+ !!(f::'A => 'B) (g::'A => 'B) x::'A.
+ (!!z::'A. u_556 z x ==> f z = g z)
+ ==> (H::('A => 'B) => 'A => 'B) f x = H g x;
+ (ALL x::'A. (f::'A => 'B) x = H f x) &
+ (ALL x::'A. (g::'A => 'B) x = H g x) |]
+==> f = g"
by (import hollight WF_UREC)
-lemma WF_UREC_WF: "(ALL H::('A::type => bool) => 'A::type => bool.
- (ALL (f::'A::type => bool) (g::'A::type => bool) x::'A::type.
- (ALL z::'A::type.
- (u_353::'A::type => 'A::type => bool) z x --> f z = g z) -->
- H f x = H g x) -->
- (ALL (f::'A::type => bool) g::'A::type => bool.
- (ALL x::'A::type. f x = H f x) & (ALL x::'A::type. g x = H g x) -->
- f = g)) -->
-WF u_353"
+lemma WF_UREC_WF: "(!!(H::('A => bool) => 'A => bool) (f::'A => bool) g::'A => bool.
+ [| !!(f::'A => bool) (g::'A => bool) x::'A.
+ (!!z::'A. (u_556::'A => 'A => bool) z x ==> f z = g z)
+ ==> H f x = H g x;
+ (ALL x::'A. f x = H f x) & (ALL x::'A. g x = H g x) |]
+ ==> f = g)
+==> wfP u_556"
by (import hollight WF_UREC_WF)
-lemma WF_REC_INVARIANT: "WF (u_353::'A::type => 'A::type => bool) -->
-(ALL (H::('A::type => 'B::type) => 'A::type => 'B::type)
- S::'A::type => 'B::type => bool.
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z & S z (f z)) -->
- H f x = H g x & S x (H f x)) -->
- (EX f::'A::type => 'B::type. ALL x::'A::type. f x = H f x))"
+lemma WF_REC_INVARIANT: "[| wfP (u_556::'A => 'A => bool);
+ !!(f::'A => 'B) (g::'A => 'B) x::'A.
+ (!!z::'A. u_556 z x ==> f z = g z & (S::'A => 'B => bool) z (f z))
+ ==> (H::('A => 'B) => 'A => 'B) f x = H g x & S x (H f x) |]
+==> EX f::'A => 'B. ALL x::'A. f x = H f x"
by (import hollight WF_REC_INVARIANT)
-lemma WF_REC: "WF (u_353::'A::type => 'A::type => bool) -->
-(ALL H::('A::type => 'B::type) => 'A::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z) --> H f x = H g x) -->
- (EX f::'A::type => 'B::type. ALL x::'A::type. f x = H f x))"
+lemma WF_REC: "[| wfP (u_556::'A => 'A => bool);
+ !!(f::'A => 'B) (g::'A => 'B) x::'A.
+ (!!z::'A. u_556 z x ==> f z = g z)
+ ==> (H::('A => 'B) => 'A => 'B) f x = H g x |]
+==> EX f::'A => 'B. ALL x::'A. f x = H f x"
by (import hollight WF_REC)
-lemma WF_REC_WF: "(ALL H::('A::type => nat) => 'A::type => nat.
- (ALL (f::'A::type => nat) (g::'A::type => nat) x::'A::type.
- (ALL z::'A::type.
- (u_353::'A::type => 'A::type => bool) z x --> f z = g z) -->
- H f x = H g x) -->
- (EX f::'A::type => nat. ALL x::'A::type. f x = H f x)) -->
-WF u_353"
+lemma WF_REC_WF: "(!!H::('A => nat) => 'A => nat.
+ (!!(f::'A => nat) (g::'A => nat) x::'A.
+ (!!z::'A. (u_556::'A => 'A => bool) z x ==> f z = g z)
+ ==> H f x = H g x)
+ ==> EX f::'A => nat. ALL x::'A. f x = H f x)
+==> wfP u_556"
by (import hollight WF_REC_WF)
-lemma WF_EREC: "WF (u_353::'A::type => 'A::type => bool) -->
-(ALL H::('A::type => 'B::type) => 'A::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z) --> H f x = H g x) -->
- (EX! f::'A::type => 'B::type. ALL x::'A::type. f x = H f x))"
+lemma WF_EREC: "[| wfP (u_556::'A => 'A => bool);
+ !!(f::'A => 'B) (g::'A => 'B) x::'A.
+ (!!z::'A. u_556 z x ==> f z = g z)
+ ==> (H::('A => 'B) => 'A => 'B) f x = H g x |]
+==> EX! f::'A => 'B. ALL x::'A. f x = H f x"
by (import hollight WF_EREC)
-lemma WF_SUBSET: "(ALL (x::'A::type) y::'A::type.
- (u_353::'A::type => 'A::type => bool) x y -->
- (u_472::'A::type => 'A::type => bool) x y) &
-WF u_472 -->
-WF u_353"
+lemma WF_SUBSET: "(ALL (x::'A) y::'A.
+ (u_556::'A => 'A => bool) x y --> (u_670::'A => 'A => bool) x y) &
+wfP u_670
+==> wfP u_556"
by (import hollight WF_SUBSET)
-lemma WF_MEASURE_GEN: "ALL m::'A::type => 'B::type.
- WF (u_353::'B::type => 'B::type => bool) -->
- WF (%(x::'A::type) x'::'A::type. u_353 (m x) (m x'))"
+lemma WF_MEASURE_GEN: "wfP (u_556::'B => 'B => bool)
+==> wfP (%(x::'A) x'::'A. u_556 ((m::'A => 'B) x) (m x'))"
by (import hollight WF_MEASURE_GEN)
-lemma WF_LEX_DEPENDENT: "ALL (R::'A::type => 'A::type => bool)
- S::'A::type => 'B::type => 'B::type => bool.
- WF R & (ALL x::'A::type. WF (S x)) -->
- WF (GABS
- (%f::'A::type * 'B::type => 'A::type * 'B::type => bool.
- ALL (r1::'A::type) s1::'B::type.
- GEQ (f (r1, s1))
- (GABS
- (%f::'A::type * 'B::type => bool.
- ALL (r2::'A::type) s2::'B::type.
- GEQ (f (r2, s2))
- (R r1 r2 | r1 = r2 & S r1 s1 s2)))))"
+lemma WF_LEX_DEPENDENT: "wfP (R::'A => 'A => bool) & (ALL x::'A. wfP ((S::'A => 'B => 'B => bool) x))
+==> wfP (SOME f::'A * 'B => 'A * 'B => bool.
+ ALL (r1::'A) s1::'B.
+ f (r1, s1) =
+ (SOME f::'A * 'B => bool.
+ ALL (r2::'A) s2::'B.
+ f (r2, s2) = (R r1 r2 | r1 = r2 & S r1 s1 s2)))"
by (import hollight WF_LEX_DEPENDENT)
-lemma WF_LEX: "ALL (x::'A::type => 'A::type => bool) xa::'B::type => 'B::type => bool.
- WF x & WF xa -->
- WF (GABS
- (%f::'A::type * 'B::type => 'A::type * 'B::type => bool.
- ALL (r1::'A::type) s1::'B::type.
- GEQ (f (r1, s1))
- (GABS
- (%f::'A::type * 'B::type => bool.
- ALL (r2::'A::type) s2::'B::type.
- GEQ (f (r2, s2)) (x r1 r2 | r1 = r2 & xa s1 s2)))))"
+lemma WF_LEX: "wfP (x::'A => 'A => bool) & wfP (xa::'B => 'B => bool)
+==> wfP (SOME f::'A * 'B => 'A * 'B => bool.
+ ALL (r1::'A) s1::'B.
+ f (r1, s1) =
+ (SOME f::'A * 'B => bool.
+ ALL (r2::'A) s2::'B.
+ f (r2, s2) = (x r1 r2 | r1 = r2 & xa s1 s2)))"
by (import hollight WF_LEX)
-lemma WF_POINTWISE: "WF (u_353::'A::type => 'A::type => bool) &
-WF (u_472::'B::type => 'B::type => bool) -->
-WF (GABS
- (%f::'A::type * 'B::type => 'A::type * 'B::type => bool.
- ALL (x1::'A::type) y1::'B::type.
- GEQ (f (x1, y1))
- (GABS
- (%f::'A::type * 'B::type => bool.
- ALL (x2::'A::type) y2::'B::type.
- GEQ (f (x2, y2)) (u_353 x1 x2 & u_472 y1 y2)))))"
+lemma WF_POINTWISE: "wfP (u_556::'A => 'A => bool) & wfP (u_670::'B => 'B => bool)
+==> wfP (SOME f::'A * 'B => 'A * 'B => bool.
+ ALL (x1::'A) y1::'B.
+ f (x1, y1) =
+ (SOME f::'A * 'B => bool.
+ ALL (x2::'A) y2::'B.
+ f (x2, y2) = (u_556 x1 x2 & u_670 y1 y2)))"
by (import hollight WF_POINTWISE)
-lemma WF_num: "WF <"
+lemma WF_num: "(wfP::(nat => nat => bool) => bool) (op <::nat => nat => bool)"
by (import hollight WF_num)
-lemma WF_REC_num: "ALL H::(nat => 'A::type) => nat => 'A::type.
- (ALL (f::nat => 'A::type) (g::nat => 'A::type) x::nat.
- (ALL z::nat. < z x --> f z = g z) --> H f x = H g x) -->
- (EX f::nat => 'A::type. ALL x::nat. f x = H f x)"
+lemma WF_REC_num: "(!!(f::nat => 'A) (g::nat => 'A) x::nat.
+ (!!z::nat. z < x ==> f z = g z)
+ ==> (H::(nat => 'A) => nat => 'A) f x = H g x)
+==> EX f::nat => 'A. ALL x::nat. f x = H f x"
by (import hollight WF_REC_num)
-consts
- measure :: "('q_11107 => nat) => 'q_11107 => 'q_11107 => bool"
-
-defs
- measure_def: "hollight.measure ==
-%(u::'q_11107::type => nat) (x::'q_11107::type) y::'q_11107::type.
- < (u x) (u y)"
-
-lemma DEF_measure: "hollight.measure =
-(%(u::'q_11107::type => nat) (x::'q_11107::type) y::'q_11107::type.
- < (u x) (u y))"
- by (import hollight DEF_measure)
-
-lemma WF_MEASURE: "ALL m::'A::type => nat. WF (hollight.measure m)"
+lemma WF_MEASURE: "wfP (%(a::'A) b::'A. measure (m::'A => nat) (a, b))"
by (import hollight WF_MEASURE)
-lemma MEASURE_LE: "(ALL x::'q_11137::type.
- hollight.measure (m::'q_11137::type => nat) x (a::'q_11137::type) -->
- hollight.measure m x (b::'q_11137::type)) =
-<= (m a) (m b)"
+lemma MEASURE_LE: "(ALL x::'q_12099.
+ measure (m::'q_12099 => nat) (x, a::'q_12099) -->
+ measure m (x, b::'q_12099)) =
+(m a <= m b)"
by (import hollight MEASURE_LE)
-lemma WF_REFL: "ALL x::'A::type. WF (u_353::'A::type => 'A::type => bool) --> ~ u_353 x x"
+lemma WF_REFL: "wfP (u_556::'A => 'A => bool) ==> ~ u_556 (x::'A) x"
by (import hollight WF_REFL)
-lemma WF_FALSE: "WF (%(x::'A::type) y::'A::type. False)"
- by (import hollight WF_FALSE)
-
-lemma WF_REC_TAIL: "ALL (P::'A::type => bool) (g::'A::type => 'A::type) h::'A::type => 'B::type.
- EX f::'A::type => 'B::type.
- ALL x::'A::type. f x = COND (P x) (f (g x)) (h x)"
+lemma WF_REC_TAIL: "EX f::'A => 'B.
+ ALL x::'A.
+ f x =
+ (if (P::'A => bool) x then f ((g::'A => 'A) x) else (h::'A => 'B) x)"
by (import hollight WF_REC_TAIL)
-lemma WF_REC_TAIL_GENERAL: "ALL (P::('A::type => 'B::type) => 'A::type => bool)
- (G::('A::type => 'B::type) => 'A::type => 'A::type)
- H::('A::type => 'B::type) => 'A::type => 'B::type.
- WF (u_353::'A::type => 'A::type => bool) &
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z) -->
- P f x = P g x & G f x = G g x & H f x = H g x) &
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
- (ALL z::'A::type. u_353 z x --> f z = g z) --> H f x = H g x) &
- (ALL (f::'A::type => 'B::type) (x::'A::type) y::'A::type.
- P f x & u_353 y (G f x) --> u_353 y x) -->
- (EX f::'A::type => 'B::type.
- ALL x::'A::type. f x = COND (P f x) (f (G f x)) (H f x))"
+lemma WF_REC_TAIL_GENERAL: "wfP (u_556::'A => 'A => bool) &
+(ALL (f::'A => 'B) (g::'A => 'B) x::'A.
+ (ALL z::'A. u_556 z x --> f z = g z) -->
+ (P::('A => 'B) => 'A => bool) f x = P g x &
+ (G::('A => 'B) => 'A => 'A) f x = G g x &
+ (H::('A => 'B) => 'A => 'B) f x = H g x) &
+(ALL (f::'A => 'B) (g::'A => 'B) x::'A.
+ (ALL z::'A. u_556 z x --> f z = g z) --> H f x = H g x) &
+(ALL (f::'A => 'B) (x::'A) y::'A. P f x & u_556 y (G f x) --> u_556 y x)
+==> EX f::'A => 'B. ALL x::'A. f x = (if P f x then f (G f x) else H f x)"
by (import hollight WF_REC_TAIL_GENERAL)
-lemma ARITH_ZERO: "(op &::bool => bool => bool) ((op =::nat => nat => bool) (0::nat) (0::nat))
- ((op =::nat => nat => bool) ((NUMERAL_BIT0::nat => nat) (0::nat)) (0::nat))"
+lemma ARITH_ZERO: "(0::nat) = (0::nat) & (0::nat) = (0::nat)"
by (import hollight ARITH_ZERO)
-lemma ARITH_SUC: "(ALL x::nat. Suc x = Suc x) &
-Suc 0 = NUMERAL_BIT1 0 &
-(ALL x::nat. Suc (NUMERAL_BIT0 x) = NUMERAL_BIT1 x) &
-(ALL x::nat. Suc (NUMERAL_BIT1 x) = NUMERAL_BIT0 (Suc x))"
+lemma ARITH_SUC: "(ALL x. Suc x = Suc x) &
+Suc 0 = 1 &
+(ALL x. Suc (2 * x) = 2 * x + 1) & (ALL x. Suc (2 * x + 1) = 2 * Suc x)"
by (import hollight ARITH_SUC)
-lemma ARITH_PRE: "(ALL x::nat. Pred x = Pred x) &
-Pred 0 = 0 &
-(ALL x::nat.
- Pred (NUMERAL_BIT0 x) = COND (x = 0) 0 (NUMERAL_BIT1 (Pred x))) &
-(ALL x::nat. Pred (NUMERAL_BIT1 x) = NUMERAL_BIT0 x)"
+lemma ARITH_PRE: "(ALL x. x - Suc 0 = x - Suc 0) &
+0 - Suc 0 = 0 &
+(ALL x. 2 * x - Suc 0 = (if x = 0 then 0 else 2 * (x - Suc 0) + 1)) &
+(ALL x. 2 * x + 1 - Suc 0 = 2 * x)"
by (import hollight ARITH_PRE)
-lemma ARITH_ADD: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool) ((plus::nat => nat => nat) x xa)
- ((plus::nat => nat => nat) x xa))))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool) ((plus::nat => nat => nat) (0::nat) (0::nat))
- (0::nat))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT0::nat => nat) x))
- ((NUMERAL_BIT0::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT1::nat => nat) x))
- ((NUMERAL_BIT1::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat) ((NUMERAL_BIT0::nat => nat) x)
- (0::nat))
- ((NUMERAL_BIT0::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat) ((NUMERAL_BIT1::nat => nat) x)
- (0::nat))
- ((NUMERAL_BIT1::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- ((NUMERAL_BIT0::nat => nat)
- ((plus::nat => nat => nat) x xa)))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- ((NUMERAL_BIT1::nat => nat)
- ((plus::nat => nat => nat) x xa)))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- ((NUMERAL_BIT1::nat => nat)
- ((plus::nat => nat => nat) x xa)))))
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- ((NUMERAL_BIT0::nat => nat)
- ((Suc::nat => nat)
- ((plus::nat => nat => nat) x
- xa))))))))))))))"
+lemma ARITH_ADD: "(ALL (x::nat) xa::nat. x + xa = x + xa) &
+(0::nat) + (0::nat) = (0::nat) &
+(ALL x::nat. (0::nat) + (2::nat) * x = (2::nat) * x) &
+(ALL x::nat.
+ (0::nat) + ((2::nat) * x + (1::nat)) = (2::nat) * x + (1::nat)) &
+(ALL x::nat. (2::nat) * x + (0::nat) = (2::nat) * x) &
+(ALL x::nat. (2::nat) * x + (1::nat) + (0::nat) = (2::nat) * x + (1::nat)) &
+(ALL (x::nat) xa::nat. (2::nat) * x + (2::nat) * xa = (2::nat) * (x + xa)) &
+(ALL (x::nat) xa::nat.
+ (2::nat) * x + ((2::nat) * xa + (1::nat)) =
+ (2::nat) * (x + xa) + (1::nat)) &
+(ALL (x::nat) xa::nat.
+ (2::nat) * x + (1::nat) + (2::nat) * xa =
+ (2::nat) * (x + xa) + (1::nat)) &
+(ALL (x::nat) xa::nat.
+ (2::nat) * x + (1::nat) + ((2::nat) * xa + (1::nat)) =
+ (2::nat) * Suc (x + xa))"
by (import hollight ARITH_ADD)
-lemma ARITH_MULT: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool) ((op *::nat => nat => nat) x xa)
- ((op *::nat => nat => nat) x xa))))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool) ((op *::nat => nat => nat) (0::nat) (0::nat))
- (0::nat))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT0::nat => nat) x))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT1::nat => nat) x))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) ((NUMERAL_BIT0::nat => nat) x)
- (0::nat))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) ((NUMERAL_BIT1::nat => nat) x)
- (0::nat))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- ((NUMERAL_BIT0::nat => nat)
- ((NUMERAL_BIT0::nat => nat)
- ((op *::nat => nat => nat) x xa))))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat)
- ((NUMERAL_BIT0::nat => nat)
- ((op *::nat => nat => nat) x xa)))))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) xa)
- ((NUMERAL_BIT0::nat => nat)
- ((NUMERAL_BIT0::nat => nat)
- ((op *::nat => nat => nat) x xa)))))))
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)
- ((plus::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) xa)
- ((NUMERAL_BIT0::nat => nat)
- ((NUMERAL_BIT0::nat => nat)
- ((op *::nat => nat => nat) x
- xa))))))))))))))))"
+lemma ARITH_MULT: "(ALL (x::nat) xa::nat. x * xa = x * xa) &
+(0::nat) * (0::nat) = (0::nat) &
+(ALL x::nat. (0::nat) * ((2::nat) * x) = (0::nat)) &
+(ALL x::nat. (0::nat) * ((2::nat) * x + (1::nat)) = (0::nat)) &
+(ALL x::nat. (2::nat) * x * (0::nat) = (0::nat)) &
+(ALL x::nat. ((2::nat) * x + (1::nat)) * (0::nat) = (0::nat)) &
+(ALL (x::nat) xa::nat.
+ (2::nat) * x * ((2::nat) * xa) = (2::nat) * ((2::nat) * (x * xa))) &
+(ALL (x::nat) xa::nat.
+ (2::nat) * x * ((2::nat) * xa + (1::nat)) =
+ (2::nat) * x + (2::nat) * ((2::nat) * (x * xa))) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat)) * ((2::nat) * xa) =
+ (2::nat) * xa + (2::nat) * ((2::nat) * (x * xa))) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat)) * ((2::nat) * xa + (1::nat)) =
+ (2::nat) * x + (1::nat) +
+ ((2::nat) * xa + (2::nat) * ((2::nat) * (x * xa))))"
by (import hollight ARITH_MULT)
-lemma ARITH_EXP: "(ALL (x::nat) xa::nat. EXP x xa = EXP x xa) &
-EXP 0 0 = NUMERAL_BIT1 0 &
-(ALL m::nat. EXP (NUMERAL_BIT0 m) 0 = NUMERAL_BIT1 0) &
-(ALL m::nat. EXP (NUMERAL_BIT1 m) 0 = NUMERAL_BIT1 0) &
-(ALL n::nat. EXP 0 (NUMERAL_BIT0 n) = EXP 0 n * EXP 0 n) &
+lemma ARITH_EXP: "(ALL (x::nat) xa::nat. x ^ xa = x ^ xa) &
+(0::nat) ^ (0::nat) = (1::nat) &
+(ALL m::nat. ((2::nat) * m) ^ (0::nat) = (1::nat)) &
+(ALL m::nat. ((2::nat) * m + (1::nat)) ^ (0::nat) = (1::nat)) &
+(ALL n::nat. (0::nat) ^ ((2::nat) * n) = (0::nat) ^ n * (0::nat) ^ n) &
(ALL (m::nat) n::nat.
- EXP (NUMERAL_BIT0 m) (NUMERAL_BIT0 n) =
- EXP (NUMERAL_BIT0 m) n * EXP (NUMERAL_BIT0 m) n) &
+ ((2::nat) * m) ^ ((2::nat) * n) =
+ ((2::nat) * m) ^ n * ((2::nat) * m) ^ n) &
+(ALL (m::nat) n::nat.
+ ((2::nat) * m + (1::nat)) ^ ((2::nat) * n) =
+ ((2::nat) * m + (1::nat)) ^ n * ((2::nat) * m + (1::nat)) ^ n) &
+(ALL n::nat. (0::nat) ^ ((2::nat) * n + (1::nat)) = (0::nat)) &
(ALL (m::nat) n::nat.
- EXP (NUMERAL_BIT1 m) (NUMERAL_BIT0 n) =
- EXP (NUMERAL_BIT1 m) n * EXP (NUMERAL_BIT1 m) n) &
-(ALL n::nat. EXP 0 (NUMERAL_BIT1 n) = 0) &
-(ALL (m::nat) n::nat.
- EXP (NUMERAL_BIT0 m) (NUMERAL_BIT1 n) =
- NUMERAL_BIT0 m * (EXP (NUMERAL_BIT0 m) n * EXP (NUMERAL_BIT0 m) n)) &
+ ((2::nat) * m) ^ ((2::nat) * n + (1::nat)) =
+ (2::nat) * m * (((2::nat) * m) ^ n * ((2::nat) * m) ^ n)) &
(ALL (m::nat) n::nat.
- EXP (NUMERAL_BIT1 m) (NUMERAL_BIT1 n) =
- NUMERAL_BIT1 m * (EXP (NUMERAL_BIT1 m) n * EXP (NUMERAL_BIT1 m) n))"
+ ((2::nat) * m + (1::nat)) ^ ((2::nat) * n + (1::nat)) =
+ ((2::nat) * m + (1::nat)) *
+ (((2::nat) * m + (1::nat)) ^ n * ((2::nat) * m + (1::nat)) ^ n))"
by (import hollight ARITH_EXP)
-lemma ARITH_EVEN: "(ALL x::nat. EVEN x = EVEN x) &
-EVEN 0 = True &
-(ALL x::nat. EVEN (NUMERAL_BIT0 x) = True) &
-(ALL x::nat. EVEN (NUMERAL_BIT1 x) = False)"
+lemma ARITH_EVEN: "(ALL x::nat. even x = even x) &
+even (0::nat) = True &
+(ALL x::nat. even ((2::nat) * x) = True) &
+(ALL x::nat. even ((2::nat) * x + (1::nat)) = False)"
by (import hollight ARITH_EVEN)
-lemma ARITH_ODD: "(ALL x::nat. ODD x = ODD x) &
-ODD 0 = False &
-(ALL x::nat. ODD (NUMERAL_BIT0 x) = False) &
-(ALL x::nat. ODD (NUMERAL_BIT1 x) = True)"
+lemma ARITH_ODD: "(ALL x::nat. odd x = odd x) &
+odd (0::nat) = False &
+(ALL x::nat. odd ((2::nat) * x) = False) &
+(ALL x::nat. odd ((2::nat) * x + (1::nat)) = True)"
by (import hollight ARITH_ODD)
-lemma ARITH_LE: "(ALL (x::nat) xa::nat. <= x xa = <= x xa) &
-<= 0 0 = True &
-(ALL x::nat. <= (NUMERAL_BIT0 x) 0 = (x = 0)) &
-(ALL x::nat. <= (NUMERAL_BIT1 x) 0 = False) &
-(ALL x::nat. <= 0 (NUMERAL_BIT0 x) = True) &
-(ALL x::nat. <= 0 (NUMERAL_BIT1 x) = True) &
-(ALL (x::nat) xa::nat. <= (NUMERAL_BIT0 x) (NUMERAL_BIT0 xa) = <= x xa) &
-(ALL (x::nat) xa::nat. <= (NUMERAL_BIT0 x) (NUMERAL_BIT1 xa) = <= x xa) &
-(ALL (x::nat) xa::nat. <= (NUMERAL_BIT1 x) (NUMERAL_BIT0 xa) = < x xa) &
-(ALL (x::nat) xa::nat. <= (NUMERAL_BIT1 x) (NUMERAL_BIT1 xa) = <= x xa)"
+lemma ARITH_LE: "(ALL (x::nat) xa::nat. (x <= xa) = (x <= xa)) &
+((0::nat) <= (0::nat)) = True &
+(ALL x::nat. ((2::nat) * x <= (0::nat)) = (x <= (0::nat))) &
+(ALL x::nat. ((2::nat) * x + (1::nat) <= (0::nat)) = False) &
+(ALL x::nat. ((0::nat) <= (2::nat) * x) = True) &
+(ALL x::nat. ((0::nat) <= (2::nat) * x + (1::nat)) = True) &
+(ALL (x::nat) xa::nat. ((2::nat) * x <= (2::nat) * xa) = (x <= xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x <= (2::nat) * xa + (1::nat)) = (x <= xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat) <= (2::nat) * xa) = (x < xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat) <= (2::nat) * xa + (1::nat)) = (x <= xa))"
by (import hollight ARITH_LE)
-lemma ARITH_LT: "(ALL (x::nat) xa::nat. < x xa = < x xa) &
-< 0 0 = False &
-(ALL x::nat. < (NUMERAL_BIT0 x) 0 = False) &
-(ALL x::nat. < (NUMERAL_BIT1 x) 0 = False) &
-(ALL x::nat. < 0 (NUMERAL_BIT0 x) = < 0 x) &
-(ALL x::nat. < 0 (NUMERAL_BIT1 x) = True) &
-(ALL (x::nat) xa::nat. < (NUMERAL_BIT0 x) (NUMERAL_BIT0 xa) = < x xa) &
-(ALL (x::nat) xa::nat. < (NUMERAL_BIT0 x) (NUMERAL_BIT1 xa) = <= x xa) &
-(ALL (x::nat) xa::nat. < (NUMERAL_BIT1 x) (NUMERAL_BIT0 xa) = < x xa) &
-(ALL (x::nat) xa::nat. < (NUMERAL_BIT1 x) (NUMERAL_BIT1 xa) = < x xa)"
+lemma ARITH_LT: "(ALL (x::nat) xa::nat. (x < xa) = (x < xa)) &
+((0::nat) < (0::nat)) = False &
+(ALL x::nat. ((2::nat) * x < (0::nat)) = False) &
+(ALL x::nat. ((2::nat) * x + (1::nat) < (0::nat)) = False) &
+(ALL x::nat. ((0::nat) < (2::nat) * x) = ((0::nat) < x)) &
+(ALL x::nat. ((0::nat) < (2::nat) * x + (1::nat)) = True) &
+(ALL (x::nat) xa::nat. ((2::nat) * x < (2::nat) * xa) = (x < xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x < (2::nat) * xa + (1::nat)) = (x <= xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat) < (2::nat) * xa) = (x < xa)) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat) < (2::nat) * xa + (1::nat)) = (x < xa))"
by (import hollight ARITH_LT)
-lemma ARITH_EQ: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool) ((op =::nat => nat => bool) x xa)
- ((op =::nat => nat => bool) x xa))))
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool)
- ((op =::nat => nat => bool) (0::nat) (0::nat)) (True::bool))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) ((NUMERAL_BIT0::nat => nat) x)
- (0::nat))
- ((op =::nat => nat => bool) x (0::nat))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) x)
- (0::nat))
- (False::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) (0::nat)
- ((NUMERAL_BIT0::nat => nat) x))
- ((op =::nat => nat => bool) (0::nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) (0::nat)
- ((NUMERAL_BIT1::nat => nat) x))
- (False::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- ((op =::nat => nat => bool) x xa))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((NUMERAL_BIT0::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- (False::bool))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT0::nat => nat) xa))
- (False::bool))))
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) x)
- ((NUMERAL_BIT1::nat => nat) xa))
- ((op =::nat => nat => bool) x xa))))))))))))"
+lemma ARITH_EQ: "(ALL (x::nat) xa::nat. (x = xa) = (x = xa)) &
+((0::nat) = (0::nat)) = True &
+(ALL x::nat. ((2::nat) * x = (0::nat)) = (x = (0::nat))) &
+(ALL x::nat. ((2::nat) * x + (1::nat) = (0::nat)) = False) &
+(ALL x::nat. ((0::nat) = (2::nat) * x) = ((0::nat) = x)) &
+(ALL x::nat. ((0::nat) = (2::nat) * x + (1::nat)) = False) &
+(ALL (x::nat) xa::nat. ((2::nat) * x = (2::nat) * xa) = (x = xa)) &
+(ALL (x::nat) xa::nat. ((2::nat) * x = (2::nat) * xa + (1::nat)) = False) &
+(ALL (x::nat) xa::nat. ((2::nat) * x + (1::nat) = (2::nat) * xa) = False) &
+(ALL (x::nat) xa::nat.
+ ((2::nat) * x + (1::nat) = (2::nat) * xa + (1::nat)) = (x = xa))"
by (import hollight ARITH_EQ)
-lemma ARITH_SUB: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool) ((op -::nat => nat => nat) x xa)
- ((op -::nat => nat => nat) x xa))))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool) ((op -::nat => nat => nat) (0::nat) (0::nat))
- (0::nat))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT0::nat => nat) x))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) (0::nat)
- ((NUMERAL_BIT1::nat => nat) x))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) ((NUMERAL_BIT0::nat => nat) x)
- (0::nat))
- ((NUMERAL_BIT0::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) ((NUMERAL_BIT1::nat => nat) x)
- (0::nat))
- ((NUMERAL_BIT1::nat => nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) m)
- ((NUMERAL_BIT0::nat => nat) n))
- ((NUMERAL_BIT0::nat => nat)
- ((op -::nat => nat => nat) m n)))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat)
- ((NUMERAL_BIT0::nat => nat) m)
- ((NUMERAL_BIT1::nat => nat) n))
- ((Pred::nat => nat)
- ((NUMERAL_BIT0::nat => nat)
- ((op -::nat => nat => nat) m n))))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) m)
- ((NUMERAL_BIT0::nat => nat) n))
- ((COND::bool => nat => nat => nat)
- ((<=::nat => nat => bool) n m)
- ((NUMERAL_BIT1::nat => nat)
- ((op -::nat => nat => nat) m n))
- (0::nat)))))
- ((All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat)
- ((NUMERAL_BIT1::nat => nat) m)
- ((NUMERAL_BIT1::nat => nat) n))
- ((NUMERAL_BIT0::nat => nat)
- ((op -::nat => nat => nat) m n)))))))))))))"
+lemma ARITH_SUB: "(ALL (x::nat) xa::nat. x - xa = x - xa) &
+(0::nat) - (0::nat) = (0::nat) &
+(ALL x::nat. (0::nat) - (2::nat) * x = (0::nat)) &
+(ALL x::nat. (0::nat) - ((2::nat) * x + (1::nat)) = (0::nat)) &
+(ALL x::nat. (2::nat) * x - (0::nat) = (2::nat) * x) &
+(ALL x::nat. (2::nat) * x + (1::nat) - (0::nat) = (2::nat) * x + (1::nat)) &
+(ALL (m::nat) n::nat. (2::nat) * m - (2::nat) * n = (2::nat) * (m - n)) &
+(ALL (m::nat) n::nat.
+ (2::nat) * m - ((2::nat) * n + (1::nat)) =
+ (2::nat) * (m - n) - Suc (0::nat)) &
+(ALL (m::nat) n::nat.
+ (2::nat) * m + (1::nat) - (2::nat) * n =
+ (if n <= m then (2::nat) * (m - n) + (1::nat) else (0::nat))) &
+(ALL (m::nat) n::nat.
+ (2::nat) * m + (1::nat) - ((2::nat) * n + (1::nat)) =
+ (2::nat) * (m - n))"
by (import hollight ARITH_SUB)
-lemma right_th: "(s::nat) * NUMERAL_BIT1 (x::nat) = s + NUMERAL_BIT0 (s * x)"
+lemma right_th: "(s::nat) * ((2::nat) * (x::nat) + (1::nat)) = s + (2::nat) * (s * x)"
by (import hollight right_th)
-lemma SEMIRING_PTHS: "(ALL (x::'A::type) (y::'A::type) z::'A::type.
- (add::'A::type => 'A::type => 'A::type) x (add y z) = add (add x y) z) &
-(ALL (x::'A::type) y::'A::type. add x y = add y x) &
-(ALL x::'A::type. add (r0::'A::type) x = x) &
-(ALL (x::'A::type) (y::'A::type) z::'A::type.
- (mul::'A::type => 'A::type => 'A::type) x (mul y z) = mul (mul x y) z) &
-(ALL (x::'A::type) y::'A::type. mul x y = mul y x) &
-(ALL x::'A::type. mul (r1::'A::type) x = x) &
-(ALL x::'A::type. mul r0 x = r0) &
-(ALL (x::'A::type) (y::'A::type) z::'A::type.
- mul x (add y z) = add (mul x y) (mul x z)) &
-(ALL x::'A::type. (pwr::'A::type => nat => 'A::type) x 0 = r1) &
-(ALL (x::'A::type) n::nat. pwr x (Suc n) = mul x (pwr x n)) -->
-mul r1 (x::'A::type) = x &
-add (mul (a::'A::type) (m::'A::type)) (mul (b::'A::type) m) =
-mul (add a b) m &
-add (mul a m) m = mul (add a r1) m &
-add m (mul a m) = mul (add a r1) m &
-add m m = mul (add r1 r1) m &
-mul r0 m = r0 &
-add r0 a = a &
-add a r0 = a &
-mul a b = mul b a &
-mul (add a b) (c::'A::type) = add (mul a c) (mul b c) &
-mul r0 a = r0 &
-mul a r0 = r0 &
-mul r1 a = a &
-mul a r1 = a &
-mul (mul (lx::'A::type) (ly::'A::type))
- (mul (rx::'A::type) (ry::'A::type)) =
-mul (mul lx rx) (mul ly ry) &
-mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry)) &
-mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry) &
-mul (mul lx ly) rx = mul (mul lx rx) ly &
-mul (mul lx ly) rx = mul lx (mul ly rx) &
-mul lx rx = mul rx lx &
-mul lx (mul rx ry) = mul (mul lx rx) ry &
-mul lx (mul rx ry) = mul rx (mul lx ry) &
-add (add a b) (add c (d::'A::type)) = add (add a c) (add b d) &
-add (add a b) c = add a (add b c) &
-add a (add c d) = add c (add a d) &
-add (add a b) c = add (add a c) b &
-add a c = add c a &
-add a (add c d) = add (add a c) d &
-mul (pwr x (p::nat)) (pwr x (q::nat)) = pwr x (p + q) &
-mul x (pwr x q) = pwr x (Suc q) &
-mul (pwr x q) x = pwr x (Suc q) &
-mul x x = pwr x (NUMERAL_BIT0 (NUMERAL_BIT1 0)) &
-pwr (mul x (y::'A::type)) q = mul (pwr x q) (pwr y q) &
-pwr (pwr x p) q = pwr x (p * q) &
-pwr x 0 = r1 &
-pwr x (NUMERAL_BIT1 0) = x &
-mul x (add y (z::'A::type)) = add (mul x y) (mul x z) &
-pwr x (Suc q) = mul x (pwr x q)"
+lemma SEMIRING_PTHS: "(ALL (x::'A) (y::'A) z::'A.
+ (add::'A => 'A => 'A) x (add y z) = add (add x y) z) &
+(ALL (x::'A) y::'A. add x y = add y x) &
+(ALL x::'A. add (r0::'A) x = x) &
+(ALL (x::'A) (y::'A) z::'A.
+ (mul::'A => 'A => 'A) x (mul y z) = mul (mul x y) z) &
+(ALL (x::'A) y::'A. mul x y = mul y x) &
+(ALL x::'A. mul (r1::'A) x = x) &
+(ALL x::'A. mul r0 x = r0) &
+(ALL (x::'A) (y::'A) z::'A. mul x (add y z) = add (mul x y) (mul x z)) &
+(ALL x::'A. (pwr::'A => nat => 'A) x (0::nat) = r1) &
+(ALL (x::'A) n::nat. pwr x (Suc n) = mul x (pwr x n))
+==> mul r1 (x::'A) = x &
+ add (mul (a::'A) (m::'A)) (mul (b::'A) m) = mul (add a b) m &
+ add (mul a m) m = mul (add a r1) m &
+ add m (mul a m) = mul (add a r1) m &
+ add m m = mul (add r1 r1) m &
+ mul r0 m = r0 &
+ add r0 a = a &
+ add a r0 = a &
+ mul a b = mul b a &
+ mul (add a b) (c::'A) = add (mul a c) (mul b c) &
+ mul r0 a = r0 &
+ mul a r0 = r0 &
+ mul r1 a = a &
+ mul a r1 = a &
+ mul (mul (lx::'A) (ly::'A)) (mul (rx::'A) (ry::'A)) =
+ mul (mul lx rx) (mul ly ry) &
+ mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry)) &
+ mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry) &
+ mul (mul lx ly) rx = mul (mul lx rx) ly &
+ mul (mul lx ly) rx = mul lx (mul ly rx) &
+ mul lx rx = mul rx lx &
+ mul lx (mul rx ry) = mul (mul lx rx) ry &
+ mul lx (mul rx ry) = mul rx (mul lx ry) &
+ add (add a b) (add c (d::'A)) = add (add a c) (add b d) &
+ add (add a b) c = add a (add b c) &
+ add a (add c d) = add c (add a d) &
+ add (add a b) c = add (add a c) b &
+ add a c = add c a &
+ add a (add c d) = add (add a c) d &
+ mul (pwr x (p::nat)) (pwr x (q::nat)) = pwr x (p + q) &
+ mul x (pwr x q) = pwr x (Suc q) &
+ mul (pwr x q) x = pwr x (Suc q) &
+ mul x x = pwr x (2::nat) &
+ pwr (mul x (y::'A)) q = mul (pwr x q) (pwr y q) &
+ pwr (pwr x p) q = pwr x (p * q) &
+ pwr x (0::nat) = r1 &
+ pwr x (1::nat) = x &
+ mul x (add y (z::'A)) = add (mul x y) (mul x z) &
+ pwr x (Suc q) = mul x (pwr x q)"
by (import hollight SEMIRING_PTHS)
-lemma sth: "(ALL (x::nat) (y::nat) z::nat. x + (y + z) = x + y + z) &
-(ALL (x::nat) y::nat. x + y = y + x) &
-(ALL x::nat. 0 + x = x) &
-(ALL (x::nat) (y::nat) z::nat. x * (y * z) = x * y * z) &
-(ALL (x::nat) y::nat. x * y = y * x) &
-(ALL x::nat. NUMERAL_BIT1 0 * x = x) &
-(ALL x::nat. 0 * x = 0) &
-(ALL (x::nat) (xa::nat) xb::nat. x * (xa + xb) = x * xa + x * xb) &
-(ALL x::nat. EXP x 0 = NUMERAL_BIT1 0) &
-(ALL (x::nat) xa::nat. EXP x (Suc xa) = x * EXP x xa)"
- by (import hollight sth)
-
-lemma NUM_INTEGRAL_LEMMA: "(w::nat) = (x::nat) + (d::nat) & (y::nat) = (z::nat) + (e::nat) -->
-(w * y + x * z = w * z + x * y) = (w = x | y = z)"
+lemma NUM_INTEGRAL_LEMMA: "(w::nat) = (x::nat) + (d::nat) & (y::nat) = (z::nat) + (e::nat)
+==> (w * y + x * z = w * z + x * y) = (w = x | y = z)"
by (import hollight NUM_INTEGRAL_LEMMA)
-lemma NUM_INTEGRAL: "(ALL x::nat. 0 * x = 0) &
+lemma NUM_INTEGRAL: "(ALL x::nat. (0::nat) * x = (0::nat)) &
(ALL (x::nat) (xa::nat) xb::nat. (x + xa = x + xb) = (xa = xb)) &
(ALL (w::nat) (x::nat) (y::nat) z::nat.
(w * y + x * z = w * z + x * y) = (w = x | y = z))"
by (import hollight NUM_INTEGRAL)
-lemma INJ_INVERSE2: "ALL P::'A::type => 'B::type => 'C::type.
- (ALL (x1::'A::type) (y1::'B::type) (x2::'A::type) y2::'B::type.
- (P x1 y1 = P x2 y2) = (x1 = x2 & y1 = y2)) -->
- (EX (x::'C::type => 'A::type) Y::'C::type => 'B::type.
- ALL (xa::'A::type) y::'B::type. x (P xa y) = xa & Y (P xa y) = y)"
+lemma INJ_INVERSE2: "(!!(x1::'A) (y1::'B) (x2::'A) y2::'B.
+ ((P::'A => 'B => 'C) x1 y1 = P x2 y2) = (x1 = x2 & y1 = y2))
+==> EX (x::'C => 'A) Y::'C => 'B.
+ ALL (xa::'A) y::'B. x (P xa y) = xa & Y (P xa y) = y"
by (import hollight INJ_INVERSE2)
-definition NUMPAIR :: "nat => nat => nat" where
- "NUMPAIR ==
-%(u::nat) ua::nat.
- EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) u *
- (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua + NUMERAL_BIT1 0)"
-
-lemma DEF_NUMPAIR: "NUMPAIR =
-(%(u::nat) ua::nat.
- EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) u *
- (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua + NUMERAL_BIT1 0))"
+definition
+ NUMPAIR :: "nat => nat => nat" where
+ "NUMPAIR == %u ua. 2 ^ u * (2 * ua + 1)"
+
+lemma DEF_NUMPAIR: "NUMPAIR = (%u ua. 2 ^ u * (2 * ua + 1))"
by (import hollight DEF_NUMPAIR)
-lemma NUMPAIR_INJ_LEMMA: "ALL (x::nat) (xa::nat) (xb::nat) xc::nat.
- NUMPAIR x xa = NUMPAIR xb xc --> x = xb"
+lemma NUMPAIR_INJ_LEMMA: "NUMPAIR x xa = NUMPAIR xb xc ==> x = xb"
by (import hollight NUMPAIR_INJ_LEMMA)
-lemma NUMPAIR_INJ: "ALL (x1::nat) (y1::nat) (x2::nat) y2::nat.
- (NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2 & y1 = y2)"
+lemma NUMPAIR_INJ: "(NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2 & y1 = y2)"
by (import hollight NUMPAIR_INJ)
-definition NUMFST :: "nat => nat" where
- "NUMFST ==
-SOME X::nat => nat.
- EX Y::nat => nat.
- ALL (x::nat) y::nat. X (NUMPAIR x y) = x & Y (NUMPAIR x y) = y"
-
-lemma DEF_NUMFST: "NUMFST =
-(SOME X::nat => nat.
- EX Y::nat => nat.
- ALL (x::nat) y::nat. X (NUMPAIR x y) = x & Y (NUMPAIR x y) = y)"
+definition
+ NUMFST :: "nat => nat" where
+ "NUMFST == SOME X. EX Y. ALL x y. X (NUMPAIR x y) = x & Y (NUMPAIR x y) = y"
+
+lemma DEF_NUMFST: "NUMFST = (SOME X. EX Y. ALL x y. X (NUMPAIR x y) = x & Y (NUMPAIR x y) = y)"
by (import hollight DEF_NUMFST)
-definition NUMSND :: "nat => nat" where
- "NUMSND ==
-SOME Y::nat => nat.
- ALL (x::nat) y::nat. NUMFST (NUMPAIR x y) = x & Y (NUMPAIR x y) = y"
-
-lemma DEF_NUMSND: "NUMSND =
-(SOME Y::nat => nat.
- ALL (x::nat) y::nat. NUMFST (NUMPAIR x y) = x & Y (NUMPAIR x y) = y)"
+definition
+ NUMSND :: "nat => nat" where
+ "NUMSND == SOME Y. ALL x y. NUMFST (NUMPAIR x y) = x & Y (NUMPAIR x y) = y"
+
+lemma DEF_NUMSND: "NUMSND = (SOME Y. ALL x y. NUMFST (NUMPAIR x y) = x & Y (NUMPAIR x y) = y)"
by (import hollight DEF_NUMSND)
-definition NUMSUM :: "bool => nat => nat" where
- "NUMSUM ==
-%(u::bool) ua::nat.
- COND u (Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua))
- (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua)"
-
-lemma DEF_NUMSUM: "NUMSUM =
-(%(u::bool) ua::nat.
- COND u (Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua))
- (NUMERAL_BIT0 (NUMERAL_BIT1 0) * ua))"
+definition
+ NUMSUM :: "bool => nat => nat" where
+ "NUMSUM == %u ua. if u then Suc (2 * ua) else 2 * ua"
+
+lemma DEF_NUMSUM: "NUMSUM = (%u ua. if u then Suc (2 * ua) else 2 * ua)"
by (import hollight DEF_NUMSUM)
-lemma NUMSUM_INJ: "ALL (b1::bool) (x1::nat) (b2::bool) x2::nat.
- (NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2 & x1 = x2)"
+lemma NUMSUM_INJ: "(NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2 & x1 = x2)"
by (import hollight NUMSUM_INJ)
-definition NUMLEFT :: "nat => bool" where
- "NUMLEFT ==
-SOME X::nat => bool.
- EX Y::nat => nat.
- ALL (x::bool) y::nat. X (NUMSUM x y) = x & Y (NUMSUM x y) = y"
-
-lemma DEF_NUMLEFT: "NUMLEFT =
-(SOME X::nat => bool.
- EX Y::nat => nat.
- ALL (x::bool) y::nat. X (NUMSUM x y) = x & Y (NUMSUM x y) = y)"
+definition
+ NUMLEFT :: "nat => bool" where
+ "NUMLEFT == SOME X. EX Y. ALL x y. X (NUMSUM x y) = x & Y (NUMSUM x y) = y"
+
+lemma DEF_NUMLEFT: "NUMLEFT = (SOME X. EX Y. ALL x y. X (NUMSUM x y) = x & Y (NUMSUM x y) = y)"
by (import hollight DEF_NUMLEFT)
-definition NUMRIGHT :: "nat => nat" where
- "NUMRIGHT ==
-SOME Y::nat => nat.
- ALL (x::bool) y::nat. NUMLEFT (NUMSUM x y) = x & Y (NUMSUM x y) = y"
-
-lemma DEF_NUMRIGHT: "NUMRIGHT =
-(SOME Y::nat => nat.
- ALL (x::bool) y::nat. NUMLEFT (NUMSUM x y) = x & Y (NUMSUM x y) = y)"
+definition
+ NUMRIGHT :: "nat => nat" where
+ "NUMRIGHT == SOME Y. ALL x y. NUMLEFT (NUMSUM x y) = x & Y (NUMSUM x y) = y"
+
+lemma DEF_NUMRIGHT: "NUMRIGHT = (SOME Y. ALL x y. NUMLEFT (NUMSUM x y) = x & Y (NUMSUM x y) = y)"
by (import hollight DEF_NUMRIGHT)
-definition INJN :: "nat => nat => 'A => bool" where
- "INJN == %(u::nat) (n::nat) a::'A::type. n = u"
-
-lemma DEF_INJN: "INJN = (%(u::nat) (n::nat) a::'A::type. n = u)"
+definition
+ INJN :: "nat => nat => 'A => bool" where
+ "INJN == %(u::nat) (n::nat) a::'A. n = u"
+
+lemma DEF_INJN: "INJN = (%(u::nat) (n::nat) a::'A. n = u)"
by (import hollight DEF_INJN)
-lemma INJN_INJ: "(All::(nat => bool) => bool)
- (%n1::nat.
- (All::(nat => bool) => bool)
- (%n2::nat.
- (op =::bool => bool => bool)
- ((op =::(nat => 'A::type => bool)
- => (nat => 'A::type => bool) => bool)
- ((INJN::nat => nat => 'A::type => bool) n1)
- ((INJN::nat => nat => 'A::type => bool) n2))
- ((op =::nat => nat => bool) n1 n2)))"
+lemma INJN_INJ: "(op =::bool => bool => bool)
+ ((op =::(nat => 'A::type => bool) => (nat => 'A::type => bool) => bool)
+ ((INJN::nat => nat => 'A::type => bool) (n1::nat))
+ ((INJN::nat => nat => 'A::type => bool) (n2::nat)))
+ ((op =::nat => nat => bool) n1 n2)"
by (import hollight INJN_INJ)
-definition INJA :: "'A => nat => 'A => bool" where
- "INJA == %(u::'A::type) (n::nat) b::'A::type. b = u"
-
-lemma DEF_INJA: "INJA = (%(u::'A::type) (n::nat) b::'A::type. b = u)"
+definition
+ INJA :: "'A => nat => 'A => bool" where
+ "INJA == %(u::'A) (n::nat) b::'A. b = u"
+
+lemma DEF_INJA: "INJA = (%(u::'A) (n::nat) b::'A. b = u)"
by (import hollight DEF_INJA)
-lemma INJA_INJ: "ALL (a1::'A::type) a2::'A::type. (INJA a1 = INJA a2) = (a1 = a2)"
+lemma INJA_INJ: "(INJA (a1::'A) = INJA (a2::'A)) = (a1 = a2)"
by (import hollight INJA_INJ)
-definition INJF :: "(nat => nat => 'A => bool) => nat => 'A => bool" where
- "INJF == %(u::nat => nat => 'A::type => bool) n::nat. u (NUMFST n) (NUMSND n)"
-
-lemma DEF_INJF: "INJF =
-(%(u::nat => nat => 'A::type => bool) n::nat. u (NUMFST n) (NUMSND n))"
+definition
+ INJF :: "(nat => nat => 'A => bool) => nat => 'A => bool" where
+ "INJF == %(u::nat => nat => 'A => bool) n::nat. u (NUMFST n) (NUMSND n)"
+
+lemma DEF_INJF: "INJF = (%(u::nat => nat => 'A => bool) n::nat. u (NUMFST n) (NUMSND n))"
by (import hollight DEF_INJF)
-lemma INJF_INJ: "ALL (f1::nat => nat => 'A::type => bool) f2::nat => nat => 'A::type => bool.
- (INJF f1 = INJF f2) = (f1 = f2)"
+lemma INJF_INJ: "(INJF (f1::nat => nat => 'A => bool) =
+ INJF (f2::nat => nat => 'A => bool)) =
+(f1 = f2)"
by (import hollight INJF_INJ)
-definition INJP :: "(nat => 'A => bool) => (nat => 'A => bool) => nat => 'A => bool" where
+definition
+ INJP :: "(nat => 'A => bool) => (nat => 'A => bool) => nat => 'A => bool" where
"INJP ==
-%(u::nat => 'A::type => bool) (ua::nat => 'A::type => bool) (n::nat)
- a::'A::type. COND (NUMLEFT n) (u (NUMRIGHT n) a) (ua (NUMRIGHT n) a)"
+%(u::nat => 'A => bool) (ua::nat => 'A => bool) (n::nat) a::'A.
+ if NUMLEFT n then u (NUMRIGHT n) a else ua (NUMRIGHT n) a"
lemma DEF_INJP: "INJP =
-(%(u::nat => 'A::type => bool) (ua::nat => 'A::type => bool) (n::nat)
- a::'A::type. COND (NUMLEFT n) (u (NUMRIGHT n) a) (ua (NUMRIGHT n) a))"
+(%(u::nat => 'A => bool) (ua::nat => 'A => bool) (n::nat) a::'A.
+ if NUMLEFT n then u (NUMRIGHT n) a else ua (NUMRIGHT n) a)"
by (import hollight DEF_INJP)
-lemma INJP_INJ: "ALL (f1::nat => 'A::type => bool) (f1'::nat => 'A::type => bool)
- (f2::nat => 'A::type => bool) f2'::nat => 'A::type => bool.
- (INJP f1 f2 = INJP f1' f2') = (f1 = f1' & f2 = f2')"
+lemma INJP_INJ: "(INJP (f1::nat => 'A => bool) (f2::nat => 'A => bool) =
+ INJP (f1'::nat => 'A => bool) (f2'::nat => 'A => bool)) =
+(f1 = f1' & f2 = f2')"
by (import hollight INJP_INJ)
-definition ZCONSTR :: "nat => 'A => (nat => nat => 'A => bool) => nat => 'A => bool" where
+definition
+ ZCONSTR :: "nat => 'A => (nat => nat => 'A => bool) => nat => 'A => bool" where
"ZCONSTR ==
-%(u::nat) (ua::'A::type) ub::nat => nat => 'A::type => bool.
+%(u::nat) (ua::'A) ub::nat => nat => 'A => bool.
INJP (INJN (Suc u)) (INJP (INJA ua) (INJF ub))"
lemma DEF_ZCONSTR: "ZCONSTR =
-(%(u::nat) (ua::'A::type) ub::nat => nat => 'A::type => bool.
+(%(u::nat) (ua::'A) ub::nat => nat => 'A => bool.
INJP (INJN (Suc u)) (INJP (INJA ua) (INJF ub)))"
by (import hollight DEF_ZCONSTR)
-definition ZBOT :: "nat => 'A => bool" where
- "ZBOT == INJP (INJN 0) (SOME z::nat => 'A::type => bool. True)"
-
-lemma DEF_ZBOT: "ZBOT = INJP (INJN 0) (SOME z::nat => 'A::type => bool. True)"
+definition
+ ZBOT :: "nat => 'A => bool" where
+ "ZBOT == INJP (INJN (0::nat)) (SOME z::nat => 'A => bool. True)"
+
+lemma DEF_ZBOT: "ZBOT = INJP (INJN (0::nat)) (SOME z::nat => 'A => bool. True)"
by (import hollight DEF_ZBOT)
-lemma ZCONSTR_ZBOT: "ALL (x::nat) (xa::'A::type) xb::nat => nat => 'A::type => bool.
- ZCONSTR x xa xb ~= ZBOT"
+lemma ZCONSTR_ZBOT: "ZCONSTR (x::nat) (xa::'A) (xb::nat => nat => 'A => bool) ~= ZBOT"
by (import hollight ZCONSTR_ZBOT)
-definition ZRECSPACE :: "(nat => 'A => bool) => bool" where
+definition
+ ZRECSPACE :: "(nat => 'A => bool) => bool" where
"ZRECSPACE ==
-%a::nat => 'A::type => bool.
- ALL ZRECSPACE'::(nat => 'A::type => bool) => bool.
- (ALL a::nat => 'A::type => bool.
+%a::nat => 'A => bool.
+ ALL ZRECSPACE'::(nat => 'A => bool) => bool.
+ (ALL a::nat => 'A => bool.
a = ZBOT |
- (EX (c::nat) (i::'A::type) r::nat => nat => 'A::type => bool.
+ (EX (c::nat) (i::'A) r::nat => nat => 'A => bool.
a = ZCONSTR c i r & (ALL n::nat. ZRECSPACE' (r n))) -->
ZRECSPACE' a) -->
ZRECSPACE' a"
lemma DEF_ZRECSPACE: "ZRECSPACE =
-(%a::nat => 'A::type => bool.
- ALL ZRECSPACE'::(nat => 'A::type => bool) => bool.
- (ALL a::nat => 'A::type => bool.
+(%a::nat => 'A => bool.
+ ALL ZRECSPACE'::(nat => 'A => bool) => bool.
+ (ALL a::nat => 'A => bool.
a = ZBOT |
- (EX (c::nat) (i::'A::type) r::nat => nat => 'A::type => bool.
+ (EX (c::nat) (i::'A) r::nat => nat => 'A => bool.
a = ZCONSTR c i r & (ALL n::nat. ZRECSPACE' (r n))) -->
ZRECSPACE' a) -->
ZRECSPACE' a)"
by (import hollight DEF_ZRECSPACE)
-typedef (open) ('A) recspace = "(Collect::((nat => 'A::type => bool) => bool)
- => (nat => 'A::type => bool) set)
- (ZRECSPACE::(nat => 'A::type => bool) => bool)" morphisms "_dest_rec" "_mk_rec"
- apply (rule light_ex_imp_nonempty[where t="ZBOT::nat => 'A::type => bool"])
+typedef (open) ('A) recspace = "Collect ZRECSPACE" morphisms "_dest_rec" "_mk_rec"
+ apply (rule light_ex_imp_nonempty[where t="ZBOT"])
by (import hollight TYDEF_recspace)
syntax
@@ -1779,1315 +1169,236 @@
[where a="a :: 'A recspace" and r=r ,
OF type_definition_recspace]
-definition BOTTOM :: "'A recspace" where
- "(op ==::'A::type recspace => 'A::type recspace => prop)
- (BOTTOM::'A::type recspace)
- ((_mk_rec::(nat => 'A::type => bool) => 'A::type recspace)
- (ZBOT::nat => 'A::type => bool))"
-
-lemma DEF_BOTTOM: "(op =::'A::type recspace => 'A::type recspace => bool)
- (BOTTOM::'A::type recspace)
- ((_mk_rec::(nat => 'A::type => bool) => 'A::type recspace)
- (ZBOT::nat => 'A::type => bool))"
+definition
+ BOTTOM :: "'A recspace" where
+ "BOTTOM == _mk_rec ZBOT"
+
+lemma DEF_BOTTOM: "BOTTOM = _mk_rec ZBOT"
by (import hollight DEF_BOTTOM)
-definition CONSTR :: "nat => 'A => (nat => 'A recspace) => 'A recspace" where
- "(op ==::(nat => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- => (nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- => prop)
- (CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
- (_mk_rec::(nat => 'A::type => bool) => 'A::type recspace)
- ((ZCONSTR::nat
- => 'A::type
- => (nat => nat => 'A::type => bool)
- => nat => 'A::type => bool)
- u ua
- (%n::nat.
- (_dest_rec::'A::type recspace => nat => 'A::type => bool)
- (ub n))))"
-
-lemma DEF_CONSTR: "(op =::(nat => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- => (nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- => bool)
- (CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
- (_mk_rec::(nat => 'A::type => bool) => 'A::type recspace)
- ((ZCONSTR::nat
- => 'A::type
- => (nat => nat => 'A::type => bool)
- => nat => 'A::type => bool)
- u ua
- (%n::nat.
- (_dest_rec::'A::type recspace => nat => 'A::type => bool)
- (ub n))))"
+definition
+ CONSTR :: "nat => 'A => (nat => 'A recspace) => 'A recspace" where
+ "CONSTR ==
+%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
+ _mk_rec (ZCONSTR u ua (%n::nat. _dest_rec (ub n)))"
+
+lemma DEF_CONSTR: "CONSTR =
+(%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
+ _mk_rec (ZCONSTR u ua (%n::nat. _dest_rec (ub n))))"
by (import hollight DEF_CONSTR)
-lemma MK_REC_INJ: "(All::((nat => 'A::type => bool) => bool) => bool)
- (%x::nat => 'A::type => bool.
- (All::((nat => 'A::type => bool) => bool) => bool)
- (%y::nat => 'A::type => bool.
- (op -->::bool => bool => bool)
- ((op =::'A::type recspace => 'A::type recspace => bool)
- ((_mk_rec::(nat => 'A::type => bool) => 'A::type recspace) x)
- ((_mk_rec::(nat => 'A::type => bool) => 'A::type recspace) y))
- ((op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((ZRECSPACE::(nat => 'A::type => bool) => bool) x)
- ((ZRECSPACE::(nat => 'A::type => bool) => bool) y))
- ((op =::(nat => 'A::type => bool)
- => (nat => 'A::type => bool) => bool)
- x y))))"
+lemma MK_REC_INJ: "[| _mk_rec (x::nat => 'A::type => bool) =
+ _mk_rec (y::nat => 'A::type => bool);
+ ZRECSPACE x & ZRECSPACE y |]
+==> x = y"
by (import hollight MK_REC_INJ)
-lemma CONSTR_BOT: "ALL (c::nat) (i::'A::type) r::nat => 'A::type recspace.
- CONSTR c i r ~= BOTTOM"
+lemma CONSTR_BOT: "CONSTR (c::nat) (i::'A) (r::nat => 'A recspace) ~= BOTTOM"
by (import hollight CONSTR_BOT)
-lemma CONSTR_INJ: "ALL (c1::nat) (i1::'A::type) (r1::nat => 'A::type recspace) (c2::nat)
- (i2::'A::type) r2::nat => 'A::type recspace.
- (CONSTR c1 i1 r1 = CONSTR c2 i2 r2) = (c1 = c2 & i1 = i2 & r1 = r2)"
+lemma CONSTR_INJ: "(CONSTR (c1::nat) (i1::'A) (r1::nat => 'A recspace) =
+ CONSTR (c2::nat) (i2::'A) (r2::nat => 'A recspace)) =
+(c1 = c2 & i1 = i2 & r1 = r2)"
by (import hollight CONSTR_INJ)
-lemma CONSTR_IND: "ALL P::'A::type recspace => bool.
- P BOTTOM &
- (ALL (c::nat) (i::'A::type) r::nat => 'A::type recspace.
- (ALL n::nat. P (r n)) --> P (CONSTR c i r)) -->
- All P"
+lemma CONSTR_IND: "(P::'A recspace => bool) BOTTOM &
+(ALL (c::nat) (i::'A) r::nat => 'A recspace.
+ (ALL n::nat. P (r n)) --> P (CONSTR c i r))
+==> P (x::'A recspace)"
by (import hollight CONSTR_IND)
-lemma CONSTR_REC: "ALL Fn::nat
- => 'A::type
- => (nat => 'A::type recspace) => (nat => 'B::type) => 'B::type.
- EX f::'A::type recspace => 'B::type.
- ALL (c::nat) (i::'A::type) r::nat => 'A::type recspace.
- f (CONSTR c i r) = Fn c i r (%n::nat. f (r n))"
+lemma CONSTR_REC: "EX f::'A recspace => 'B.
+ ALL (c::nat) (i::'A) r::nat => 'A recspace.
+ f (CONSTR c i r) =
+ (Fn::nat => 'A => (nat => 'A recspace) => (nat => 'B) => 'B) c i r
+ (%n::nat. f (r n))"
by (import hollight CONSTR_REC)
-definition FCONS :: "'A => (nat => 'A) => nat => 'A" where
+definition
+ FCONS :: "'A => (nat => 'A) => nat => 'A" where
"FCONS ==
-SOME FCONS::'A::type => (nat => 'A::type) => nat => 'A::type.
- (ALL (a::'A::type) f::nat => 'A::type. FCONS a f 0 = a) &
- (ALL (a::'A::type) (f::nat => 'A::type) n::nat. FCONS a f (Suc n) = f n)"
+SOME FCONS::'A => (nat => 'A) => nat => 'A.
+ (ALL (a::'A) f::nat => 'A. FCONS a f (0::nat) = a) &
+ (ALL (a::'A) (f::nat => 'A) n::nat. FCONS a f (Suc n) = f n)"
lemma DEF_FCONS: "FCONS =
-(SOME FCONS::'A::type => (nat => 'A::type) => nat => 'A::type.
- (ALL (a::'A::type) f::nat => 'A::type. FCONS a f 0 = a) &
- (ALL (a::'A::type) (f::nat => 'A::type) n::nat.
- FCONS a f (Suc n) = f n))"
+(SOME FCONS::'A => (nat => 'A) => nat => 'A.
+ (ALL (a::'A) f::nat => 'A. FCONS a f (0::nat) = a) &
+ (ALL (a::'A) (f::nat => 'A) n::nat. FCONS a f (Suc n) = f n))"
by (import hollight DEF_FCONS)
-lemma FCONS_UNDO: "ALL f::nat => 'A::type. f = FCONS (f 0) (f o Suc)"
+lemma FCONS_UNDO: "(f::nat => 'A) = FCONS (f (0::nat)) (f o Suc)"
by (import hollight FCONS_UNDO)
-definition FNIL :: "nat => 'A" where
- "FNIL == %u::nat. SOME x::'A::type. True"
-
-lemma DEF_FNIL: "FNIL = (%u::nat. SOME x::'A::type. True)"
+definition
+ FNIL :: "nat => 'A" where
+ "FNIL == %u::nat. SOME x::'A. True"
+
+lemma DEF_FNIL: "FNIL = (%u::nat. SOME x::'A. True)"
by (import hollight DEF_FNIL)
-consts
- OUTL :: "'A + 'B => 'A"
-
-defs
- OUTL_def: "hollight.OUTL ==
-SOME OUTL::'A::type + 'B::type => 'A::type.
- ALL x::'A::type. OUTL (Inl x) = x"
-
-lemma DEF_OUTL: "hollight.OUTL =
-(SOME OUTL::'A::type + 'B::type => 'A::type.
- ALL x::'A::type. OUTL (Inl x) = x)"
- by (import hollight DEF_OUTL)
-
-consts
- OUTR :: "'A + 'B => 'B"
-
-defs
- OUTR_def: "hollight.OUTR ==
-SOME OUTR::'A::type + 'B::type => 'B::type.
- ALL y::'B::type. OUTR (Inr y) = y"
-
-lemma DEF_OUTR: "hollight.OUTR =
-(SOME OUTR::'A::type + 'B::type => 'B::type.
- ALL y::'B::type. OUTR (Inr y) = y)"
- by (import hollight DEF_OUTR)
-
-typedef (open) ('A) option = "(Collect::('A::type recspace => bool) => 'A::type recspace set)
- (%a::'A::type recspace.
- (All::(('A::type recspace => bool) => bool) => bool)
- (%option'::'A::type recspace => bool.
- (op -->::bool => bool => bool)
- ((All::('A::type recspace => bool) => bool)
- (%a::'A::type recspace.
- (op -->::bool => bool => bool)
- ((op |::bool => bool => bool)
- ((op =::'A::type recspace => 'A::type recspace => bool)
- a ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace)
- => 'A::type recspace)
- ((NUMERAL::nat => nat) (0::nat))
- ((Eps::('A::type => bool) => 'A::type)
- (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))
- ((Ex::('A::type => bool) => bool)
- (%aa::'A::type.
- (op =::'A::type recspace
- => 'A::type recspace => bool)
- a ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- ((Suc::nat => nat)
- ((NUMERAL::nat => nat) (0::nat)))
- aa (%n::nat. BOTTOM::'A::type recspace)))))
- (option' a)))
- (option' a)))" morphisms "_dest_option" "_mk_option"
- apply (rule light_ex_imp_nonempty[where t="(CONSTR::nat => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- ((NUMERAL::nat => nat) (0::nat))
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)"])
- by (import hollight TYDEF_option)
+definition
+ ISO :: "('A => 'B) => ('B => 'A) => bool" where
+ "ISO ==
+%(u::'A => 'B) ua::'B => 'A.
+ (ALL x::'B. u (ua x) = x) & (ALL y::'A. ua (u y) = y)"
+
+lemma DEF_ISO: "ISO =
+(%(u::'A => 'B) ua::'B => 'A.
+ (ALL x::'B. u (ua x) = x) & (ALL y::'A. ua (u y) = y))"
+ by (import hollight DEF_ISO)
+
+lemma ISO_REFL: "ISO (%x::'A. x) (%x::'A. x)"
+ by (import hollight ISO_REFL)
+
+lemma ISO_FUN: "ISO (f::'A => 'A') (f'::'A' => 'A) & ISO (g::'B => 'B') (g'::'B' => 'B)
+==> ISO (%(h::'A => 'B) a'::'A'. g (h (f' a')))
+ (%(h::'A' => 'B') a::'A. g' (h (f a)))"
+ by (import hollight ISO_FUN)
+
+lemma ISO_USAGE: "ISO (f::'q_17485 => 'q_17482) (g::'q_17482 => 'q_17485)
+==> (ALL P::'q_17485 => bool. All P = (ALL x::'q_17482. P (g x))) &
+ (ALL P::'q_17485 => bool. Ex P = (EX x::'q_17482. P (g x))) &
+ (ALL (a::'q_17485) b::'q_17482. (a = g b) = (f a = b))"
+ by (import hollight ISO_USAGE)
+
+typedef (open) char = "{a. ALL char'.
+ (ALL a.
+ (EX a0 a1 a2 a3 a4 a5 a6 a7.
+ a =
+ CONSTR (NUMERAL 0) (a0, a1, a2, a3, a4, a5, a6, a7)
+ (%n. BOTTOM)) -->
+ char' a) -->
+ char' a}" morphisms "_dest_char" "_mk_char"
+ apply (rule light_ex_imp_nonempty[where t="CONSTR (NUMERAL 0) (a0, a1, a2, a3, a4, a5, a6, a7) (%n. BOTTOM)"])
+ by (import hollight TYDEF_char)
syntax
- "_dest_option" :: _ ("'_dest'_option")
+ "_dest_char" :: _ ("'_dest'_char")
syntax
- "_mk_option" :: _ ("'_mk'_option")
-
-lemmas "TYDEF_option_@intern" = typedef_hol2hollight
- [where a="a :: 'A hollight.option" and r=r ,
- OF type_definition_option]
-
-definition NONE :: "'A hollight.option" where
- "(op ==::'A::type hollight.option => 'A::type hollight.option => prop)
- (NONE::'A::type hollight.option)
- ((_mk_option::'A::type recspace => 'A::type hollight.option)
- ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (0::nat)
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))"
-
-lemma DEF_NONE: "(op =::'A::type hollight.option => 'A::type hollight.option => bool)
- (NONE::'A::type hollight.option)
- ((_mk_option::'A::type recspace => 'A::type hollight.option)
- ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (0::nat)
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))"
- by (import hollight DEF_NONE)
+ "_mk_char" :: _ ("'_mk'_char")
+
+lemmas "TYDEF_char_@intern" = typedef_hol2hollight
+ [where a="a :: hollight.char" and r=r ,
+ OF type_definition_char]
consts
- SOME :: "'A => 'A hollight.option" ("SOME")
-
-defs
- SOME_def: "(op ==::('A::type => 'A::type hollight.option)
- => ('A::type => 'A::type hollight.option) => prop)
- (SOME::'A::type => 'A::type hollight.option)
- (%a::'A::type.
- (_mk_option::'A::type recspace => 'A::type hollight.option)
- ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace) => 'A::type recspace)
- ((Suc::nat => nat) (0::nat)) a
- (%n::nat. BOTTOM::'A::type recspace)))"
-
-lemma DEF_SOME: "(op =::('A::type => 'A::type hollight.option)
- => ('A::type => 'A::type hollight.option) => bool)
- (SOME::'A::type => 'A::type hollight.option)
- (%a::'A::type.
- (_mk_option::'A::type recspace => 'A::type hollight.option)
- ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace) => 'A::type recspace)
- ((Suc::nat => nat) (0::nat)) a
- (%n::nat. BOTTOM::'A::type recspace)))"
- by (import hollight DEF_SOME)
-
-typedef (open) ('A) list = "(Collect::('A::type recspace => bool) => 'A::type recspace set)
- (%a::'A::type recspace.
- (All::(('A::type recspace => bool) => bool) => bool)
- (%list'::'A::type recspace => bool.
- (op -->::bool => bool => bool)
- ((All::('A::type recspace => bool) => bool)
- (%a::'A::type recspace.
- (op -->::bool => bool => bool)
- ((op |::bool => bool => bool)
- ((op =::'A::type recspace => 'A::type recspace => bool)
- a ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace)
- => 'A::type recspace)
- ((NUMERAL::nat => nat) (0::nat))
- ((Eps::('A::type => bool) => 'A::type)
- (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))
- ((Ex::('A::type => bool) => bool)
- (%a0::'A::type.
- (Ex::('A::type recspace => bool) => bool)
- (%a1::'A::type recspace.
- (op &::bool => bool => bool)
- ((op =::'A::type recspace
- => 'A::type recspace => bool)
- a ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
-((Suc::nat => nat) ((NUMERAL::nat => nat) (0::nat))) a0
-((FCONS::'A::type recspace
- => (nat => 'A::type recspace) => nat => 'A::type recspace)
- a1 (%n::nat. BOTTOM::'A::type recspace))))
- (list' a1)))))
- (list' a)))
- (list' a)))" morphisms "_dest_list" "_mk_list"
- apply (rule light_ex_imp_nonempty[where t="(CONSTR::nat => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- ((NUMERAL::nat => nat) (0::nat))
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)"])
- by (import hollight TYDEF_list)
-
-syntax
- "_dest_list" :: _ ("'_dest'_list")
-
-syntax
- "_mk_list" :: _ ("'_mk'_list")
-
-lemmas "TYDEF_list_@intern" = typedef_hol2hollight
- [where a="a :: 'A hollight.list" and r=r ,
- OF type_definition_list]
-
-definition NIL :: "'A hollight.list" where
- "(op ==::'A::type hollight.list => 'A::type hollight.list => prop)
- (NIL::'A::type hollight.list)
- ((_mk_list::'A::type recspace => 'A::type hollight.list)
- ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (0::nat)
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))"
-
-lemma DEF_NIL: "(op =::'A::type hollight.list => 'A::type hollight.list => bool)
- (NIL::'A::type hollight.list)
- ((_mk_list::'A::type recspace => 'A::type hollight.list)
- ((CONSTR::nat
- => 'A::type => (nat => 'A::type recspace) => 'A::type recspace)
- (0::nat)
- ((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
- (%n::nat. BOTTOM::'A::type recspace)))"
- by (import hollight DEF_NIL)
-
-definition CONS :: "'A => 'A hollight.list => 'A hollight.list" where
- "(op ==::('A::type => 'A::type hollight.list => 'A::type hollight.list)
- => ('A::type => 'A::type hollight.list => 'A::type hollight.list)
- => prop)
- (CONS::'A::type => 'A::type hollight.list => 'A::type hollight.list)
- (%(a0::'A::type) a1::'A::type hollight.list.
- (_mk_list::'A::type recspace => 'A::type hollight.list)
- ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace) => 'A::type recspace)
- ((Suc::nat => nat) (0::nat)) a0
- ((FCONS::'A::type recspace
- => (nat => 'A::type recspace) => nat => 'A::type recspace)
- ((_dest_list::'A::type hollight.list => 'A::type recspace) a1)
- (%n::nat. BOTTOM::'A::type recspace))))"
-
-lemma DEF_CONS: "(op =::('A::type => 'A::type hollight.list => 'A::type hollight.list)
- => ('A::type => 'A::type hollight.list => 'A::type hollight.list)
- => bool)
- (CONS::'A::type => 'A::type hollight.list => 'A::type hollight.list)
- (%(a0::'A::type) a1::'A::type hollight.list.
- (_mk_list::'A::type recspace => 'A::type hollight.list)
- ((CONSTR::nat
- => 'A::type
- => (nat => 'A::type recspace) => 'A::type recspace)
- ((Suc::nat => nat) (0::nat)) a0
- ((FCONS::'A::type recspace
- => (nat => 'A::type recspace) => nat => 'A::type recspace)
- ((_dest_list::'A::type hollight.list => 'A::type recspace) a1)
- (%n::nat. BOTTOM::'A::type recspace))))"
- by (import hollight DEF_CONS)
-
-lemma pair_RECURSION: "ALL PAIR'::'A::type => 'B::type => 'C::type.
- EX x::'A::type * 'B::type => 'C::type.
- ALL (a0::'A::type) a1::'B::type. x (a0, a1) = PAIR' a0 a1"
- by (import hollight pair_RECURSION)
-
-lemma num_RECURSION_STD: "ALL (e::'Z::type) f::nat => 'Z::type => 'Z::type.
- EX fn::nat => 'Z::type. fn 0 = e & (ALL n::nat. fn (Suc n) = f n (fn n))"
- by (import hollight num_RECURSION_STD)
-
-lemma bool_RECURSION: "ALL (a::'A::type) b::'A::type.
- EX x::bool => 'A::type. x False = a & x True = b"
- by (import hollight bool_RECURSION)
-
-definition ISO :: "('A => 'B) => ('B => 'A) => bool" where
- "ISO ==
-%(u::'A::type => 'B::type) ua::'B::type => 'A::type.
- (ALL x::'B::type. u (ua x) = x) & (ALL y::'A::type. ua (u y) = y)"
-
-lemma DEF_ISO: "ISO =
-(%(u::'A::type => 'B::type) ua::'B::type => 'A::type.
- (ALL x::'B::type. u (ua x) = x) & (ALL y::'A::type. ua (u y) = y))"
- by (import hollight DEF_ISO)
-
-lemma ISO_REFL: "ISO (%x::'A::type. x) (%x::'A::type. x)"
- by (import hollight ISO_REFL)
-
-lemma ISO_FUN: "ISO (f::'A::type => 'A'::type) (f'::'A'::type => 'A::type) &
-ISO (g::'B::type => 'B'::type) (g'::'B'::type => 'B::type) -->
-ISO (%(h::'A::type => 'B::type) a'::'A'::type. g (h (f' a')))
- (%(h::'A'::type => 'B'::type) a::'A::type. g' (h (f a)))"
- by (import hollight ISO_FUN)
-
-lemma ISO_USAGE: "ISO (f::'q_16636::type => 'q_16633::type)
- (g::'q_16633::type => 'q_16636::type) -->
-(ALL P::'q_16636::type => bool. All P = (ALL x::'q_16633::type. P (g x))) &
-(ALL P::'q_16636::type => bool. Ex P = (EX x::'q_16633::type. P (g x))) &
-(ALL (a::'q_16636::type) b::'q_16633::type. (a = g b) = (f a = b))"
- by (import hollight ISO_USAGE)
-
-typedef (open) N_2 = "{a::bool recspace.
- ALL u::bool recspace => bool.
- (ALL a::bool recspace.
- a = CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM) |
- a =
- CONSTR (Suc (NUMERAL 0)) (SOME x::bool. True) (%n::nat. BOTTOM) -->
- u a) -->
- u a}" morphisms "_dest_2" "_mk_2"
- apply (rule light_ex_imp_nonempty[where t="CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM)"])
- by (import hollight TYDEF_2)
-
-syntax
- "_dest_2" :: _ ("'_dest'_2")
-
-syntax
- "_mk_2" :: _ ("'_mk'_2")
-
-lemmas "TYDEF_2_@intern" = typedef_hol2hollight
- [where a="a :: N_2" and r=r ,
- OF type_definition_N_2]
-
-consts
- "_10302" :: "N_2" ("'_10302")
-
-defs
- "_10302_def": "(op ==::N_2 => N_2 => prop) (_10302::N_2)
- ((_mk_2::bool recspace => N_2)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- (0::nat) ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
-
-lemma DEF__10302: "(op =::N_2 => N_2 => bool) (_10302::N_2)
- ((_mk_2::bool recspace => N_2)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- (0::nat) ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
- by (import hollight DEF__10302)
-
-consts
- "_10303" :: "N_2" ("'_10303")
-
-defs
- "_10303_def": "(op ==::N_2 => N_2 => prop) (_10303::N_2)
- ((_mk_2::bool recspace => N_2)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) (0::nat))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
-
-lemma DEF__10303: "(op =::N_2 => N_2 => bool) (_10303::N_2)
- ((_mk_2::bool recspace => N_2)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) (0::nat))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
- by (import hollight DEF__10303)
-
-definition two_1 :: "N_2" where
- "two_1 == _10302"
-
-lemma DEF_two_1: "two_1 = _10302"
- by (import hollight DEF_two_1)
-
-definition two_2 :: "N_2" where
- "two_2 == _10303"
-
-lemma DEF_two_2: "two_2 = _10303"
- by (import hollight DEF_two_2)
-
-typedef (open) N_3 = "{a::bool recspace.
- ALL u::bool recspace => bool.
- (ALL a::bool recspace.
- a = CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM) |
- a =
- CONSTR (Suc (NUMERAL 0)) (SOME x::bool. True) (%n::nat. BOTTOM) |
- a =
- CONSTR (Suc (Suc (NUMERAL 0))) (SOME x::bool. True)
- (%n::nat. BOTTOM) -->
- u a) -->
- u a}" morphisms "_dest_3" "_mk_3"
- apply (rule light_ex_imp_nonempty[where t="CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM)"])
- by (import hollight TYDEF_3)
-
-syntax
- "_dest_3" :: _ ("'_dest'_3")
-
-syntax
- "_mk_3" :: _ ("'_mk'_3")
-
-lemmas "TYDEF_3_@intern" = typedef_hol2hollight
- [where a="a :: N_3" and r=r ,
- OF type_definition_N_3]
-
-consts
- "_10326" :: "N_3" ("'_10326")
-
-defs
- "_10326_def": "(op ==::N_3 => N_3 => prop) (_10326::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- (0::nat) ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
-
-lemma DEF__10326: "(op =::N_3 => N_3 => bool) (_10326::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- (0::nat) ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
- by (import hollight DEF__10326)
-
-consts
- "_10327" :: "N_3" ("'_10327")
-
-defs
- "_10327_def": "(op ==::N_3 => N_3 => prop) (_10327::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) (0::nat))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
-
-lemma DEF__10327: "(op =::N_3 => N_3 => bool) (_10327::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) (0::nat))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
- by (import hollight DEF__10327)
-
-consts
- "_10328" :: "N_3" ("'_10328")
+ "_11937" :: "bool
+=> bool => bool => bool => bool => bool => bool => bool => hollight.char" ("'_11937")
defs
- "_10328_def": "(op ==::N_3 => N_3 => prop) (_10328::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) ((Suc::nat => nat) (0::nat)))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
-
-lemma DEF__10328: "(op =::N_3 => N_3 => bool) (_10328::N_3)
- ((_mk_3::bool recspace => N_3)
- ((CONSTR::nat => bool => (nat => bool recspace) => bool recspace)
- ((Suc::nat => nat) ((Suc::nat => nat) (0::nat)))
- ((Eps::(bool => bool) => bool) (%x::bool. True::bool))
- (%n::nat. BOTTOM::bool recspace)))"
- by (import hollight DEF__10328)
-
-definition three_1 :: "N_3" where
- "three_1 == _10326"
-
-lemma DEF_three_1: "three_1 = _10326"
- by (import hollight DEF_three_1)
-
-definition three_2 :: "N_3" where
- "three_2 == _10327"
-
-lemma DEF_three_2: "three_2 = _10327"
- by (import hollight DEF_three_2)
-
-definition three_3 :: "N_3" where
- "three_3 == _10328"
-
-lemma DEF_three_3: "three_3 = _10328"
- by (import hollight DEF_three_3)
-
-lemma list_INDUCT: "ALL P::'A::type hollight.list => bool.
- P NIL &
- (ALL (a0::'A::type) a1::'A::type hollight.list.
- P a1 --> P (CONS a0 a1)) -->
- All P"
- by (import hollight list_INDUCT)
-
-definition HD :: "'A hollight.list => 'A" where
- "HD ==
-SOME HD::'A::type hollight.list => 'A::type.
- ALL (t::'A::type hollight.list) h::'A::type. HD (CONS h t) = h"
-
-lemma DEF_HD: "HD =
-(SOME HD::'A::type hollight.list => 'A::type.
- ALL (t::'A::type hollight.list) h::'A::type. HD (CONS h t) = h)"
- by (import hollight DEF_HD)
-
-definition TL :: "'A hollight.list => 'A hollight.list" where
- "TL ==
-SOME TL::'A::type hollight.list => 'A::type hollight.list.
- ALL (h::'A::type) t::'A::type hollight.list. TL (CONS h t) = t"
-
-lemma DEF_TL: "TL =
-(SOME TL::'A::type hollight.list => 'A::type hollight.list.
- ALL (h::'A::type) t::'A::type hollight.list. TL (CONS h t) = t)"
- by (import hollight DEF_TL)
-
-definition APPEND :: "'A hollight.list => 'A hollight.list => 'A hollight.list" where
- "APPEND ==
-SOME APPEND::'A::type hollight.list
- => 'A::type hollight.list => 'A::type hollight.list.
- (ALL l::'A::type hollight.list. APPEND NIL l = l) &
- (ALL (h::'A::type) (t::'A::type hollight.list) l::'A::type hollight.list.
- APPEND (CONS h t) l = CONS h (APPEND t l))"
-
-lemma DEF_APPEND: "APPEND =
-(SOME APPEND::'A::type hollight.list
- => 'A::type hollight.list => 'A::type hollight.list.
- (ALL l::'A::type hollight.list. APPEND NIL l = l) &
- (ALL (h::'A::type) (t::'A::type hollight.list)
- l::'A::type hollight.list.
- APPEND (CONS h t) l = CONS h (APPEND t l)))"
- by (import hollight DEF_APPEND)
-
-definition REVERSE :: "'A hollight.list => 'A hollight.list" where
- "REVERSE ==
-SOME REVERSE::'A::type hollight.list => 'A::type hollight.list.
- REVERSE NIL = NIL &
- (ALL (l::'A::type hollight.list) x::'A::type.
- REVERSE (CONS x l) = APPEND (REVERSE l) (CONS x NIL))"
-
-lemma DEF_REVERSE: "REVERSE =
-(SOME REVERSE::'A::type hollight.list => 'A::type hollight.list.
- REVERSE NIL = NIL &
- (ALL (l::'A::type hollight.list) x::'A::type.
- REVERSE (CONS x l) = APPEND (REVERSE l) (CONS x NIL)))"
- by (import hollight DEF_REVERSE)
-
-definition LENGTH :: "'A hollight.list => nat" where
- "LENGTH ==
-SOME LENGTH::'A::type hollight.list => nat.
- LENGTH NIL = 0 &
- (ALL (h::'A::type) t::'A::type hollight.list.
- LENGTH (CONS h t) = Suc (LENGTH t))"
-
-lemma DEF_LENGTH: "LENGTH =
-(SOME LENGTH::'A::type hollight.list => nat.
- LENGTH NIL = 0 &
- (ALL (h::'A::type) t::'A::type hollight.list.
- LENGTH (CONS h t) = Suc (LENGTH t)))"
- by (import hollight DEF_LENGTH)
-
-definition MAP :: "('A => 'B) => 'A hollight.list => 'B hollight.list" where
- "MAP ==
-SOME MAP::('A::type => 'B::type)
- => 'A::type hollight.list => 'B::type hollight.list.
- (ALL f::'A::type => 'B::type. MAP f NIL = NIL) &
- (ALL (f::'A::type => 'B::type) (h::'A::type) t::'A::type hollight.list.
- MAP f (CONS h t) = CONS (f h) (MAP f t))"
-
-lemma DEF_MAP: "MAP =
-(SOME MAP::('A::type => 'B::type)
- => 'A::type hollight.list => 'B::type hollight.list.
- (ALL f::'A::type => 'B::type. MAP f NIL = NIL) &
- (ALL (f::'A::type => 'B::type) (h::'A::type) t::'A::type hollight.list.
- MAP f (CONS h t) = CONS (f h) (MAP f t)))"
- by (import hollight DEF_MAP)
-
-definition LAST :: "'A hollight.list => 'A" where
- "LAST ==
-SOME LAST::'A::type hollight.list => 'A::type.
- ALL (h::'A::type) t::'A::type hollight.list.
- LAST (CONS h t) = COND (t = NIL) h (LAST t)"
-
-lemma DEF_LAST: "LAST =
-(SOME LAST::'A::type hollight.list => 'A::type.
- ALL (h::'A::type) t::'A::type hollight.list.
- LAST (CONS h t) = COND (t = NIL) h (LAST t))"
- by (import hollight DEF_LAST)
-
-definition REPLICATE :: "nat => 'q_16860 => 'q_16860 hollight.list" where
- "REPLICATE ==
-SOME REPLICATE::nat => 'q_16860::type => 'q_16860::type hollight.list.
- (ALL x::'q_16860::type. REPLICATE 0 x = NIL) &
- (ALL (n::nat) x::'q_16860::type.
- REPLICATE (Suc n) x = CONS x (REPLICATE n x))"
-
-lemma DEF_REPLICATE: "REPLICATE =
-(SOME REPLICATE::nat => 'q_16860::type => 'q_16860::type hollight.list.
- (ALL x::'q_16860::type. REPLICATE 0 x = NIL) &
- (ALL (n::nat) x::'q_16860::type.
- REPLICATE (Suc n) x = CONS x (REPLICATE n x)))"
- by (import hollight DEF_REPLICATE)
-
-definition NULL :: "'q_16875 hollight.list => bool" where
- "NULL ==
-SOME NULL::'q_16875::type hollight.list => bool.
- NULL NIL = True &
- (ALL (h::'q_16875::type) t::'q_16875::type hollight.list.
- NULL (CONS h t) = False)"
-
-lemma DEF_NULL: "NULL =
-(SOME NULL::'q_16875::type hollight.list => bool.
- NULL NIL = True &
- (ALL (h::'q_16875::type) t::'q_16875::type hollight.list.
- NULL (CONS h t) = False))"
- by (import hollight DEF_NULL)
-
-definition ALL_list :: "('q_16895 => bool) => 'q_16895 hollight.list => bool" where
- "ALL_list ==
-SOME u::('q_16895::type => bool) => 'q_16895::type hollight.list => bool.
- (ALL P::'q_16895::type => bool. u P NIL = True) &
- (ALL (h::'q_16895::type) (P::'q_16895::type => bool)
- t::'q_16895::type hollight.list. u P (CONS h t) = (P h & u P t))"
-
-lemma DEF_ALL: "ALL_list =
-(SOME u::('q_16895::type => bool) => 'q_16895::type hollight.list => bool.
- (ALL P::'q_16895::type => bool. u P NIL = True) &
- (ALL (h::'q_16895::type) (P::'q_16895::type => bool)
- t::'q_16895::type hollight.list. u P (CONS h t) = (P h & u P t)))"
- by (import hollight DEF_ALL)
-
-consts
- EX :: "('q_16916 => bool) => 'q_16916 hollight.list => bool" ("EX")
-
-defs
- EX_def: "EX ==
-SOME u::('q_16916::type => bool) => 'q_16916::type hollight.list => bool.
- (ALL P::'q_16916::type => bool. u P NIL = False) &
- (ALL (h::'q_16916::type) (P::'q_16916::type => bool)
- t::'q_16916::type hollight.list. u P (CONS h t) = (P h | u P t))"
-
-lemma DEF_EX: "EX =
-(SOME u::('q_16916::type => bool) => 'q_16916::type hollight.list => bool.
- (ALL P::'q_16916::type => bool. u P NIL = False) &
- (ALL (h::'q_16916::type) (P::'q_16916::type => bool)
- t::'q_16916::type hollight.list. u P (CONS h t) = (P h | u P t)))"
- by (import hollight DEF_EX)
-
-definition ITLIST :: "('q_16939 => 'q_16938 => 'q_16938)
-=> 'q_16939 hollight.list => 'q_16938 => 'q_16938" where
- "ITLIST ==
-SOME ITLIST::('q_16939::type => 'q_16938::type => 'q_16938::type)
- => 'q_16939::type hollight.list
- => 'q_16938::type => 'q_16938::type.
- (ALL (f::'q_16939::type => 'q_16938::type => 'q_16938::type)
- b::'q_16938::type. ITLIST f NIL b = b) &
- (ALL (h::'q_16939::type)
- (f::'q_16939::type => 'q_16938::type => 'q_16938::type)
- (t::'q_16939::type hollight.list) b::'q_16938::type.
- ITLIST f (CONS h t) b = f h (ITLIST f t b))"
-
-lemma DEF_ITLIST: "ITLIST =
-(SOME ITLIST::('q_16939::type => 'q_16938::type => 'q_16938::type)
- => 'q_16939::type hollight.list
- => 'q_16938::type => 'q_16938::type.
- (ALL (f::'q_16939::type => 'q_16938::type => 'q_16938::type)
- b::'q_16938::type. ITLIST f NIL b = b) &
- (ALL (h::'q_16939::type)
- (f::'q_16939::type => 'q_16938::type => 'q_16938::type)
- (t::'q_16939::type hollight.list) b::'q_16938::type.
- ITLIST f (CONS h t) b = f h (ITLIST f t b)))"
- by (import hollight DEF_ITLIST)
-
-definition MEM :: "'q_16964 => 'q_16964 hollight.list => bool" where
- "MEM ==
-SOME MEM::'q_16964::type => 'q_16964::type hollight.list => bool.
- (ALL x::'q_16964::type. MEM x NIL = False) &
- (ALL (h::'q_16964::type) (x::'q_16964::type)
- t::'q_16964::type hollight.list.
- MEM x (CONS h t) = (x = h | MEM x t))"
-
-lemma DEF_MEM: "MEM =
-(SOME MEM::'q_16964::type => 'q_16964::type hollight.list => bool.
- (ALL x::'q_16964::type. MEM x NIL = False) &
- (ALL (h::'q_16964::type) (x::'q_16964::type)
- t::'q_16964::type hollight.list.
- MEM x (CONS h t) = (x = h | MEM x t)))"
- by (import hollight DEF_MEM)
-
-definition ALL2 :: "('q_16997 => 'q_17004 => bool)
-=> 'q_16997 hollight.list => 'q_17004 hollight.list => bool" where
- "ALL2 ==
-SOME ALL2::('q_16997::type => 'q_17004::type => bool)
- => 'q_16997::type hollight.list
- => 'q_17004::type hollight.list => bool.
- (ALL (P::'q_16997::type => 'q_17004::type => bool)
- l2::'q_17004::type hollight.list. ALL2 P NIL l2 = (l2 = NIL)) &
- (ALL (h1::'q_16997::type) (P::'q_16997::type => 'q_17004::type => bool)
- (t1::'q_16997::type hollight.list) l2::'q_17004::type hollight.list.
- ALL2 P (CONS h1 t1) l2 =
- COND (l2 = NIL) False (P h1 (HD l2) & ALL2 P t1 (TL l2)))"
-
-lemma DEF_ALL2: "ALL2 =
-(SOME ALL2::('q_16997::type => 'q_17004::type => bool)
- => 'q_16997::type hollight.list
- => 'q_17004::type hollight.list => bool.
- (ALL (P::'q_16997::type => 'q_17004::type => bool)
- l2::'q_17004::type hollight.list. ALL2 P NIL l2 = (l2 = NIL)) &
- (ALL (h1::'q_16997::type) (P::'q_16997::type => 'q_17004::type => bool)
- (t1::'q_16997::type hollight.list) l2::'q_17004::type hollight.list.
- ALL2 P (CONS h1 t1) l2 =
- COND (l2 = NIL) False (P h1 (HD l2) & ALL2 P t1 (TL l2))))"
- by (import hollight DEF_ALL2)
-
-lemma ALL2: "ALL2 (P::'q_17059::type => 'q_17058::type => bool) NIL NIL = True &
-ALL2 P (CONS (h1::'q_17059::type) (t1::'q_17059::type hollight.list)) NIL =
-False &
-ALL2 P NIL (CONS (h2::'q_17058::type) (t2::'q_17058::type hollight.list)) =
-False &
-ALL2 P (CONS h1 t1) (CONS h2 t2) = (P h1 h2 & ALL2 P t1 t2)"
- by (import hollight ALL2)
-
-definition MAP2 :: "('q_17089 => 'q_17096 => 'q_17086)
-=> 'q_17089 hollight.list
- => 'q_17096 hollight.list => 'q_17086 hollight.list" where
- "MAP2 ==
-SOME MAP2::('q_17089::type => 'q_17096::type => 'q_17086::type)
- => 'q_17089::type hollight.list
- => 'q_17096::type hollight.list
- => 'q_17086::type hollight.list.
- (ALL (f::'q_17089::type => 'q_17096::type => 'q_17086::type)
- l::'q_17096::type hollight.list. MAP2 f NIL l = NIL) &
- (ALL (h1::'q_17089::type)
- (f::'q_17089::type => 'q_17096::type => 'q_17086::type)
- (t1::'q_17089::type hollight.list) l::'q_17096::type hollight.list.
- MAP2 f (CONS h1 t1) l = CONS (f h1 (HD l)) (MAP2 f t1 (TL l)))"
-
-lemma DEF_MAP2: "MAP2 =
-(SOME MAP2::('q_17089::type => 'q_17096::type => 'q_17086::type)
- => 'q_17089::type hollight.list
- => 'q_17096::type hollight.list
- => 'q_17086::type hollight.list.
- (ALL (f::'q_17089::type => 'q_17096::type => 'q_17086::type)
- l::'q_17096::type hollight.list. MAP2 f NIL l = NIL) &
- (ALL (h1::'q_17089::type)
- (f::'q_17089::type => 'q_17096::type => 'q_17086::type)
- (t1::'q_17089::type hollight.list) l::'q_17096::type hollight.list.
- MAP2 f (CONS h1 t1) l = CONS (f h1 (HD l)) (MAP2 f t1 (TL l))))"
- by (import hollight DEF_MAP2)
-
-lemma MAP2: "MAP2 (f::'q_17131::type => 'q_17130::type => 'q_17137::type) NIL NIL = NIL &
-MAP2 f (CONS (h1::'q_17131::type) (t1::'q_17131::type hollight.list))
- (CONS (h2::'q_17130::type) (t2::'q_17130::type hollight.list)) =
-CONS (f h1 h2) (MAP2 f t1 t2)"
- by (import hollight MAP2)
-
-definition EL :: "nat => 'q_17157 hollight.list => 'q_17157" where
- "EL ==
-SOME EL::nat => 'q_17157::type hollight.list => 'q_17157::type.
- (ALL l::'q_17157::type hollight.list. EL 0 l = HD l) &
- (ALL (n::nat) l::'q_17157::type hollight.list.
- EL (Suc n) l = EL n (TL l))"
-
-lemma DEF_EL: "EL =
-(SOME EL::nat => 'q_17157::type hollight.list => 'q_17157::type.
- (ALL l::'q_17157::type hollight.list. EL 0 l = HD l) &
- (ALL (n::nat) l::'q_17157::type hollight.list.
- EL (Suc n) l = EL n (TL l)))"
- by (import hollight DEF_EL)
-
-definition FILTER :: "('q_17182 => bool) => 'q_17182 hollight.list => 'q_17182 hollight.list" where
- "FILTER ==
-SOME FILTER::('q_17182::type => bool)
- => 'q_17182::type hollight.list
- => 'q_17182::type hollight.list.
- (ALL P::'q_17182::type => bool. FILTER P NIL = NIL) &
- (ALL (h::'q_17182::type) (P::'q_17182::type => bool)
- t::'q_17182::type hollight.list.
- FILTER P (CONS h t) = COND (P h) (CONS h (FILTER P t)) (FILTER P t))"
-
-lemma DEF_FILTER: "FILTER =
-(SOME FILTER::('q_17182::type => bool)
- => 'q_17182::type hollight.list
- => 'q_17182::type hollight.list.
- (ALL P::'q_17182::type => bool. FILTER P NIL = NIL) &
- (ALL (h::'q_17182::type) (P::'q_17182::type => bool)
- t::'q_17182::type hollight.list.
- FILTER P (CONS h t) =
- COND (P h) (CONS h (FILTER P t)) (FILTER P t)))"
- by (import hollight DEF_FILTER)
-
-definition ASSOC :: "'q_17211 => ('q_17211 * 'q_17205) hollight.list => 'q_17205" where
- "ASSOC ==
-SOME ASSOC::'q_17211::type
- => ('q_17211::type * 'q_17205::type) hollight.list
- => 'q_17205::type.
- ALL (h::'q_17211::type * 'q_17205::type) (a::'q_17211::type)
- t::('q_17211::type * 'q_17205::type) hollight.list.
- ASSOC a (CONS h t) = COND (fst h = a) (snd h) (ASSOC a t)"
-
-lemma DEF_ASSOC: "ASSOC =
-(SOME ASSOC::'q_17211::type
- => ('q_17211::type * 'q_17205::type) hollight.list
- => 'q_17205::type.
- ALL (h::'q_17211::type * 'q_17205::type) (a::'q_17211::type)
- t::('q_17211::type * 'q_17205::type) hollight.list.
- ASSOC a (CONS h t) = COND (fst h = a) (snd h) (ASSOC a t))"
- by (import hollight DEF_ASSOC)
-
-definition ITLIST2 :: "('q_17235 => 'q_17243 => 'q_17233 => 'q_17233)
-=> 'q_17235 hollight.list => 'q_17243 hollight.list => 'q_17233 => 'q_17233" where
- "ITLIST2 ==
-SOME ITLIST2::('q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- => 'q_17235::type hollight.list
- => 'q_17243::type hollight.list
- => 'q_17233::type => 'q_17233::type.
- (ALL (f::'q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- (l2::'q_17243::type hollight.list) b::'q_17233::type.
- ITLIST2 f NIL l2 b = b) &
- (ALL (h1::'q_17235::type)
- (f::'q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- (t1::'q_17235::type hollight.list) (l2::'q_17243::type hollight.list)
- b::'q_17233::type.
- ITLIST2 f (CONS h1 t1) l2 b = f h1 (HD l2) (ITLIST2 f t1 (TL l2) b))"
-
-lemma DEF_ITLIST2: "ITLIST2 =
-(SOME ITLIST2::('q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- => 'q_17235::type hollight.list
- => 'q_17243::type hollight.list
- => 'q_17233::type => 'q_17233::type.
- (ALL (f::'q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- (l2::'q_17243::type hollight.list) b::'q_17233::type.
- ITLIST2 f NIL l2 b = b) &
- (ALL (h1::'q_17235::type)
- (f::'q_17235::type
- => 'q_17243::type => 'q_17233::type => 'q_17233::type)
- (t1::'q_17235::type hollight.list)
- (l2::'q_17243::type hollight.list) b::'q_17233::type.
- ITLIST2 f (CONS h1 t1) l2 b =
- f h1 (HD l2) (ITLIST2 f t1 (TL l2) b)))"
- by (import hollight DEF_ITLIST2)
-
-lemma ITLIST2: "ITLIST2
- (f::'q_17277::type => 'q_17276::type => 'q_17275::type => 'q_17275::type)
- NIL NIL (b::'q_17275::type) =
-b &
-ITLIST2 f (CONS (h1::'q_17277::type) (t1::'q_17277::type hollight.list))
- (CONS (h2::'q_17276::type) (t2::'q_17276::type hollight.list)) b =
-f h1 h2 (ITLIST2 f t1 t2 b)"
- by (import hollight ITLIST2)
+ "_11937_def": "_11937 ==
+%(a0::bool) (a1::bool) (a2::bool) (a3::bool) (a4::bool) (a5::bool)
+ (a6::bool) a7::bool.
+ _mk_char
+ (CONSTR (0::nat) (a0, a1, a2, a3, a4, a5, a6, a7) (%n::nat. BOTTOM))"
+
+lemma DEF__11937: "_11937 =
+(%(a0::bool) (a1::bool) (a2::bool) (a3::bool) (a4::bool) (a5::bool)
+ (a6::bool) a7::bool.
+ _mk_char
+ (CONSTR (0::nat) (a0, a1, a2, a3, a4, a5, a6, a7) (%n::nat. BOTTOM)))"
+ by (import hollight DEF__11937)
+
+definition
+ ASCII :: "bool
+=> bool => bool => bool => bool => bool => bool => bool => hollight.char" where
+ "ASCII == _11937"
+
+lemma DEF_ASCII: "ASCII = _11937"
+ by (import hollight DEF_ASCII)
consts
- ZIP :: "'q_17307 hollight.list
-=> 'q_17315 hollight.list => ('q_17307 * 'q_17315) hollight.list"
+ dist :: "nat * nat => nat"
defs
- ZIP_def: "hollight.ZIP ==
-SOME ZIP::'q_17307::type hollight.list
- => 'q_17315::type hollight.list
- => ('q_17307::type * 'q_17315::type) hollight.list.
- (ALL l2::'q_17315::type hollight.list. ZIP NIL l2 = NIL) &
- (ALL (h1::'q_17307::type) (t1::'q_17307::type hollight.list)
- l2::'q_17315::type hollight.list.
- ZIP (CONS h1 t1) l2 = CONS (h1, HD l2) (ZIP t1 (TL l2)))"
-
-lemma DEF_ZIP: "hollight.ZIP =
-(SOME ZIP::'q_17307::type hollight.list
- => 'q_17315::type hollight.list
- => ('q_17307::type * 'q_17315::type) hollight.list.
- (ALL l2::'q_17315::type hollight.list. ZIP NIL l2 = NIL) &
- (ALL (h1::'q_17307::type) (t1::'q_17307::type hollight.list)
- l2::'q_17315::type hollight.list.
- ZIP (CONS h1 t1) l2 = CONS (h1, HD l2) (ZIP t1 (TL l2))))"
- by (import hollight DEF_ZIP)
-
-lemma ZIP: "(op &::bool => bool => bool)
- ((op =::('q_17326::type * 'q_17327::type) hollight.list
- => ('q_17326::type * 'q_17327::type) hollight.list => bool)
- ((hollight.ZIP::'q_17326::type hollight.list
- => 'q_17327::type hollight.list
- => ('q_17326::type * 'q_17327::type) hollight.list)
- (NIL::'q_17326::type hollight.list)
- (NIL::'q_17327::type hollight.list))
- (NIL::('q_17326::type * 'q_17327::type) hollight.list))
- ((op =::('q_17351::type * 'q_17352::type) hollight.list
- => ('q_17351::type * 'q_17352::type) hollight.list => bool)
- ((hollight.ZIP::'q_17351::type hollight.list
- => 'q_17352::type hollight.list
- => ('q_17351::type * 'q_17352::type) hollight.list)
- ((CONS::'q_17351::type
- => 'q_17351::type hollight.list
- => 'q_17351::type hollight.list)
- (h1::'q_17351::type) (t1::'q_17351::type hollight.list))
- ((CONS::'q_17352::type
- => 'q_17352::type hollight.list
- => 'q_17352::type hollight.list)
- (h2::'q_17352::type) (t2::'q_17352::type hollight.list)))
- ((CONS::'q_17351::type * 'q_17352::type
- => ('q_17351::type * 'q_17352::type) hollight.list
- => ('q_17351::type * 'q_17352::type) hollight.list)
- ((Pair::'q_17351::type
- => 'q_17352::type => 'q_17351::type * 'q_17352::type)
- h1 h2)
- ((hollight.ZIP::'q_17351::type hollight.list
- => 'q_17352::type hollight.list
- => ('q_17351::type * 'q_17352::type) hollight.list)
- t1 t2)))"
- by (import hollight ZIP)
-
-lemma NOT_CONS_NIL: "ALL (x::'A::type) xa::'A::type hollight.list. CONS x xa ~= NIL"
- by (import hollight NOT_CONS_NIL)
-
-lemma LAST_CLAUSES: "LAST (CONS (h::'A::type) NIL) = h &
-LAST (CONS h (CONS (k::'A::type) (t::'A::type hollight.list))) =
-LAST (CONS k t)"
- by (import hollight LAST_CLAUSES)
-
-lemma APPEND_NIL: "ALL l::'A::type hollight.list. APPEND l NIL = l"
- by (import hollight APPEND_NIL)
-
-lemma APPEND_ASSOC: "ALL (l::'A::type hollight.list) (m::'A::type hollight.list)
- n::'A::type hollight.list. APPEND l (APPEND m n) = APPEND (APPEND l m) n"
- by (import hollight APPEND_ASSOC)
-
-lemma REVERSE_APPEND: "ALL (l::'A::type hollight.list) m::'A::type hollight.list.
- REVERSE (APPEND l m) = APPEND (REVERSE m) (REVERSE l)"
- by (import hollight REVERSE_APPEND)
-
-lemma REVERSE_REVERSE: "ALL l::'A::type hollight.list. REVERSE (REVERSE l) = l"
- by (import hollight REVERSE_REVERSE)
-
-lemma CONS_11: "ALL (x::'A::type) (xa::'A::type) (xb::'A::type hollight.list)
- xc::'A::type hollight.list. (CONS x xb = CONS xa xc) = (x = xa & xb = xc)"
- by (import hollight CONS_11)
-
-lemma list_CASES: "ALL l::'A::type hollight.list.
- l = NIL | (EX (h::'A::type) t::'A::type hollight.list. l = CONS h t)"
- by (import hollight list_CASES)
-
-lemma LENGTH_APPEND: "ALL (l::'A::type hollight.list) m::'A::type hollight.list.
- LENGTH (APPEND l m) = LENGTH l + LENGTH m"
- by (import hollight LENGTH_APPEND)
-
-lemma MAP_APPEND: "ALL (f::'A::type => 'B::type) (l1::'A::type hollight.list)
- l2::'A::type hollight.list.
- MAP f (APPEND l1 l2) = APPEND (MAP f l1) (MAP f l2)"
- by (import hollight MAP_APPEND)
-
-lemma LENGTH_MAP: "ALL (l::'A::type hollight.list) f::'A::type => 'B::type.
- LENGTH (MAP f l) = LENGTH l"
- by (import hollight LENGTH_MAP)
-
-lemma LENGTH_EQ_NIL: "ALL l::'A::type hollight.list. (LENGTH l = 0) = (l = NIL)"
- by (import hollight LENGTH_EQ_NIL)
-
-lemma LENGTH_EQ_CONS: "ALL (l::'q_17659::type hollight.list) n::nat.
- (LENGTH l = Suc n) =
- (EX (h::'q_17659::type) t::'q_17659::type hollight.list.
- l = CONS h t & LENGTH t = n)"
- by (import hollight LENGTH_EQ_CONS)
-
-lemma MAP_o: "ALL (f::'A::type => 'B::type) (g::'B::type => 'C::type)
- l::'A::type hollight.list. MAP (g o f) l = MAP g (MAP f l)"
- by (import hollight MAP_o)
-
-lemma MAP_EQ: "ALL (f::'q_17723::type => 'q_17734::type)
- (g::'q_17723::type => 'q_17734::type) l::'q_17723::type hollight.list.
- ALL_list (%x::'q_17723::type. f x = g x) l --> MAP f l = MAP g l"
- by (import hollight MAP_EQ)
-
-lemma ALL_IMP: "ALL (P::'q_17764::type => bool) (Q::'q_17764::type => bool)
- l::'q_17764::type hollight.list.
- (ALL x::'q_17764::type. MEM x l & P x --> Q x) & ALL_list P l -->
- ALL_list Q l"
- by (import hollight ALL_IMP)
-
-lemma NOT_EX: "ALL (P::'q_17792::type => bool) l::'q_17792::type hollight.list.
- (~ EX P l) = ALL_list (%x::'q_17792::type. ~ P x) l"
- by (import hollight NOT_EX)
-
-lemma NOT_ALL: "ALL (P::'q_17814::type => bool) l::'q_17814::type hollight.list.
- (~ ALL_list P l) = EX (%x::'q_17814::type. ~ P x) l"
- by (import hollight NOT_ALL)
-
-lemma ALL_MAP: "ALL (P::'q_17836::type => bool) (f::'q_17835::type => 'q_17836::type)
- l::'q_17835::type hollight.list.
- ALL_list P (MAP f l) = ALL_list (P o f) l"
- by (import hollight ALL_MAP)
-
-lemma ALL_T: "All (ALL_list (%x::'q_17854::type. True))"
- by (import hollight ALL_T)
-
-lemma MAP_EQ_ALL2: "ALL (l::'q_17879::type hollight.list) m::'q_17879::type hollight.list.
- ALL2
- (%(x::'q_17879::type) y::'q_17879::type.
- (f::'q_17879::type => 'q_17890::type) x = f y)
- l m -->
- MAP f l = MAP f m"
- by (import hollight MAP_EQ_ALL2)
-
-lemma ALL2_MAP: "ALL (P::'q_17921::type => 'q_17922::type => bool)
- (f::'q_17922::type => 'q_17921::type) l::'q_17922::type hollight.list.
- ALL2 P (MAP f l) l = ALL_list (%a::'q_17922::type. P (f a) a) l"
- by (import hollight ALL2_MAP)
-
-lemma MAP_EQ_DEGEN: "ALL (l::'q_17939::type hollight.list) f::'q_17939::type => 'q_17939::type.
- ALL_list (%x::'q_17939::type. f x = x) l --> MAP f l = l"
- by (import hollight MAP_EQ_DEGEN)
-
-lemma ALL2_AND_RIGHT: "ALL (l::'q_17982::type hollight.list) (m::'q_17981::type hollight.list)
- (P::'q_17982::type => bool) Q::'q_17982::type => 'q_17981::type => bool.
- ALL2 (%(x::'q_17982::type) y::'q_17981::type. P x & Q x y) l m =
- (ALL_list P l & ALL2 Q l m)"
- by (import hollight ALL2_AND_RIGHT)
-
-lemma ITLIST_EXTRA: "ALL l::'q_18019::type hollight.list.
- ITLIST (f::'q_18019::type => 'q_18018::type => 'q_18018::type)
- (APPEND l (CONS (a::'q_18019::type) NIL)) (b::'q_18018::type) =
- ITLIST f l (f a b)"
- by (import hollight ITLIST_EXTRA)
-
-lemma ALL_MP: "ALL (P::'q_18045::type => bool) (Q::'q_18045::type => bool)
- l::'q_18045::type hollight.list.
- ALL_list (%x::'q_18045::type. P x --> Q x) l & ALL_list P l -->
- ALL_list Q l"
- by (import hollight ALL_MP)
-
-lemma AND_ALL: "ALL x::'q_18075::type hollight.list.
- (ALL_list (P::'q_18075::type => bool) x &
- ALL_list (Q::'q_18075::type => bool) x) =
- ALL_list (%x::'q_18075::type. P x & Q x) x"
- by (import hollight AND_ALL)
-
-lemma EX_IMP: "ALL (P::'q_18105::type => bool) (Q::'q_18105::type => bool)
- l::'q_18105::type hollight.list.
- (ALL x::'q_18105::type. MEM x l & P x --> Q x) & EX P l --> EX Q l"
- by (import hollight EX_IMP)
-
-lemma ALL_MEM: "ALL (P::'q_18132::type => bool) l::'q_18132::type hollight.list.
- (ALL x::'q_18132::type. MEM x l --> P x) = ALL_list P l"
- by (import hollight ALL_MEM)
-
-lemma LENGTH_REPLICATE: "ALL (n::nat) x::'q_18150::type. LENGTH (REPLICATE n x) = n"
- by (import hollight LENGTH_REPLICATE)
-
-lemma EX_MAP: "ALL (P::'q_18174::type => bool) (f::'q_18173::type => 'q_18174::type)
- l::'q_18173::type hollight.list. EX P (MAP f l) = EX (P o f) l"
- by (import hollight EX_MAP)
-
-lemma EXISTS_EX: "ALL (P::'q_18212::type => 'q_18211::type => bool)
- l::'q_18211::type hollight.list.
- (EX x::'q_18212::type. EX (P x) l) =
- EX (%s::'q_18211::type. EX x::'q_18212::type. P x s) l"
- by (import hollight EXISTS_EX)
-
-lemma FORALL_ALL: "ALL (P::'q_18242::type => 'q_18241::type => bool)
- l::'q_18241::type hollight.list.
- (ALL x::'q_18242::type. ALL_list (P x) l) =
- ALL_list (%s::'q_18241::type. ALL x::'q_18242::type. P x s) l"
- by (import hollight FORALL_ALL)
-
-lemma MEM_APPEND: "ALL (x::'q_18270::type) (l1::'q_18270::type hollight.list)
- l2::'q_18270::type hollight.list.
- MEM x (APPEND l1 l2) = (MEM x l1 | MEM x l2)"
- by (import hollight MEM_APPEND)
-
-lemma MEM_MAP: "ALL (f::'q_18306::type => 'q_18303::type) (y::'q_18303::type)
- l::'q_18306::type hollight.list.
- MEM y (MAP f l) = (EX x::'q_18306::type. MEM x l & y = f x)"
- by (import hollight MEM_MAP)
-
-lemma FILTER_APPEND: "ALL (P::'q_18337::type => bool) (l1::'q_18337::type hollight.list)
- l2::'q_18337::type hollight.list.
- FILTER P (APPEND l1 l2) = APPEND (FILTER P l1) (FILTER P l2)"
- by (import hollight FILTER_APPEND)
-
-lemma FILTER_MAP: "ALL (P::'q_18364::type => bool) (f::'q_18371::type => 'q_18364::type)
- l::'q_18371::type hollight.list.
- FILTER P (MAP f l) = MAP f (FILTER (P o f) l)"
- by (import hollight FILTER_MAP)
-
-lemma MEM_FILTER: "ALL (P::'q_18398::type => bool) (l::'q_18398::type hollight.list)
- x::'q_18398::type. MEM x (FILTER P l) = (P x & MEM x l)"
- by (import hollight MEM_FILTER)
-
-lemma EX_MEM: "ALL (P::'q_18419::type => bool) l::'q_18419::type hollight.list.
- (EX x::'q_18419::type. P x & MEM x l) = EX P l"
- by (import hollight EX_MEM)
-
-lemma MAP_FST_ZIP: "ALL (l1::'q_18439::type hollight.list) l2::'q_18441::type hollight.list.
- LENGTH l1 = LENGTH l2 --> MAP fst (hollight.ZIP l1 l2) = l1"
- by (import hollight MAP_FST_ZIP)
-
-lemma MAP_SND_ZIP: "ALL (l1::'q_18470::type hollight.list) l2::'q_18472::type hollight.list.
- LENGTH l1 = LENGTH l2 --> MAP snd (hollight.ZIP l1 l2) = l2"
- by (import hollight MAP_SND_ZIP)
-
-lemma MEM_ASSOC: "ALL (l::('q_18516::type * 'q_18500::type) hollight.list) x::'q_18516::type.
- MEM (x, ASSOC x l) l = MEM x (MAP fst l)"
- by (import hollight MEM_ASSOC)
-
-lemma ALL_APPEND: "ALL (P::'q_18537::type => bool) (l1::'q_18537::type hollight.list)
- l2::'q_18537::type hollight.list.
- ALL_list P (APPEND l1 l2) = (ALL_list P l1 & ALL_list P l2)"
- by (import hollight ALL_APPEND)
-
-lemma MEM_EL: "ALL (l::'q_18560::type hollight.list) n::nat.
- < n (LENGTH l) --> MEM (EL n l) l"
- by (import hollight MEM_EL)
-
-lemma ALL2_MAP2: "ALL (l::'q_18603::type hollight.list) m::'q_18604::type hollight.list.
- ALL2 (P::'q_18602::type => 'q_18601::type => bool)
- (MAP (f::'q_18603::type => 'q_18602::type) l)
- (MAP (g::'q_18604::type => 'q_18601::type) m) =
- ALL2 (%(x::'q_18603::type) y::'q_18604::type. P (f x) (g y)) l m"
- by (import hollight ALL2_MAP2)
-
-lemma AND_ALL2: "ALL (P::'q_18650::type => 'q_18649::type => bool)
- (Q::'q_18650::type => 'q_18649::type => bool)
- (x::'q_18650::type hollight.list) xa::'q_18649::type hollight.list.
- (ALL2 P x xa & ALL2 Q x xa) =
- ALL2 (%(x::'q_18650::type) y::'q_18649::type. P x y & Q x y) x xa"
- by (import hollight AND_ALL2)
-
-lemma ALL2_ALL: "ALL (P::'q_18672::type => 'q_18672::type => bool)
- l::'q_18672::type hollight.list.
- ALL2 P l l = ALL_list (%x::'q_18672::type. P x x) l"
- by (import hollight ALL2_ALL)
-
-lemma APPEND_EQ_NIL: "ALL (x::'q_18701::type hollight.list) xa::'q_18701::type hollight.list.
- (APPEND x xa = NIL) = (x = NIL & xa = NIL)"
- by (import hollight APPEND_EQ_NIL)
-
-lemma LENGTH_MAP2: "ALL (f::'q_18721::type => 'q_18723::type => 'q_18734::type)
- (l::'q_18721::type hollight.list) m::'q_18723::type hollight.list.
- LENGTH l = LENGTH m --> LENGTH (MAP2 f l m) = LENGTH m"
- by (import hollight LENGTH_MAP2)
-
-lemma MONO_ALL: "(ALL x::'A::type. (P::'A::type => bool) x --> (Q::'A::type => bool) x) -->
-ALL_list P (l::'A::type hollight.list) --> ALL_list Q l"
- by (import hollight MONO_ALL)
-
-lemma MONO_ALL2: "(ALL (x::'A::type) y::'B::type.
- (P::'A::type => 'B::type => bool) x y -->
- (Q::'A::type => 'B::type => bool) x y) -->
-ALL2 P (l::'A::type hollight.list) (l'::'B::type hollight.list) -->
-ALL2 Q l l'"
- by (import hollight MONO_ALL2)
-
-definition dist :: "nat * nat => nat" where
- "dist == %u::nat * nat. fst u - snd u + (snd u - fst u)"
-
-lemma DEF_dist: "dist = (%u::nat * nat. fst u - snd u + (snd u - fst u))"
+ dist_def: "hollight.dist == %u. fst u - snd u + (snd u - fst u)"
+
+lemma DEF_dist: "hollight.dist = (%u. fst u - snd u + (snd u - fst u))"
by (import hollight DEF_dist)
-lemma DIST_REFL: "ALL x::nat. dist (x, x) = 0"
+lemma DIST_REFL: "hollight.dist (x, x) = 0"
by (import hollight DIST_REFL)
-lemma DIST_LZERO: "ALL x::nat. dist (0, x) = x"
+lemma DIST_LZERO: "hollight.dist (0, x) = x"
by (import hollight DIST_LZERO)
-lemma DIST_RZERO: "ALL x::nat. dist (x, 0) = x"
+lemma DIST_RZERO: "hollight.dist (x, 0) = x"
by (import hollight DIST_RZERO)
-lemma DIST_SYM: "ALL (x::nat) xa::nat. dist (x, xa) = dist (xa, x)"
+lemma DIST_SYM: "hollight.dist (x, xa) = hollight.dist (xa, x)"
by (import hollight DIST_SYM)
-lemma DIST_LADD: "ALL (x::nat) (xa::nat) xb::nat. dist (x + xb, x + xa) = dist (xb, xa)"
+lemma DIST_LADD: "hollight.dist (x + xb, x + xa) = hollight.dist (xb, xa)"
by (import hollight DIST_LADD)
-lemma DIST_RADD: "ALL (x::nat) (xa::nat) xb::nat. dist (x + xa, xb + xa) = dist (x, xb)"
+lemma DIST_RADD: "hollight.dist (x + xa, xb + xa) = hollight.dist (x, xb)"
by (import hollight DIST_RADD)
-lemma DIST_LADD_0: "ALL (x::nat) xa::nat. dist (x + xa, x) = xa"
+lemma DIST_LADD_0: "hollight.dist (x + xa, x) = xa"
by (import hollight DIST_LADD_0)
-lemma DIST_RADD_0: "ALL (x::nat) xa::nat. dist (x, x + xa) = xa"
+lemma DIST_RADD_0: "hollight.dist (x, x + xa) = xa"
by (import hollight DIST_RADD_0)
-lemma DIST_LMUL: "ALL (x::nat) (xa::nat) xb::nat. x * dist (xa, xb) = dist (x * xa, x * xb)"
+lemma DIST_LMUL: "x * hollight.dist (xa, xb) = hollight.dist (x * xa, x * xb)"
by (import hollight DIST_LMUL)
-lemma DIST_RMUL: "ALL (x::nat) (xa::nat) xb::nat. dist (x, xa) * xb = dist (x * xb, xa * xb)"
+lemma DIST_RMUL: "hollight.dist (x, xa) * xb = hollight.dist (x * xb, xa * xb)"
by (import hollight DIST_RMUL)
-lemma DIST_EQ_0: "ALL (x::nat) xa::nat. (dist (x, xa) = 0) = (x = xa)"
+lemma DIST_EQ_0: "(hollight.dist (x, xa) = 0) = (x = xa)"
by (import hollight DIST_EQ_0)
-lemma DIST_ELIM_THM: "(P::nat => bool) (dist (x::nat, y::nat)) =
-(ALL d::nat. (x = y + d --> P d) & (y = x + d --> P d))"
+lemma DIST_ELIM_THM: "P (hollight.dist (x, y)) =
+(ALL d. (x = y + d --> P d) & (y = x + d --> P d))"
by (import hollight DIST_ELIM_THM)
-lemma DIST_LE_CASES: "ALL (m::nat) (n::nat) p::nat.
- <= (dist (m, n)) p = (<= m (n + p) & <= n (m + p))"
+lemma DIST_LE_CASES: "(hollight.dist (m, n) <= p) = (m <= n + p & n <= m + p)"
by (import hollight DIST_LE_CASES)
-lemma DIST_ADDBOUND: "ALL (m::nat) n::nat. <= (dist (m, n)) (m + n)"
- by (import hollight DIST_ADDBOUND)
-
-lemma DIST_TRIANGLE: "ALL (m::nat) (n::nat) p::nat. <= (dist (m, p)) (dist (m, n) + dist (n, p))"
- by (import hollight DIST_TRIANGLE)
-
-lemma DIST_ADD2: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- <= (dist (m + n, p + q)) (dist (m, p) + dist (n, q))"
- by (import hollight DIST_ADD2)
-
-lemma DIST_ADD2_REV: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- <= (dist (m, p)) (dist (m + n, p + q) + dist (n, q))"
- by (import hollight DIST_ADD2_REV)
-
-lemma DIST_TRIANGLE_LE: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- <= (dist (m, n) + dist (n, p)) q --> <= (dist (m, p)) q"
+lemma DIST_TRIANGLE_LE: "hollight.dist (m, n) + hollight.dist (n, p) <= q
+==> hollight.dist (m, p) <= q"
by (import hollight DIST_TRIANGLE_LE)
-lemma DIST_TRIANGLES_LE: "ALL (m::nat) (n::nat) (p::nat) (q::nat) (r::nat) s::nat.
- <= (dist (m, n)) r & <= (dist (p, q)) s -->
- <= (dist (m, p)) (dist (n, q) + (r + s))"
+lemma DIST_TRIANGLES_LE: "hollight.dist (m, n) <= r & hollight.dist (p, q) <= s
+==> hollight.dist (m, p) <= hollight.dist (n, q) + (r + s)"
by (import hollight DIST_TRIANGLES_LE)
-lemma BOUNDS_LINEAR: "ALL (A::nat) (B::nat) C::nat. (ALL n::nat. <= (A * n) (B * n + C)) = <= A B"
+lemma BOUNDS_LINEAR: "(ALL n::nat. (A::nat) * n <= (B::nat) * n + (C::nat)) = (A <= B)"
by (import hollight BOUNDS_LINEAR)
-lemma BOUNDS_LINEAR_0: "ALL (A::nat) B::nat. (ALL n::nat. <= (A * n) B) = (A = 0)"
+lemma BOUNDS_LINEAR_0: "(ALL n::nat. (A::nat) * n <= (B::nat)) = (A = (0::nat))"
by (import hollight BOUNDS_LINEAR_0)
-lemma BOUNDS_DIVIDED: "ALL P::nat => nat.
- (EX B::nat. ALL n::nat. <= (P n) B) =
- (EX (x::nat) B::nat. ALL n::nat. <= (n * P n) (x * n + B))"
+lemma BOUNDS_DIVIDED: "(EX B::nat. ALL n::nat. (P::nat => nat) n <= B) =
+(EX (x::nat) B::nat. ALL n::nat. n * P n <= x * n + B)"
by (import hollight BOUNDS_DIVIDED)
-lemma BOUNDS_NOTZERO: "ALL (P::nat => nat => nat) (A::nat) B::nat.
- P 0 0 = 0 & (ALL (m::nat) n::nat. <= (P m n) (A * (m + n) + B)) -->
- (EX x::nat. ALL (m::nat) n::nat. <= (P m n) (x * (m + n)))"
+lemma BOUNDS_NOTZERO: "(P::nat => nat => nat) (0::nat) (0::nat) = (0::nat) &
+(ALL (m::nat) n::nat. P m n <= (A::nat) * (m + n) + (B::nat))
+==> EX x::nat. ALL (m::nat) n::nat. P m n <= x * (m + n)"
by (import hollight BOUNDS_NOTZERO)
-lemma BOUNDS_IGNORE: "ALL (P::nat => nat) Q::nat => nat.
- (EX B::nat. ALL i::nat. <= (P i) (Q i + B)) =
- (EX (x::nat) N::nat. ALL i::nat. <= N i --> <= (P i) (Q i + x))"
+lemma BOUNDS_IGNORE: "(EX B::nat. ALL i::nat. (P::nat => nat) i <= (Q::nat => nat) i + B) =
+(EX (x::nat) N::nat. ALL i>=N. P i <= Q i + x)"
by (import hollight BOUNDS_IGNORE)
-definition is_nadd :: "(nat => nat) => bool" where
+definition
+ is_nadd :: "(nat => nat) => bool" where
"is_nadd ==
-%u::nat => nat.
- EX B::nat.
- ALL (m::nat) n::nat. <= (dist (m * u n, n * u m)) (B * (m + n))"
+%u. EX B. ALL m n. hollight.dist (m * u n, n * u m) <= B * (m + n)"
lemma DEF_is_nadd: "is_nadd =
-(%u::nat => nat.
- EX B::nat.
- ALL (m::nat) n::nat. <= (dist (m * u n, n * u m)) (B * (m + n)))"
+(%u. EX B. ALL m n. hollight.dist (m * u n, n * u m) <= B * (m + n))"
by (import hollight DEF_is_nadd)
-lemma is_nadd_0: "is_nadd (%n::nat. 0)"
+lemma is_nadd_0: "is_nadd (%n. 0)"
by (import hollight is_nadd_0)
typedef (open) nadd = "Collect is_nadd" morphisms "dest_nadd" "mk_nadd"
- apply (rule light_ex_imp_nonempty[where t="%n::nat. NUMERAL 0"])
+ apply (rule light_ex_imp_nonempty[where t="%n. NUMERAL 0"])
by (import hollight TYDEF_nadd)
syntax
@@ -3100,394 +1411,329 @@
[where a="a :: nadd" and r=r ,
OF type_definition_nadd]
-lemma NADD_CAUCHY: "ALL x::nadd.
- EX xa::nat.
- ALL (xb::nat) xc::nat.
- <= (dist (xb * dest_nadd x xc, xc * dest_nadd x xb))
- (xa * (xb + xc))"
+lemma NADD_CAUCHY: "EX xa.
+ ALL xb xc.
+ hollight.dist (xb * dest_nadd x xc, xc * dest_nadd x xb)
+ <= xa * (xb + xc)"
by (import hollight NADD_CAUCHY)
-lemma NADD_BOUND: "ALL x::nadd.
- EX (xa::nat) B::nat. ALL n::nat. <= (dest_nadd x n) (xa * n + B)"
+lemma NADD_BOUND: "EX xa B. ALL n. dest_nadd x n <= xa * n + B"
by (import hollight NADD_BOUND)
-lemma NADD_MULTIPLICATIVE: "ALL x::nadd.
- EX xa::nat.
- ALL (m::nat) n::nat.
- <= (dist (dest_nadd x (m * n), m * dest_nadd x n)) (xa * m + xa)"
+lemma NADD_MULTIPLICATIVE: "EX xa.
+ ALL m n.
+ hollight.dist (dest_nadd x (m * n), m * dest_nadd x n) <= xa * m + xa"
by (import hollight NADD_MULTIPLICATIVE)
-lemma NADD_ADDITIVE: "ALL x::nadd.
- EX xa::nat.
- ALL (m::nat) n::nat.
- <= (dist (dest_nadd x (m + n), dest_nadd x m + dest_nadd x n)) xa"
+lemma NADD_ADDITIVE: "EX xa.
+ ALL m n.
+ hollight.dist (dest_nadd x (m + n), dest_nadd x m + dest_nadd x n)
+ <= xa"
by (import hollight NADD_ADDITIVE)
-lemma NADD_SUC: "ALL x::nadd.
- EX xa::nat. ALL n::nat. <= (dist (dest_nadd x (Suc n), dest_nadd x n)) xa"
+lemma NADD_SUC: "EX xa. ALL n. hollight.dist (dest_nadd x (Suc n), dest_nadd x n) <= xa"
by (import hollight NADD_SUC)
-lemma NADD_DIST_LEMMA: "ALL x::nadd.
- EX xa::nat.
- ALL (m::nat) n::nat.
- <= (dist (dest_nadd x (m + n), dest_nadd x m)) (xa * n)"
+lemma NADD_DIST_LEMMA: "EX xa. ALL m n. hollight.dist (dest_nadd x (m + n), dest_nadd x m) <= xa * n"
by (import hollight NADD_DIST_LEMMA)
-lemma NADD_DIST: "ALL x::nadd.
- EX xa::nat.
- ALL (m::nat) n::nat.
- <= (dist (dest_nadd x m, dest_nadd x n)) (xa * dist (m, n))"
+lemma NADD_DIST: "EX xa.
+ ALL m n.
+ hollight.dist (dest_nadd x m, dest_nadd x n)
+ <= xa * hollight.dist (m, n)"
by (import hollight NADD_DIST)
-lemma NADD_ALTMUL: "ALL (x::nadd) y::nadd.
- EX (A::nat) B::nat.
- ALL n::nat.
- <= (dist
- (n * dest_nadd x (dest_nadd y n),
- dest_nadd x n * dest_nadd y n))
- (A * n + B)"
+lemma NADD_ALTMUL: "EX A B.
+ ALL n.
+ hollight.dist
+ (n * dest_nadd x (dest_nadd y n), dest_nadd x n * dest_nadd y n)
+ <= A * n + B"
by (import hollight NADD_ALTMUL)
-definition nadd_eq :: "nadd => nadd => bool" where
+definition
+ nadd_eq :: "nadd => nadd => bool" where
"nadd_eq ==
-%(u::nadd) ua::nadd.
- EX B::nat. ALL n::nat. <= (dist (dest_nadd u n, dest_nadd ua n)) B"
+%u ua. EX B. ALL n. hollight.dist (dest_nadd u n, dest_nadd ua n) <= B"
lemma DEF_nadd_eq: "nadd_eq =
-(%(u::nadd) ua::nadd.
- EX B::nat. ALL n::nat. <= (dist (dest_nadd u n, dest_nadd ua n)) B)"
+(%u ua. EX B. ALL n. hollight.dist (dest_nadd u n, dest_nadd ua n) <= B)"
by (import hollight DEF_nadd_eq)
-lemma NADD_EQ_REFL: "ALL x::nadd. nadd_eq x x"
+lemma NADD_EQ_REFL: "nadd_eq x x"
by (import hollight NADD_EQ_REFL)
-lemma NADD_EQ_SYM: "ALL (x::nadd) y::nadd. nadd_eq x y = nadd_eq y x"
+lemma NADD_EQ_SYM: "nadd_eq x y = nadd_eq y x"
by (import hollight NADD_EQ_SYM)
-lemma NADD_EQ_TRANS: "ALL (x::nadd) (y::nadd) z::nadd. nadd_eq x y & nadd_eq y z --> nadd_eq x z"
+lemma NADD_EQ_TRANS: "nadd_eq x y & nadd_eq y z ==> nadd_eq x z"
by (import hollight NADD_EQ_TRANS)
-definition nadd_of_num :: "nat => nadd" where
- "nadd_of_num == %u::nat. mk_nadd (op * u)"
-
-lemma DEF_nadd_of_num: "nadd_of_num = (%u::nat. mk_nadd (op * u))"
+definition
+ nadd_of_num :: "nat => nadd" where
+ "nadd_of_num == %u. mk_nadd (op * u)"
+
+lemma DEF_nadd_of_num: "nadd_of_num = (%u. mk_nadd (op * u))"
by (import hollight DEF_nadd_of_num)
-lemma NADD_OF_NUM: "ALL x::nat. dest_nadd (nadd_of_num x) = op * x"
+lemma NADD_OF_NUM: "dest_nadd (nadd_of_num x) = op * x"
by (import hollight NADD_OF_NUM)
-lemma NADD_OF_NUM_WELLDEF: "ALL (m::nat) n::nat. m = n --> nadd_eq (nadd_of_num m) (nadd_of_num n)"
+lemma NADD_OF_NUM_WELLDEF: "m = n ==> nadd_eq (nadd_of_num m) (nadd_of_num n)"
by (import hollight NADD_OF_NUM_WELLDEF)
-lemma NADD_OF_NUM_EQ: "ALL (m::nat) n::nat. nadd_eq (nadd_of_num m) (nadd_of_num n) = (m = n)"
+lemma NADD_OF_NUM_EQ: "nadd_eq (nadd_of_num m) (nadd_of_num n) = (m = n)"
by (import hollight NADD_OF_NUM_EQ)
-definition nadd_le :: "nadd => nadd => bool" where
- "nadd_le ==
-%(u::nadd) ua::nadd.
- EX B::nat. ALL n::nat. <= (dest_nadd u n) (dest_nadd ua n + B)"
-
-lemma DEF_nadd_le: "nadd_le =
-(%(u::nadd) ua::nadd.
- EX B::nat. ALL n::nat. <= (dest_nadd u n) (dest_nadd ua n + B))"
+definition
+ nadd_le :: "nadd => nadd => bool" where
+ "nadd_le == %u ua. EX B. ALL n. dest_nadd u n <= dest_nadd ua n + B"
+
+lemma DEF_nadd_le: "nadd_le = (%u ua. EX B. ALL n. dest_nadd u n <= dest_nadd ua n + B)"
by (import hollight DEF_nadd_le)
-lemma NADD_LE_WELLDEF_LEMMA: "ALL (x::nadd) (x'::nadd) (y::nadd) y'::nadd.
- nadd_eq x x' & nadd_eq y y' & nadd_le x y --> nadd_le x' y'"
+lemma NADD_LE_WELLDEF_LEMMA: "nadd_eq x x' & nadd_eq y y' & nadd_le x y ==> nadd_le x' y'"
by (import hollight NADD_LE_WELLDEF_LEMMA)
-lemma NADD_LE_WELLDEF: "ALL (x::nadd) (x'::nadd) (y::nadd) y'::nadd.
- nadd_eq x x' & nadd_eq y y' --> nadd_le x y = nadd_le x' y'"
+lemma NADD_LE_WELLDEF: "nadd_eq x x' & nadd_eq y y' ==> nadd_le x y = nadd_le x' y'"
by (import hollight NADD_LE_WELLDEF)
-lemma NADD_LE_REFL: "ALL x::nadd. nadd_le x x"
+lemma NADD_LE_REFL: "nadd_le x x"
by (import hollight NADD_LE_REFL)
-lemma NADD_LE_TRANS: "ALL (x::nadd) (y::nadd) z::nadd. nadd_le x y & nadd_le y z --> nadd_le x z"
+lemma NADD_LE_TRANS: "nadd_le x y & nadd_le y z ==> nadd_le x z"
by (import hollight NADD_LE_TRANS)
-lemma NADD_LE_ANTISYM: "ALL (x::nadd) y::nadd. (nadd_le x y & nadd_le y x) = nadd_eq x y"
+lemma NADD_LE_ANTISYM: "(nadd_le x y & nadd_le y x) = nadd_eq x y"
by (import hollight NADD_LE_ANTISYM)
-lemma NADD_LE_TOTAL_LEMMA: "ALL (x::nadd) y::nadd.
- ~ nadd_le x y -->
- (ALL B::nat. EX n::nat. n ~= 0 & < (dest_nadd y n + B) (dest_nadd x n))"
+lemma NADD_LE_TOTAL_LEMMA: "~ nadd_le x y ==> EX n. n ~= 0 & dest_nadd y n + B < dest_nadd x n"
by (import hollight NADD_LE_TOTAL_LEMMA)
-lemma NADD_LE_TOTAL: "ALL (x::nadd) y::nadd. nadd_le x y | nadd_le y x"
+lemma NADD_LE_TOTAL: "nadd_le x y | nadd_le y x"
by (import hollight NADD_LE_TOTAL)
-lemma NADD_ARCH: "ALL x::nadd. EX xa::nat. nadd_le x (nadd_of_num xa)"
+lemma NADD_ARCH: "EX xa. nadd_le x (nadd_of_num xa)"
by (import hollight NADD_ARCH)
-lemma NADD_OF_NUM_LE: "ALL (m::nat) n::nat. nadd_le (nadd_of_num m) (nadd_of_num n) = <= m n"
+lemma NADD_OF_NUM_LE: "nadd_le (nadd_of_num m) (nadd_of_num n) = (m <= n)"
by (import hollight NADD_OF_NUM_LE)
-definition nadd_add :: "nadd => nadd => nadd" where
- "nadd_add ==
-%(u::nadd) ua::nadd. mk_nadd (%n::nat. dest_nadd u n + dest_nadd ua n)"
-
-lemma DEF_nadd_add: "nadd_add =
-(%(u::nadd) ua::nadd. mk_nadd (%n::nat. dest_nadd u n + dest_nadd ua n))"
+definition
+ nadd_add :: "nadd => nadd => nadd" where
+ "nadd_add == %u ua. mk_nadd (%n. dest_nadd u n + dest_nadd ua n)"
+
+lemma DEF_nadd_add: "nadd_add = (%u ua. mk_nadd (%n. dest_nadd u n + dest_nadd ua n))"
by (import hollight DEF_nadd_add)
-lemma NADD_ADD: "ALL (x::nadd) y::nadd.
- dest_nadd (nadd_add x y) = (%n::nat. dest_nadd x n + dest_nadd y n)"
+lemma NADD_ADD: "dest_nadd (nadd_add x y) = (%n. dest_nadd x n + dest_nadd y n)"
by (import hollight NADD_ADD)
-lemma NADD_ADD_WELLDEF: "ALL (x::nadd) (x'::nadd) (y::nadd) y'::nadd.
- nadd_eq x x' & nadd_eq y y' --> nadd_eq (nadd_add x y) (nadd_add x' y')"
+lemma NADD_ADD_WELLDEF: "nadd_eq x x' & nadd_eq y y' ==> nadd_eq (nadd_add x y) (nadd_add x' y')"
by (import hollight NADD_ADD_WELLDEF)
-lemma NADD_ADD_SYM: "ALL (x::nadd) y::nadd. nadd_eq (nadd_add x y) (nadd_add y x)"
+lemma NADD_ADD_SYM: "nadd_eq (nadd_add x y) (nadd_add y x)"
by (import hollight NADD_ADD_SYM)
-lemma NADD_ADD_ASSOC: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_eq (nadd_add x (nadd_add y z)) (nadd_add (nadd_add x y) z)"
+lemma NADD_ADD_ASSOC: "nadd_eq (nadd_add x (nadd_add y z)) (nadd_add (nadd_add x y) z)"
by (import hollight NADD_ADD_ASSOC)
-lemma NADD_ADD_LID: "ALL x::nadd. nadd_eq (nadd_add (nadd_of_num 0) x) x"
+lemma NADD_ADD_LID: "nadd_eq (nadd_add (nadd_of_num 0) x) x"
by (import hollight NADD_ADD_LID)
-lemma NADD_ADD_LCANCEL: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_eq (nadd_add x y) (nadd_add x z) --> nadd_eq y z"
+lemma NADD_ADD_LCANCEL: "nadd_eq (nadd_add x y) (nadd_add x z) ==> nadd_eq y z"
by (import hollight NADD_ADD_LCANCEL)
-lemma NADD_LE_ADD: "ALL (x::nadd) y::nadd. nadd_le x (nadd_add x y)"
+lemma NADD_LE_ADD: "nadd_le x (nadd_add x y)"
by (import hollight NADD_LE_ADD)
-lemma NADD_LE_EXISTS: "ALL (x::nadd) y::nadd.
- nadd_le x y --> (EX d::nadd. nadd_eq y (nadd_add x d))"
+lemma NADD_LE_EXISTS: "nadd_le x y ==> EX d. nadd_eq y (nadd_add x d)"
by (import hollight NADD_LE_EXISTS)
-lemma NADD_OF_NUM_ADD: "ALL (x::nat) xa::nat.
- nadd_eq (nadd_add (nadd_of_num x) (nadd_of_num xa))
- (nadd_of_num (x + xa))"
+lemma NADD_OF_NUM_ADD: "nadd_eq (nadd_add (nadd_of_num x) (nadd_of_num xa)) (nadd_of_num (x + xa))"
by (import hollight NADD_OF_NUM_ADD)
-definition nadd_mul :: "nadd => nadd => nadd" where
- "nadd_mul ==
-%(u::nadd) ua::nadd. mk_nadd (%n::nat. dest_nadd u (dest_nadd ua n))"
-
-lemma DEF_nadd_mul: "nadd_mul =
-(%(u::nadd) ua::nadd. mk_nadd (%n::nat. dest_nadd u (dest_nadd ua n)))"
+definition
+ nadd_mul :: "nadd => nadd => nadd" where
+ "nadd_mul == %u ua. mk_nadd (%n. dest_nadd u (dest_nadd ua n))"
+
+lemma DEF_nadd_mul: "nadd_mul = (%u ua. mk_nadd (%n. dest_nadd u (dest_nadd ua n)))"
by (import hollight DEF_nadd_mul)
-lemma NADD_MUL: "ALL (x::nadd) y::nadd.
- dest_nadd (nadd_mul x y) = (%n::nat. dest_nadd x (dest_nadd y n))"
+lemma NADD_MUL: "dest_nadd (nadd_mul x y) = (%n. dest_nadd x (dest_nadd y n))"
by (import hollight NADD_MUL)
-lemma NADD_MUL_SYM: "ALL (x::nadd) y::nadd. nadd_eq (nadd_mul x y) (nadd_mul y x)"
+lemma NADD_MUL_SYM: "nadd_eq (nadd_mul x y) (nadd_mul y x)"
by (import hollight NADD_MUL_SYM)
-lemma NADD_MUL_ASSOC: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_eq (nadd_mul x (nadd_mul y z)) (nadd_mul (nadd_mul x y) z)"
+lemma NADD_MUL_ASSOC: "nadd_eq (nadd_mul x (nadd_mul y z)) (nadd_mul (nadd_mul x y) z)"
by (import hollight NADD_MUL_ASSOC)
-lemma NADD_MUL_LID: "ALL x::nadd. nadd_eq (nadd_mul (nadd_of_num (NUMERAL_BIT1 0)) x) x"
+lemma NADD_MUL_LID: "nadd_eq (nadd_mul (nadd_of_num 1) x) x"
by (import hollight NADD_MUL_LID)
-lemma NADD_LDISTRIB: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_eq (nadd_mul x (nadd_add y z))
- (nadd_add (nadd_mul x y) (nadd_mul x z))"
+lemma NADD_LDISTRIB: "nadd_eq (nadd_mul x (nadd_add y z)) (nadd_add (nadd_mul x y) (nadd_mul x z))"
by (import hollight NADD_LDISTRIB)
-lemma NADD_MUL_WELLDEF_LEMMA: "ALL (x::nadd) (y::nadd) y'::nadd.
- nadd_eq y y' --> nadd_eq (nadd_mul x y) (nadd_mul x y')"
+lemma NADD_MUL_WELLDEF_LEMMA: "nadd_eq y y' ==> nadd_eq (nadd_mul x y) (nadd_mul x y')"
by (import hollight NADD_MUL_WELLDEF_LEMMA)
-lemma NADD_MUL_WELLDEF: "ALL (x::nadd) (x'::nadd) (y::nadd) y'::nadd.
- nadd_eq x x' & nadd_eq y y' --> nadd_eq (nadd_mul x y) (nadd_mul x' y')"
+lemma NADD_MUL_WELLDEF: "nadd_eq x x' & nadd_eq y y' ==> nadd_eq (nadd_mul x y) (nadd_mul x' y')"
by (import hollight NADD_MUL_WELLDEF)
-lemma NADD_OF_NUM_MUL: "ALL (x::nat) xa::nat.
- nadd_eq (nadd_mul (nadd_of_num x) (nadd_of_num xa))
- (nadd_of_num (x * xa))"
+lemma NADD_OF_NUM_MUL: "nadd_eq (nadd_mul (nadd_of_num x) (nadd_of_num xa)) (nadd_of_num (x * xa))"
by (import hollight NADD_OF_NUM_MUL)
-lemma NADD_LE_0: "All (nadd_le (nadd_of_num 0))"
+lemma NADD_LE_0: "nadd_le (nadd_of_num 0) x"
by (import hollight NADD_LE_0)
-lemma NADD_EQ_IMP_LE: "ALL (x::nadd) y::nadd. nadd_eq x y --> nadd_le x y"
+lemma NADD_EQ_IMP_LE: "nadd_eq x y ==> nadd_le x y"
by (import hollight NADD_EQ_IMP_LE)
-lemma NADD_LE_LMUL: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_le y z --> nadd_le (nadd_mul x y) (nadd_mul x z)"
+lemma NADD_LE_LMUL: "nadd_le y z ==> nadd_le (nadd_mul x y) (nadd_mul x z)"
by (import hollight NADD_LE_LMUL)
-lemma NADD_LE_RMUL: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_le x y --> nadd_le (nadd_mul x z) (nadd_mul y z)"
+lemma NADD_LE_RMUL: "nadd_le x y ==> nadd_le (nadd_mul x z) (nadd_mul y z)"
by (import hollight NADD_LE_RMUL)
-lemma NADD_LE_RADD: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_le (nadd_add x z) (nadd_add y z) = nadd_le x y"
+lemma NADD_LE_RADD: "nadd_le (nadd_add x z) (nadd_add y z) = nadd_le x y"
by (import hollight NADD_LE_RADD)
-lemma NADD_LE_LADD: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_le (nadd_add x y) (nadd_add x z) = nadd_le y z"
+lemma NADD_LE_LADD: "nadd_le (nadd_add x y) (nadd_add x z) = nadd_le y z"
by (import hollight NADD_LE_LADD)
-lemma NADD_RDISTRIB: "ALL (x::nadd) (y::nadd) z::nadd.
- nadd_eq (nadd_mul (nadd_add x y) z)
- (nadd_add (nadd_mul x z) (nadd_mul y z))"
+lemma NADD_RDISTRIB: "nadd_eq (nadd_mul (nadd_add x y) z) (nadd_add (nadd_mul x z) (nadd_mul y z))"
by (import hollight NADD_RDISTRIB)
-lemma NADD_ARCH_MULT: "ALL (x::nadd) k::nat.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX xa::nat. nadd_le (nadd_of_num k) (nadd_mul (nadd_of_num xa) x))"
+lemma NADD_ARCH_MULT: "~ nadd_eq x (nadd_of_num 0)
+==> EX xa. nadd_le (nadd_of_num k) (nadd_mul (nadd_of_num xa) x)"
by (import hollight NADD_ARCH_MULT)
-lemma NADD_ARCH_ZERO: "ALL (x::nadd) k::nadd.
- (ALL n::nat. nadd_le (nadd_mul (nadd_of_num n) x) k) -->
- nadd_eq x (nadd_of_num 0)"
+lemma NADD_ARCH_ZERO: "(!!n. nadd_le (nadd_mul (nadd_of_num n) x) k) ==> nadd_eq x (nadd_of_num 0)"
by (import hollight NADD_ARCH_ZERO)
-lemma NADD_ARCH_LEMMA: "ALL (x::nadd) (y::nadd) z::nadd.
- (ALL n::nat.
- nadd_le (nadd_mul (nadd_of_num n) x)
- (nadd_add (nadd_mul (nadd_of_num n) y) z)) -->
- nadd_le x y"
+lemma NADD_ARCH_LEMMA: "(!!n. nadd_le (nadd_mul (nadd_of_num n) x)
+ (nadd_add (nadd_mul (nadd_of_num n) y) z))
+==> nadd_le x y"
by (import hollight NADD_ARCH_LEMMA)
-lemma NADD_COMPLETE: "ALL P::nadd => bool.
- Ex P & (EX M::nadd. ALL x::nadd. P x --> nadd_le x M) -->
- (EX M::nadd.
- (ALL x::nadd. P x --> nadd_le x M) &
- (ALL M'::nadd. (ALL x::nadd. P x --> nadd_le x M') --> nadd_le M M'))"
+lemma NADD_COMPLETE: "Ex P & (EX M. ALL x. P x --> nadd_le x M)
+==> EX M. (ALL x. P x --> nadd_le x M) &
+ (ALL M'. (ALL x. P x --> nadd_le x M') --> nadd_le M M')"
by (import hollight NADD_COMPLETE)
-lemma NADD_UBOUND: "ALL x::nadd.
- EX (xa::nat) N::nat. ALL n::nat. <= N n --> <= (dest_nadd x n) (xa * n)"
+lemma NADD_UBOUND: "EX xa N. ALL n>=N. dest_nadd x n <= xa * n"
by (import hollight NADD_UBOUND)
-lemma NADD_NONZERO: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX N::nat. ALL n::nat. <= N n --> dest_nadd x n ~= 0)"
+lemma NADD_NONZERO: "~ nadd_eq x (nadd_of_num 0) ==> EX N. ALL n>=N. dest_nadd x n ~= 0"
by (import hollight NADD_NONZERO)
-lemma NADD_LBOUND: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX (A::nat) N::nat. ALL n::nat. <= N n --> <= n (A * dest_nadd x n))"
+lemma NADD_LBOUND: "~ nadd_eq x (nadd_of_num 0) ==> EX A N. ALL n>=N. n <= A * dest_nadd x n"
by (import hollight NADD_LBOUND)
-definition nadd_rinv :: "nadd => nat => nat" where
- "nadd_rinv == %(u::nadd) n::nat. DIV (n * n) (dest_nadd u n)"
-
-lemma DEF_nadd_rinv: "nadd_rinv = (%(u::nadd) n::nat. DIV (n * n) (dest_nadd u n))"
+definition
+ nadd_rinv :: "nadd => nat => nat" where
+ "nadd_rinv == %u n. n * n div dest_nadd u n"
+
+lemma DEF_nadd_rinv: "nadd_rinv = (%u n. n * n div dest_nadd u n)"
by (import hollight DEF_nadd_rinv)
-lemma NADD_MUL_LINV_LEMMA0: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX (xa::nat) B::nat. ALL i::nat. <= (nadd_rinv x i) (xa * i + B))"
+lemma NADD_MUL_LINV_LEMMA0: "~ nadd_eq x (nadd_of_num 0) ==> EX xa B. ALL i. nadd_rinv x i <= xa * i + B"
by (import hollight NADD_MUL_LINV_LEMMA0)
-lemma NADD_MUL_LINV_LEMMA1: "ALL (x::nadd) n::nat.
- dest_nadd x n ~= 0 -->
- <= (dist (dest_nadd x n * nadd_rinv x n, n * n)) (dest_nadd x n)"
+lemma NADD_MUL_LINV_LEMMA1: "dest_nadd x n ~= 0
+==> hollight.dist (dest_nadd x n * nadd_rinv x n, n * n) <= dest_nadd x n"
by (import hollight NADD_MUL_LINV_LEMMA1)
-lemma NADD_MUL_LINV_LEMMA2: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX N::nat.
- ALL n::nat.
- <= N n -->
- <= (dist (dest_nadd x n * nadd_rinv x n, n * n)) (dest_nadd x n))"
+lemma NADD_MUL_LINV_LEMMA2: "~ nadd_eq x (nadd_of_num 0)
+==> EX N. ALL n>=N.
+ hollight.dist (dest_nadd x n * nadd_rinv x n, n * n)
+ <= dest_nadd x n"
by (import hollight NADD_MUL_LINV_LEMMA2)
-lemma NADD_MUL_LINV_LEMMA3: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX N::nat.
- ALL (m::nat) n::nat.
- <= N n -->
- <= (dist
- (m * (dest_nadd x m * (dest_nadd x n * nadd_rinv x n)),
- m * (dest_nadd x m * (n * n))))
- (m * (dest_nadd x m * dest_nadd x n)))"
+lemma NADD_MUL_LINV_LEMMA3: "~ nadd_eq x (nadd_of_num 0)
+==> EX N. ALL m n.
+ N <= n -->
+ hollight.dist
+ (m * (dest_nadd x m * (dest_nadd x n * nadd_rinv x n)),
+ m * (dest_nadd x m * (n * n)))
+ <= m * (dest_nadd x m * dest_nadd x n)"
by (import hollight NADD_MUL_LINV_LEMMA3)
-lemma NADD_MUL_LINV_LEMMA4: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX N::nat.
- ALL (m::nat) n::nat.
- <= N m & <= N n -->
- <= (dest_nadd x m * dest_nadd x n *
- dist (m * nadd_rinv x n, n * nadd_rinv x m))
- (m * n * dist (m * dest_nadd x n, n * dest_nadd x m) +
- dest_nadd x m * dest_nadd x n * (m + n)))"
+lemma NADD_MUL_LINV_LEMMA4: "~ nadd_eq x (nadd_of_num 0)
+==> EX N. ALL m n.
+ N <= m & N <= n -->
+ dest_nadd x m * dest_nadd x n *
+ hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m)
+ <= m * n *
+ hollight.dist (m * dest_nadd x n, n * dest_nadd x m) +
+ dest_nadd x m * dest_nadd x n * (m + n)"
by (import hollight NADD_MUL_LINV_LEMMA4)
-lemma NADD_MUL_LINV_LEMMA5: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX (B::nat) N::nat.
- ALL (m::nat) n::nat.
- <= N m & <= N n -->
- <= (dest_nadd x m * dest_nadd x n *
- dist (m * nadd_rinv x n, n * nadd_rinv x m))
- (B * (m * n * (m + n))))"
+lemma NADD_MUL_LINV_LEMMA5: "~ nadd_eq x (nadd_of_num 0)
+==> EX B N.
+ ALL m n.
+ N <= m & N <= n -->
+ dest_nadd x m * dest_nadd x n *
+ hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m)
+ <= B * (m * n * (m + n))"
by (import hollight NADD_MUL_LINV_LEMMA5)
-lemma NADD_MUL_LINV_LEMMA6: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX (B::nat) N::nat.
- ALL (m::nat) n::nat.
- <= N m & <= N n -->
- <= (m * n * dist (m * nadd_rinv x n, n * nadd_rinv x m))
- (B * (m * n * (m + n))))"
+lemma NADD_MUL_LINV_LEMMA6: "~ nadd_eq x (nadd_of_num 0)
+==> EX B N.
+ ALL m n.
+ N <= m & N <= n -->
+ m * n * hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m)
+ <= B * (m * n * (m + n))"
by (import hollight NADD_MUL_LINV_LEMMA6)
-lemma NADD_MUL_LINV_LEMMA7: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX (B::nat) N::nat.
- ALL (m::nat) n::nat.
- <= N m & <= N n -->
- <= (dist (m * nadd_rinv x n, n * nadd_rinv x m)) (B * (m + n)))"
+lemma NADD_MUL_LINV_LEMMA7: "~ nadd_eq x (nadd_of_num 0)
+==> EX B N.
+ ALL m n.
+ N <= m & N <= n -->
+ hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m)
+ <= B * (m + n)"
by (import hollight NADD_MUL_LINV_LEMMA7)
-lemma NADD_MUL_LINV_LEMMA7a: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (ALL N::nat.
- EX (A::nat) B::nat.
- ALL (m::nat) n::nat.
- <= m N -->
- <= (dist (m * nadd_rinv x n, n * nadd_rinv x m)) (A * n + B))"
+lemma NADD_MUL_LINV_LEMMA7a: "~ nadd_eq x (nadd_of_num 0)
+==> EX A B.
+ ALL m n.
+ m <= N -->
+ hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m) <= A * n + B"
by (import hollight NADD_MUL_LINV_LEMMA7a)
-lemma NADD_MUL_LINV_LEMMA8: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- (EX B::nat.
- ALL (m::nat) n::nat.
- <= (dist (m * nadd_rinv x n, n * nadd_rinv x m)) (B * (m + n)))"
+lemma NADD_MUL_LINV_LEMMA8: "~ nadd_eq x (nadd_of_num 0)
+==> EX B. ALL m n.
+ hollight.dist (m * nadd_rinv x n, n * nadd_rinv x m)
+ <= B * (m + n)"
by (import hollight NADD_MUL_LINV_LEMMA8)
-definition nadd_inv :: "nadd => nadd" where
+definition
+ nadd_inv :: "nadd => nadd" where
"nadd_inv ==
-%u::nadd.
- COND (nadd_eq u (nadd_of_num 0)) (nadd_of_num 0) (mk_nadd (nadd_rinv u))"
+%u. if nadd_eq u (nadd_of_num 0) then nadd_of_num 0
+ else mk_nadd (nadd_rinv u)"
lemma DEF_nadd_inv: "nadd_inv =
-(%u::nadd.
- COND (nadd_eq u (nadd_of_num 0)) (nadd_of_num 0)
- (mk_nadd (nadd_rinv u)))"
+(%u. if nadd_eq u (nadd_of_num 0) then nadd_of_num 0
+ else mk_nadd (nadd_rinv u))"
by (import hollight DEF_nadd_inv)
-lemma NADD_INV: "ALL x::nadd.
- dest_nadd (nadd_inv x) =
- COND (nadd_eq x (nadd_of_num 0)) (%n::nat. 0) (nadd_rinv x)"
+lemma NADD_INV: "dest_nadd (nadd_inv x) =
+(if nadd_eq x (nadd_of_num 0) then %n. 0 else nadd_rinv x)"
by (import hollight NADD_INV)
-lemma NADD_MUL_LINV: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num 0) -->
- nadd_eq (nadd_mul (nadd_inv x) x) (nadd_of_num (NUMERAL_BIT1 0))"
+lemma NADD_MUL_LINV: "~ nadd_eq x (nadd_of_num 0)
+==> nadd_eq (nadd_mul (nadd_inv x) x) (nadd_of_num 1)"
by (import hollight NADD_MUL_LINV)
lemma NADD_INV_0: "nadd_eq (nadd_inv (nadd_of_num 0)) (nadd_of_num 0)"
by (import hollight NADD_INV_0)
-lemma NADD_INV_WELLDEF: "ALL (x::nadd) y::nadd. nadd_eq x y --> nadd_eq (nadd_inv x) (nadd_inv y)"
+lemma NADD_INV_WELLDEF: "nadd_eq x y ==> nadd_eq (nadd_inv x) (nadd_inv y)"
by (import hollight NADD_INV_WELLDEF)
-typedef (open) hreal = "{s::nadd => bool. EX x::nadd. s = nadd_eq x}" morphisms "dest_hreal" "mk_hreal"
- apply (rule light_ex_imp_nonempty[where t="nadd_eq (x::nadd)"])
+typedef (open) hreal = "{s. EX x. s = nadd_eq x}" morphisms "dest_hreal" "mk_hreal"
+ apply (rule light_ex_imp_nonempty[where t="nadd_eq x"])
by (import hollight TYDEF_hreal)
syntax
@@ -3500,328 +1746,285 @@
[where a="a :: hreal" and r=r ,
OF type_definition_hreal]
-definition hreal_of_num :: "nat => hreal" where
- "hreal_of_num == %m::nat. mk_hreal (nadd_eq (nadd_of_num m))"
-
-lemma DEF_hreal_of_num: "hreal_of_num = (%m::nat. mk_hreal (nadd_eq (nadd_of_num m)))"
+definition
+ hreal_of_num :: "nat => hreal" where
+ "hreal_of_num == %m. mk_hreal (nadd_eq (nadd_of_num m))"
+
+lemma DEF_hreal_of_num: "hreal_of_num = (%m. mk_hreal (nadd_eq (nadd_of_num m)))"
by (import hollight DEF_hreal_of_num)
-definition hreal_add :: "hreal => hreal => hreal" where
+definition
+ hreal_add :: "hreal => hreal => hreal" where
"hreal_add ==
-%(x::hreal) y::hreal.
- mk_hreal
- (%u::nadd.
- EX (xa::nadd) ya::nadd.
- nadd_eq (nadd_add xa ya) u & dest_hreal x xa & dest_hreal y ya)"
+%x y. mk_hreal
+ (%u. EX xa ya.
+ nadd_eq (nadd_add xa ya) u &
+ dest_hreal x xa & dest_hreal y ya)"
lemma DEF_hreal_add: "hreal_add =
-(%(x::hreal) y::hreal.
- mk_hreal
- (%u::nadd.
- EX (xa::nadd) ya::nadd.
- nadd_eq (nadd_add xa ya) u & dest_hreal x xa & dest_hreal y ya))"
+(%x y. mk_hreal
+ (%u. EX xa ya.
+ nadd_eq (nadd_add xa ya) u &
+ dest_hreal x xa & dest_hreal y ya))"
by (import hollight DEF_hreal_add)
-definition hreal_mul :: "hreal => hreal => hreal" where
+definition
+ hreal_mul :: "hreal => hreal => hreal" where
"hreal_mul ==
-%(x::hreal) y::hreal.
- mk_hreal
- (%u::nadd.
- EX (xa::nadd) ya::nadd.
- nadd_eq (nadd_mul xa ya) u & dest_hreal x xa & dest_hreal y ya)"
+%x y. mk_hreal
+ (%u. EX xa ya.
+ nadd_eq (nadd_mul xa ya) u &
+ dest_hreal x xa & dest_hreal y ya)"
lemma DEF_hreal_mul: "hreal_mul =
-(%(x::hreal) y::hreal.
- mk_hreal
- (%u::nadd.
- EX (xa::nadd) ya::nadd.
- nadd_eq (nadd_mul xa ya) u & dest_hreal x xa & dest_hreal y ya))"
+(%x y. mk_hreal
+ (%u. EX xa ya.
+ nadd_eq (nadd_mul xa ya) u &
+ dest_hreal x xa & dest_hreal y ya))"
by (import hollight DEF_hreal_mul)
-definition hreal_le :: "hreal => hreal => bool" where
+definition
+ hreal_le :: "hreal => hreal => bool" where
"hreal_le ==
-%(x::hreal) y::hreal.
- SOME u::bool.
- EX (xa::nadd) ya::nadd.
- nadd_le xa ya = u & dest_hreal x xa & dest_hreal y ya"
+%x y. SOME u.
+ EX xa ya. nadd_le xa ya = u & dest_hreal x xa & dest_hreal y ya"
lemma DEF_hreal_le: "hreal_le =
-(%(x::hreal) y::hreal.
- SOME u::bool.
- EX (xa::nadd) ya::nadd.
- nadd_le xa ya = u & dest_hreal x xa & dest_hreal y ya)"
+(%x y. SOME u.
+ EX xa ya. nadd_le xa ya = u & dest_hreal x xa & dest_hreal y ya)"
by (import hollight DEF_hreal_le)
-definition hreal_inv :: "hreal => hreal" where
+definition
+ hreal_inv :: "hreal => hreal" where
"hreal_inv ==
-%x::hreal.
- mk_hreal
- (%u::nadd. EX xa::nadd. nadd_eq (nadd_inv xa) u & dest_hreal x xa)"
+%x. mk_hreal (%u. EX xa. nadd_eq (nadd_inv xa) u & dest_hreal x xa)"
lemma DEF_hreal_inv: "hreal_inv =
-(%x::hreal.
- mk_hreal
- (%u::nadd. EX xa::nadd. nadd_eq (nadd_inv xa) u & dest_hreal x xa))"
+(%x. mk_hreal (%u. EX xa. nadd_eq (nadd_inv xa) u & dest_hreal x xa))"
by (import hollight DEF_hreal_inv)
-lemma HREAL_LE_EXISTS_DEF: "ALL (m::hreal) n::hreal. hreal_le m n = (EX d::hreal. n = hreal_add m d)"
+lemma HREAL_LE_EXISTS_DEF: "hreal_le m n = (EX d. n = hreal_add m d)"
by (import hollight HREAL_LE_EXISTS_DEF)
-lemma HREAL_EQ_ADD_LCANCEL: "ALL (m::hreal) (n::hreal) p::hreal.
- (hreal_add m n = hreal_add m p) = (n = p)"
+lemma HREAL_EQ_ADD_LCANCEL: "(hreal_add m n = hreal_add m p) = (n = p)"
by (import hollight HREAL_EQ_ADD_LCANCEL)
-lemma HREAL_EQ_ADD_RCANCEL: "ALL (x::hreal) (xa::hreal) xb::hreal.
- (hreal_add x xb = hreal_add xa xb) = (x = xa)"
+lemma HREAL_EQ_ADD_RCANCEL: "(hreal_add x xb = hreal_add xa xb) = (x = xa)"
by (import hollight HREAL_EQ_ADD_RCANCEL)
-lemma HREAL_LE_ADD_LCANCEL: "ALL (x::hreal) (xa::hreal) xb::hreal.
- hreal_le (hreal_add x xa) (hreal_add x xb) = hreal_le xa xb"
+lemma HREAL_LE_ADD_LCANCEL: "hreal_le (hreal_add x xa) (hreal_add x xb) = hreal_le xa xb"
by (import hollight HREAL_LE_ADD_LCANCEL)
-lemma HREAL_LE_ADD_RCANCEL: "ALL (x::hreal) (xa::hreal) xb::hreal.
- hreal_le (hreal_add x xb) (hreal_add xa xb) = hreal_le x xa"
+lemma HREAL_LE_ADD_RCANCEL: "hreal_le (hreal_add x xb) (hreal_add xa xb) = hreal_le x xa"
by (import hollight HREAL_LE_ADD_RCANCEL)
-lemma HREAL_ADD_RID: "ALL x::hreal. hreal_add x (hreal_of_num 0) = x"
+lemma HREAL_ADD_RID: "hreal_add x (hreal_of_num 0) = x"
by (import hollight HREAL_ADD_RID)
-lemma HREAL_ADD_RDISTRIB: "ALL (x::hreal) (xa::hreal) xb::hreal.
- hreal_mul (hreal_add x xa) xb =
- hreal_add (hreal_mul x xb) (hreal_mul xa xb)"
+lemma HREAL_ADD_RDISTRIB: "hreal_mul (hreal_add x xa) xb = hreal_add (hreal_mul x xb) (hreal_mul xa xb)"
by (import hollight HREAL_ADD_RDISTRIB)
-lemma HREAL_MUL_LZERO: "ALL m::hreal. hreal_mul (hreal_of_num 0) m = hreal_of_num 0"
+lemma HREAL_MUL_LZERO: "hreal_mul (hreal_of_num 0) m = hreal_of_num 0"
by (import hollight HREAL_MUL_LZERO)
-lemma HREAL_MUL_RZERO: "ALL x::hreal. hreal_mul x (hreal_of_num 0) = hreal_of_num 0"
+lemma HREAL_MUL_RZERO: "hreal_mul x (hreal_of_num 0) = hreal_of_num 0"
by (import hollight HREAL_MUL_RZERO)
-lemma HREAL_ADD_AC: "hreal_add (m::hreal) (n::hreal) = hreal_add n m &
-hreal_add (hreal_add m n) (p::hreal) = hreal_add m (hreal_add n p) &
+lemma HREAL_ADD_AC: "hreal_add m n = hreal_add n m &
+hreal_add (hreal_add m n) p = hreal_add m (hreal_add n p) &
hreal_add m (hreal_add n p) = hreal_add n (hreal_add m p)"
by (import hollight HREAL_ADD_AC)
-lemma HREAL_LE_ADD2: "ALL (a::hreal) (b::hreal) (c::hreal) d::hreal.
- hreal_le a b & hreal_le c d --> hreal_le (hreal_add a c) (hreal_add b d)"
+lemma HREAL_LE_ADD2: "hreal_le a b & hreal_le c d ==> hreal_le (hreal_add a c) (hreal_add b d)"
by (import hollight HREAL_LE_ADD2)
-lemma HREAL_LE_MUL_RCANCEL_IMP: "ALL (a::hreal) (b::hreal) c::hreal.
- hreal_le a b --> hreal_le (hreal_mul a c) (hreal_mul b c)"
+lemma HREAL_LE_MUL_RCANCEL_IMP: "hreal_le a b ==> hreal_le (hreal_mul a c) (hreal_mul b c)"
by (import hollight HREAL_LE_MUL_RCANCEL_IMP)
-definition treal_of_num :: "nat => hreal * hreal" where
- "treal_of_num == %u::nat. (hreal_of_num u, hreal_of_num 0)"
-
-lemma DEF_treal_of_num: "treal_of_num = (%u::nat. (hreal_of_num u, hreal_of_num 0))"
+definition
+ treal_of_num :: "nat => hreal * hreal" where
+ "treal_of_num == %u. (hreal_of_num u, hreal_of_num 0)"
+
+lemma DEF_treal_of_num: "treal_of_num = (%u. (hreal_of_num u, hreal_of_num 0))"
by (import hollight DEF_treal_of_num)
-definition treal_neg :: "hreal * hreal => hreal * hreal" where
- "treal_neg == %u::hreal * hreal. (snd u, fst u)"
-
-lemma DEF_treal_neg: "treal_neg = (%u::hreal * hreal. (snd u, fst u))"
+definition
+ treal_neg :: "hreal * hreal => hreal * hreal" where
+ "treal_neg == %u. (snd u, fst u)"
+
+lemma DEF_treal_neg: "treal_neg = (%u. (snd u, fst u))"
by (import hollight DEF_treal_neg)
-definition treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal" where
- "treal_add ==
-%(u::hreal * hreal) ua::hreal * hreal.
- (hreal_add (fst u) (fst ua), hreal_add (snd u) (snd ua))"
+definition
+ treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal" where
+ "treal_add == %u ua. (hreal_add (fst u) (fst ua), hreal_add (snd u) (snd ua))"
lemma DEF_treal_add: "treal_add =
-(%(u::hreal * hreal) ua::hreal * hreal.
- (hreal_add (fst u) (fst ua), hreal_add (snd u) (snd ua)))"
+(%u ua. (hreal_add (fst u) (fst ua), hreal_add (snd u) (snd ua)))"
by (import hollight DEF_treal_add)
-definition treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal" where
+definition
+ treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal" where
"treal_mul ==
-%(u::hreal * hreal) ua::hreal * hreal.
+%u ua.
(hreal_add (hreal_mul (fst u) (fst ua)) (hreal_mul (snd u) (snd ua)),
hreal_add (hreal_mul (fst u) (snd ua)) (hreal_mul (snd u) (fst ua)))"
lemma DEF_treal_mul: "treal_mul =
-(%(u::hreal * hreal) ua::hreal * hreal.
+(%u ua.
(hreal_add (hreal_mul (fst u) (fst ua)) (hreal_mul (snd u) (snd ua)),
hreal_add (hreal_mul (fst u) (snd ua)) (hreal_mul (snd u) (fst ua))))"
by (import hollight DEF_treal_mul)
-definition treal_le :: "hreal * hreal => hreal * hreal => bool" where
+definition
+ treal_le :: "hreal * hreal => hreal * hreal => bool" where
"treal_le ==
-%(u::hreal * hreal) ua::hreal * hreal.
- hreal_le (hreal_add (fst u) (snd ua)) (hreal_add (fst ua) (snd u))"
+%u ua. hreal_le (hreal_add (fst u) (snd ua)) (hreal_add (fst ua) (snd u))"
lemma DEF_treal_le: "treal_le =
-(%(u::hreal * hreal) ua::hreal * hreal.
- hreal_le (hreal_add (fst u) (snd ua)) (hreal_add (fst ua) (snd u)))"
+(%u ua. hreal_le (hreal_add (fst u) (snd ua)) (hreal_add (fst ua) (snd u)))"
by (import hollight DEF_treal_le)
-definition treal_inv :: "hreal * hreal => hreal * hreal" where
+definition
+ treal_inv :: "hreal * hreal => hreal * hreal" where
"treal_inv ==
-%u::hreal * hreal.
- COND (fst u = snd u) (hreal_of_num 0, hreal_of_num 0)
- (COND (hreal_le (snd u) (fst u))
- (hreal_inv (SOME d::hreal. fst u = hreal_add (snd u) d),
- hreal_of_num 0)
- (hreal_of_num 0,
- hreal_inv (SOME d::hreal. snd u = hreal_add (fst u) d)))"
+%u. if fst u = snd u then (hreal_of_num 0, hreal_of_num 0)
+ else if hreal_le (snd u) (fst u)
+ then (hreal_inv (SOME d. fst u = hreal_add (snd u) d),
+ hreal_of_num 0)
+ else (hreal_of_num 0,
+ hreal_inv (SOME d. snd u = hreal_add (fst u) d))"
lemma DEF_treal_inv: "treal_inv =
-(%u::hreal * hreal.
- COND (fst u = snd u) (hreal_of_num 0, hreal_of_num 0)
- (COND (hreal_le (snd u) (fst u))
- (hreal_inv (SOME d::hreal. fst u = hreal_add (snd u) d),
- hreal_of_num 0)
- (hreal_of_num 0,
- hreal_inv (SOME d::hreal. snd u = hreal_add (fst u) d))))"
+(%u. if fst u = snd u then (hreal_of_num 0, hreal_of_num 0)
+ else if hreal_le (snd u) (fst u)
+ then (hreal_inv (SOME d. fst u = hreal_add (snd u) d),
+ hreal_of_num 0)
+ else (hreal_of_num 0,
+ hreal_inv (SOME d. snd u = hreal_add (fst u) d)))"
by (import hollight DEF_treal_inv)
-definition treal_eq :: "hreal * hreal => hreal * hreal => bool" where
- "treal_eq ==
-%(u::hreal * hreal) ua::hreal * hreal.
- hreal_add (fst u) (snd ua) = hreal_add (fst ua) (snd u)"
-
-lemma DEF_treal_eq: "treal_eq =
-(%(u::hreal * hreal) ua::hreal * hreal.
- hreal_add (fst u) (snd ua) = hreal_add (fst ua) (snd u))"
+definition
+ treal_eq :: "hreal * hreal => hreal * hreal => bool" where
+ "treal_eq == %u ua. hreal_add (fst u) (snd ua) = hreal_add (fst ua) (snd u)"
+
+lemma DEF_treal_eq: "treal_eq = (%u ua. hreal_add (fst u) (snd ua) = hreal_add (fst ua) (snd u))"
by (import hollight DEF_treal_eq)
-lemma TREAL_EQ_REFL: "ALL x::hreal * hreal. treal_eq x x"
+lemma TREAL_EQ_REFL: "treal_eq x x"
by (import hollight TREAL_EQ_REFL)
-lemma TREAL_EQ_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_eq x y = treal_eq y x"
+lemma TREAL_EQ_SYM: "treal_eq x y = treal_eq y x"
by (import hollight TREAL_EQ_SYM)
-lemma TREAL_EQ_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_eq x y & treal_eq y z --> treal_eq x z"
+lemma TREAL_EQ_TRANS: "treal_eq x y & treal_eq y z ==> treal_eq x z"
by (import hollight TREAL_EQ_TRANS)
-lemma TREAL_EQ_AP: "ALL (x::hreal * hreal) y::hreal * hreal. x = y --> treal_eq x y"
+lemma TREAL_EQ_AP: "x = xa ==> treal_eq x xa"
by (import hollight TREAL_EQ_AP)
-lemma TREAL_OF_NUM_EQ: "ALL (x::nat) xa::nat. treal_eq (treal_of_num x) (treal_of_num xa) = (x = xa)"
+lemma TREAL_OF_NUM_EQ: "treal_eq (treal_of_num x) (treal_of_num xa) = (x = xa)"
by (import hollight TREAL_OF_NUM_EQ)
-lemma TREAL_OF_NUM_LE: "ALL (x::nat) xa::nat. treal_le (treal_of_num x) (treal_of_num xa) = <= x xa"
+lemma TREAL_OF_NUM_LE: "treal_le (treal_of_num x) (treal_of_num xa) = (x <= xa)"
by (import hollight TREAL_OF_NUM_LE)
-lemma TREAL_OF_NUM_ADD: "ALL (x::nat) xa::nat.
- treal_eq (treal_add (treal_of_num x) (treal_of_num xa))
- (treal_of_num (x + xa))"
+lemma TREAL_OF_NUM_ADD: "treal_eq (treal_add (treal_of_num x) (treal_of_num xa))
+ (treal_of_num (x + xa))"
by (import hollight TREAL_OF_NUM_ADD)
-lemma TREAL_OF_NUM_MUL: "ALL (x::nat) xa::nat.
- treal_eq (treal_mul (treal_of_num x) (treal_of_num xa))
- (treal_of_num (x * xa))"
+lemma TREAL_OF_NUM_MUL: "treal_eq (treal_mul (treal_of_num x) (treal_of_num xa))
+ (treal_of_num (x * xa))"
by (import hollight TREAL_OF_NUM_MUL)
-lemma TREAL_ADD_SYM_EQ: "ALL (x::hreal * hreal) y::hreal * hreal. treal_add x y = treal_add y x"
+lemma TREAL_ADD_SYM_EQ: "treal_add x y = treal_add y x"
by (import hollight TREAL_ADD_SYM_EQ)
-lemma TREAL_MUL_SYM_EQ: "ALL (x::hreal * hreal) y::hreal * hreal. treal_mul x y = treal_mul y x"
+lemma TREAL_MUL_SYM_EQ: "treal_mul x y = treal_mul y x"
by (import hollight TREAL_MUL_SYM_EQ)
-lemma TREAL_ADD_SYM: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_eq (treal_add x y) (treal_add y x)"
+lemma TREAL_ADD_SYM: "treal_eq (treal_add x y) (treal_add y x)"
by (import hollight TREAL_ADD_SYM)
-lemma TREAL_ADD_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_eq (treal_add x (treal_add y z)) (treal_add (treal_add x y) z)"
+lemma TREAL_ADD_ASSOC: "treal_eq (treal_add x (treal_add y z)) (treal_add (treal_add x y) z)"
by (import hollight TREAL_ADD_ASSOC)
-lemma TREAL_ADD_LID: "ALL x::hreal * hreal. treal_eq (treal_add (treal_of_num 0) x) x"
+lemma TREAL_ADD_LID: "treal_eq (treal_add (treal_of_num 0) x) x"
by (import hollight TREAL_ADD_LID)
-lemma TREAL_ADD_LINV: "ALL x::hreal * hreal. treal_eq (treal_add (treal_neg x) x) (treal_of_num 0)"
+lemma TREAL_ADD_LINV: "treal_eq (treal_add (treal_neg x) x) (treal_of_num 0)"
by (import hollight TREAL_ADD_LINV)
-lemma TREAL_MUL_SYM: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_eq (treal_mul x y) (treal_mul y x)"
+lemma TREAL_MUL_SYM: "treal_eq (treal_mul x xa) (treal_mul xa x)"
by (import hollight TREAL_MUL_SYM)
-lemma TREAL_MUL_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_eq (treal_mul x (treal_mul y z)) (treal_mul (treal_mul x y) z)"
+lemma TREAL_MUL_ASSOC: "treal_eq (treal_mul x (treal_mul y z)) (treal_mul (treal_mul x y) z)"
by (import hollight TREAL_MUL_ASSOC)
-lemma TREAL_MUL_LID: "ALL x::hreal * hreal.
- treal_eq (treal_mul (treal_of_num (NUMERAL_BIT1 0)) x) x"
+lemma TREAL_MUL_LID: "treal_eq (treal_mul (treal_of_num 1) x) x"
by (import hollight TREAL_MUL_LID)
-lemma TREAL_ADD_LDISTRIB: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_eq (treal_mul x (treal_add y z))
- (treal_add (treal_mul x y) (treal_mul x z))"
+lemma TREAL_ADD_LDISTRIB: "treal_eq (treal_mul x (treal_add y z))
+ (treal_add (treal_mul x y) (treal_mul x z))"
by (import hollight TREAL_ADD_LDISTRIB)
-lemma TREAL_LE_REFL: "ALL x::hreal * hreal. treal_le x x"
+lemma TREAL_LE_REFL: "treal_le x x"
by (import hollight TREAL_LE_REFL)
-lemma TREAL_LE_ANTISYM: "ALL (x::hreal * hreal) y::hreal * hreal.
- (treal_le x y & treal_le y x) = treal_eq x y"
+lemma TREAL_LE_ANTISYM: "(treal_le x y & treal_le y x) = treal_eq x y"
by (import hollight TREAL_LE_ANTISYM)
-lemma TREAL_LE_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_le x y & treal_le y z --> treal_le x z"
+lemma TREAL_LE_TRANS: "treal_le x y & treal_le y z ==> treal_le x z"
by (import hollight TREAL_LE_TRANS)
-lemma TREAL_LE_TOTAL: "ALL (x::hreal * hreal) y::hreal * hreal. treal_le x y | treal_le y x"
+lemma TREAL_LE_TOTAL: "treal_le x y | treal_le y x"
by (import hollight TREAL_LE_TOTAL)
-lemma TREAL_LE_LADD_IMP: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_le y z --> treal_le (treal_add x y) (treal_add x z)"
+lemma TREAL_LE_LADD_IMP: "treal_le y z ==> treal_le (treal_add x y) (treal_add x z)"
by (import hollight TREAL_LE_LADD_IMP)
-lemma TREAL_LE_MUL: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_le (treal_of_num 0) x & treal_le (treal_of_num 0) y -->
- treal_le (treal_of_num 0) (treal_mul x y)"
+lemma TREAL_LE_MUL: "treal_le (treal_of_num 0) x & treal_le (treal_of_num 0) y
+==> treal_le (treal_of_num 0) (treal_mul x y)"
by (import hollight TREAL_LE_MUL)
lemma TREAL_INV_0: "treal_eq (treal_inv (treal_of_num 0)) (treal_of_num 0)"
by (import hollight TREAL_INV_0)
-lemma TREAL_MUL_LINV: "ALL x::hreal * hreal.
- ~ treal_eq x (treal_of_num 0) -->
- treal_eq (treal_mul (treal_inv x) x) (treal_of_num (NUMERAL_BIT1 0))"
+lemma TREAL_MUL_LINV: "~ treal_eq x (treal_of_num 0)
+==> treal_eq (treal_mul (treal_inv x) x) (treal_of_num 1)"
by (import hollight TREAL_MUL_LINV)
-lemma TREAL_OF_NUM_WELLDEF: "ALL (m::nat) n::nat. m = n --> treal_eq (treal_of_num m) (treal_of_num n)"
+lemma TREAL_OF_NUM_WELLDEF: "m = n ==> treal_eq (treal_of_num m) (treal_of_num n)"
by (import hollight TREAL_OF_NUM_WELLDEF)
-lemma TREAL_NEG_WELLDEF: "ALL (x1::hreal * hreal) x2::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_neg x1) (treal_neg x2)"
+lemma TREAL_NEG_WELLDEF: "treal_eq x1 x2 ==> treal_eq (treal_neg x1) (treal_neg x2)"
by (import hollight TREAL_NEG_WELLDEF)
-lemma TREAL_ADD_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_add x1 y) (treal_add x2 y)"
+lemma TREAL_ADD_WELLDEFR: "treal_eq x1 x2 ==> treal_eq (treal_add x1 y) (treal_add x2 y)"
by (import hollight TREAL_ADD_WELLDEFR)
-lemma TREAL_ADD_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 -->
- treal_eq (treal_add x1 y1) (treal_add x2 y2)"
+lemma TREAL_ADD_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2
+==> treal_eq (treal_add x1 y1) (treal_add x2 y2)"
by (import hollight TREAL_ADD_WELLDEF)
-lemma TREAL_MUL_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_mul x1 y) (treal_mul x2 y)"
+lemma TREAL_MUL_WELLDEFR: "treal_eq x1 x2 ==> treal_eq (treal_mul x1 y) (treal_mul x2 y)"
by (import hollight TREAL_MUL_WELLDEFR)
-lemma TREAL_MUL_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 -->
- treal_eq (treal_mul x1 y1) (treal_mul x2 y2)"
+lemma TREAL_MUL_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2
+==> treal_eq (treal_mul x1 y1) (treal_mul x2 y2)"
by (import hollight TREAL_MUL_WELLDEF)
-lemma TREAL_EQ_IMP_LE: "ALL (x::hreal * hreal) y::hreal * hreal. treal_eq x y --> treal_le x y"
+lemma TREAL_EQ_IMP_LE: "treal_eq x y ==> treal_le x y"
by (import hollight TREAL_EQ_IMP_LE)
-lemma TREAL_LE_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 --> treal_le x1 y1 = treal_le x2 y2"
+lemma TREAL_LE_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2 ==> treal_le x1 y1 = treal_le x2 y2"
by (import hollight TREAL_LE_WELLDEF)
-lemma TREAL_INV_WELLDEF: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_eq x y --> treal_eq (treal_inv x) (treal_inv y)"
+lemma TREAL_INV_WELLDEF: "treal_eq x y ==> treal_eq (treal_inv x) (treal_inv y)"
by (import hollight TREAL_INV_WELLDEF)
-typedef (open) real = "{s::hreal * hreal => bool. EX x::hreal * hreal. s = treal_eq x}" morphisms "dest_real" "mk_real"
- apply (rule light_ex_imp_nonempty[where t="treal_eq (x::hreal * hreal)"])
+typedef (open) real = "{s. EX x. s = treal_eq x}" morphisms "dest_real" "mk_real"
+ apply (rule light_ex_imp_nonempty[where t="treal_eq x"])
by (import hollight TYDEF_real)
syntax
@@ -3834,3079 +2037,4606 @@
[where a="a :: hollight.real" and r=r ,
OF type_definition_real]
-definition real_of_num :: "nat => hollight.real" where
- "real_of_num == %m::nat. mk_real (treal_eq (treal_of_num m))"
-
-lemma DEF_real_of_num: "real_of_num = (%m::nat. mk_real (treal_eq (treal_of_num m)))"
+definition
+ real_of_num :: "nat => hollight.real" where
+ "real_of_num == %m. mk_real (treal_eq (treal_of_num m))"
+
+lemma DEF_real_of_num: "real_of_num = (%m. mk_real (treal_eq (treal_of_num m)))"
by (import hollight DEF_real_of_num)
-definition real_neg :: "hollight.real => hollight.real" where
+definition
+ real_neg :: "hollight.real => hollight.real" where
"real_neg ==
-%x1::hollight.real.
- mk_real
- (%u::hreal * hreal.
- EX x1a::hreal * hreal.
- treal_eq (treal_neg x1a) u & dest_real x1 x1a)"
+%x1. mk_real (%u. EX x1a. treal_eq (treal_neg x1a) u & dest_real x1 x1a)"
lemma DEF_real_neg: "real_neg =
-(%x1::hollight.real.
- mk_real
- (%u::hreal * hreal.
- EX x1a::hreal * hreal.
- treal_eq (treal_neg x1a) u & dest_real x1 x1a))"
+(%x1. mk_real (%u. EX x1a. treal_eq (treal_neg x1a) u & dest_real x1 x1a))"
by (import hollight DEF_real_neg)
-definition real_add :: "hollight.real => hollight.real => hollight.real" where
+definition
+ real_add :: "hollight.real => hollight.real => hollight.real" where
"real_add ==
-%(x1::hollight.real) y1::hollight.real.
+%x1 y1.
mk_real
- (%u::hreal * hreal.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
- treal_eq (treal_add x1a y1a) u &
- dest_real x1 x1a & dest_real y1 y1a)"
+ (%u. EX x1a y1a.
+ treal_eq (treal_add x1a y1a) u &
+ dest_real x1 x1a & dest_real y1 y1a)"
lemma DEF_real_add: "real_add =
-(%(x1::hollight.real) y1::hollight.real.
+(%x1 y1.
mk_real
- (%u::hreal * hreal.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
- treal_eq (treal_add x1a y1a) u &
- dest_real x1 x1a & dest_real y1 y1a))"
+ (%u. EX x1a y1a.
+ treal_eq (treal_add x1a y1a) u &
+ dest_real x1 x1a & dest_real y1 y1a))"
by (import hollight DEF_real_add)
-definition real_mul :: "hollight.real => hollight.real => hollight.real" where
+definition
+ real_mul :: "hollight.real => hollight.real => hollight.real" where
"real_mul ==
-%(x1::hollight.real) y1::hollight.real.
+%x1 y1.
mk_real
- (%u::hreal * hreal.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
- treal_eq (treal_mul x1a y1a) u &
- dest_real x1 x1a & dest_real y1 y1a)"
+ (%u. EX x1a y1a.
+ treal_eq (treal_mul x1a y1a) u &
+ dest_real x1 x1a & dest_real y1 y1a)"
lemma DEF_real_mul: "real_mul =
-(%(x1::hollight.real) y1::hollight.real.
+(%x1 y1.
mk_real
- (%u::hreal * hreal.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
- treal_eq (treal_mul x1a y1a) u &
- dest_real x1 x1a & dest_real y1 y1a))"
+ (%u. EX x1a y1a.
+ treal_eq (treal_mul x1a y1a) u &
+ dest_real x1 x1a & dest_real y1 y1a))"
by (import hollight DEF_real_mul)
-definition real_le :: "hollight.real => hollight.real => bool" where
+definition
+ real_le :: "hollight.real => hollight.real => bool" where
"real_le ==
-%(x1::hollight.real) y1::hollight.real.
- SOME u::bool.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
- treal_le x1a y1a = u & dest_real x1 x1a & dest_real y1 y1a"
+%x1 y1.
+ SOME u.
+ EX x1a y1a. treal_le x1a y1a = u & dest_real x1 x1a & dest_real y1 y1a"
lemma DEF_real_le: "real_le =
-(%(x1::hollight.real) y1::hollight.real.
- SOME u::bool.
- EX (x1a::hreal * hreal) y1a::hreal * hreal.
+(%x1 y1.
+ SOME u.
+ EX x1a y1a.
treal_le x1a y1a = u & dest_real x1 x1a & dest_real y1 y1a)"
by (import hollight DEF_real_le)
-definition real_inv :: "hollight.real => hollight.real" where
+definition
+ real_inv :: "hollight.real => hollight.real" where
"real_inv ==
-%x::hollight.real.
- mk_real
- (%u::hreal * hreal.
- EX xa::hreal * hreal. treal_eq (treal_inv xa) u & dest_real x xa)"
+%x. mk_real (%u. EX xa. treal_eq (treal_inv xa) u & dest_real x xa)"
lemma DEF_real_inv: "real_inv =
-(%x::hollight.real.
- mk_real
- (%u::hreal * hreal.
- EX xa::hreal * hreal. treal_eq (treal_inv xa) u & dest_real x xa))"
+(%x. mk_real (%u. EX xa. treal_eq (treal_inv xa) u & dest_real x xa))"
by (import hollight DEF_real_inv)
-definition real_sub :: "hollight.real => hollight.real => hollight.real" where
- "real_sub == %(u::hollight.real) ua::hollight.real. real_add u (real_neg ua)"
-
-lemma DEF_real_sub: "real_sub = (%(u::hollight.real) ua::hollight.real. real_add u (real_neg ua))"
+definition
+ real_sub :: "hollight.real => hollight.real => hollight.real" where
+ "real_sub == %u ua. real_add u (real_neg ua)"
+
+lemma DEF_real_sub: "real_sub = (%u ua. real_add u (real_neg ua))"
by (import hollight DEF_real_sub)
-definition real_lt :: "hollight.real => hollight.real => bool" where
- "real_lt == %(u::hollight.real) ua::hollight.real. ~ real_le ua u"
-
-lemma DEF_real_lt: "real_lt = (%(u::hollight.real) ua::hollight.real. ~ real_le ua u)"
+definition
+ real_lt :: "hollight.real => hollight.real => bool" where
+ "real_lt == %u ua. ~ real_le ua u"
+
+lemma DEF_real_lt: "real_lt = (%u ua. ~ real_le ua u)"
by (import hollight DEF_real_lt)
-consts
- real_ge :: "hollight.real => hollight.real => bool"
-
-defs
- real_ge_def: "hollight.real_ge == %(u::hollight.real) ua::hollight.real. real_le ua u"
-
-lemma DEF_real_ge: "hollight.real_ge = (%(u::hollight.real) ua::hollight.real. real_le ua u)"
+definition
+ real_ge :: "hollight.real => hollight.real => bool" where
+ "real_ge == %u ua. real_le ua u"
+
+lemma DEF_real_ge: "real_ge = (%u ua. real_le ua u)"
by (import hollight DEF_real_ge)
-consts
- real_gt :: "hollight.real => hollight.real => bool"
-
-defs
- real_gt_def: "hollight.real_gt == %(u::hollight.real) ua::hollight.real. real_lt ua u"
-
-lemma DEF_real_gt: "hollight.real_gt = (%(u::hollight.real) ua::hollight.real. real_lt ua u)"
+definition
+ real_gt :: "hollight.real => hollight.real => bool" where
+ "real_gt == %u ua. real_lt ua u"
+
+lemma DEF_real_gt: "real_gt = (%u ua. real_lt ua u)"
by (import hollight DEF_real_gt)
-definition real_abs :: "hollight.real => hollight.real" where
- "real_abs ==
-%u::hollight.real. COND (real_le (real_of_num 0) u) u (real_neg u)"
-
-lemma DEF_real_abs: "real_abs =
-(%u::hollight.real. COND (real_le (real_of_num 0) u) u (real_neg u))"
+definition
+ real_abs :: "hollight.real => hollight.real" where
+ "real_abs == %u. if real_le (real_of_num 0) u then u else real_neg u"
+
+lemma DEF_real_abs: "real_abs = (%u. if real_le (real_of_num 0) u then u else real_neg u)"
by (import hollight DEF_real_abs)
-definition real_pow :: "hollight.real => nat => hollight.real" where
+definition
+ real_pow :: "hollight.real => nat => hollight.real" where
"real_pow ==
-SOME real_pow::hollight.real => nat => hollight.real.
- (ALL x::hollight.real. real_pow x 0 = real_of_num (NUMERAL_BIT1 0)) &
- (ALL (x::hollight.real) n::nat.
- real_pow x (Suc n) = real_mul x (real_pow x n))"
+SOME real_pow.
+ (ALL x. real_pow x 0 = real_of_num 1) &
+ (ALL x n. real_pow x (Suc n) = real_mul x (real_pow x n))"
lemma DEF_real_pow: "real_pow =
-(SOME real_pow::hollight.real => nat => hollight.real.
- (ALL x::hollight.real. real_pow x 0 = real_of_num (NUMERAL_BIT1 0)) &
- (ALL (x::hollight.real) n::nat.
- real_pow x (Suc n) = real_mul x (real_pow x n)))"
+(SOME real_pow.
+ (ALL x. real_pow x 0 = real_of_num 1) &
+ (ALL x n. real_pow x (Suc n) = real_mul x (real_pow x n)))"
by (import hollight DEF_real_pow)
-definition real_div :: "hollight.real => hollight.real => hollight.real" where
- "real_div == %(u::hollight.real) ua::hollight.real. real_mul u (real_inv ua)"
-
-lemma DEF_real_div: "real_div = (%(u::hollight.real) ua::hollight.real. real_mul u (real_inv ua))"
+definition
+ real_div :: "hollight.real => hollight.real => hollight.real" where
+ "real_div == %u ua. real_mul u (real_inv ua)"
+
+lemma DEF_real_div: "real_div = (%u ua. real_mul u (real_inv ua))"
by (import hollight DEF_real_div)
-definition real_max :: "hollight.real => hollight.real => hollight.real" where
- "real_max == %(u::hollight.real) ua::hollight.real. COND (real_le u ua) ua u"
-
-lemma DEF_real_max: "real_max = (%(u::hollight.real) ua::hollight.real. COND (real_le u ua) ua u)"
+definition
+ real_max :: "hollight.real => hollight.real => hollight.real" where
+ "real_max == %u ua. if real_le u ua then ua else u"
+
+lemma DEF_real_max: "real_max = (%u ua. if real_le u ua then ua else u)"
by (import hollight DEF_real_max)
-definition real_min :: "hollight.real => hollight.real => hollight.real" where
- "real_min == %(u::hollight.real) ua::hollight.real. COND (real_le u ua) u ua"
-
-lemma DEF_real_min: "real_min = (%(u::hollight.real) ua::hollight.real. COND (real_le u ua) u ua)"
+definition
+ real_min :: "hollight.real => hollight.real => hollight.real" where
+ "real_min == %u ua. if real_le u ua then u else ua"
+
+lemma DEF_real_min: "real_min = (%u ua. if real_le u ua then u else ua)"
by (import hollight DEF_real_min)
-lemma REAL_HREAL_LEMMA1: "EX x::hreal => hollight.real.
- (ALL xa::hollight.real.
- real_le (real_of_num 0) xa = (EX y::hreal. xa = x y)) &
- (ALL (y::hreal) z::hreal. hreal_le y z = real_le (x y) (x z))"
+lemma REAL_HREAL_LEMMA1: "EX x. (ALL xa. real_le (real_of_num 0) xa = (EX y. xa = x y)) &
+ (ALL y z. hreal_le y z = real_le (x y) (x z))"
by (import hollight REAL_HREAL_LEMMA1)
-lemma REAL_HREAL_LEMMA2: "EX (x::hollight.real => hreal) r::hreal => hollight.real.
- (ALL xa::hreal. x (r xa) = xa) &
- (ALL xa::hollight.real. real_le (real_of_num 0) xa --> r (x xa) = xa) &
- (ALL x::hreal. real_le (real_of_num 0) (r x)) &
- (ALL (x::hreal) y::hreal. hreal_le x y = real_le (r x) (r y))"
+lemma REAL_HREAL_LEMMA2: "EX x r.
+ (ALL xa. x (r xa) = xa) &
+ (ALL xa. real_le (real_of_num 0) xa --> r (x xa) = xa) &
+ (ALL x. real_le (real_of_num 0) (r x)) &
+ (ALL x y. hreal_le x y = real_le (r x) (r y))"
by (import hollight REAL_HREAL_LEMMA2)
-lemma REAL_COMPLETE_SOMEPOS: "ALL P::hollight.real => bool.
- (EX x::hollight.real. P x & real_le (real_of_num 0) x) &
- (EX M::hollight.real. ALL x::hollight.real. P x --> real_le x M) -->
- (EX M::hollight.real.
- (ALL x::hollight.real. P x --> real_le x M) &
- (ALL M'::hollight.real.
- (ALL x::hollight.real. P x --> real_le x M') --> real_le M M'))"
+lemma REAL_COMPLETE_SOMEPOS: "(EX x. P x & real_le (real_of_num 0) x) & (EX M. ALL x. P x --> real_le x M)
+==> EX M. (ALL x. P x --> real_le x M) &
+ (ALL M'. (ALL x. P x --> real_le x M') --> real_le M M')"
by (import hollight REAL_COMPLETE_SOMEPOS)
-lemma REAL_COMPLETE: "ALL P::hollight.real => bool.
- Ex P &
- (EX M::hollight.real. ALL x::hollight.real. P x --> real_le x M) -->
- (EX M::hollight.real.
- (ALL x::hollight.real. P x --> real_le x M) &
- (ALL M'::hollight.real.
- (ALL x::hollight.real. P x --> real_le x M') --> real_le M M'))"
+lemma REAL_COMPLETE: "Ex P & (EX M. ALL x. P x --> real_le x M)
+==> EX M. (ALL x. P x --> real_le x M) &
+ (ALL M'. (ALL x. P x --> real_le x M') --> real_le M M')"
by (import hollight REAL_COMPLETE)
-lemma REAL_ADD_AC: "real_add (m::hollight.real) (n::hollight.real) = real_add n m &
-real_add (real_add m n) (p::hollight.real) = real_add m (real_add n p) &
+lemma REAL_ADD_AC: "real_add m n = real_add n m &
+real_add (real_add m n) p = real_add m (real_add n p) &
real_add m (real_add n p) = real_add n (real_add m p)"
by (import hollight REAL_ADD_AC)
-lemma REAL_ADD_RINV: "ALL x::hollight.real. real_add x (real_neg x) = real_of_num 0"
+lemma REAL_ADD_RINV: "real_add x (real_neg x) = real_of_num 0"
by (import hollight REAL_ADD_RINV)
-lemma REAL_EQ_ADD_LCANCEL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- (real_add x y = real_add x z) = (y = z)"
+lemma REAL_EQ_ADD_LCANCEL: "(real_add x y = real_add x z) = (y = z)"
by (import hollight REAL_EQ_ADD_LCANCEL)
-lemma REAL_EQ_ADD_RCANCEL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- (real_add x z = real_add y z) = (x = y)"
+lemma REAL_EQ_ADD_RCANCEL: "(real_add x z = real_add y z) = (x = y)"
by (import hollight REAL_EQ_ADD_RCANCEL)
-lemma REAL_MUL_RZERO: "ALL x::hollight.real. real_mul x (real_of_num 0) = real_of_num 0"
+lemma REAL_MUL_RZERO: "real_mul x (real_of_num 0) = real_of_num 0"
by (import hollight REAL_MUL_RZERO)
-lemma REAL_MUL_LZERO: "ALL x::hollight.real. real_mul (real_of_num 0) x = real_of_num 0"
+lemma REAL_MUL_LZERO: "real_mul (real_of_num 0) x = real_of_num 0"
by (import hollight REAL_MUL_LZERO)
-lemma REAL_NEG_NEG: "ALL x::hollight.real. real_neg (real_neg x) = x"
+lemma REAL_NEG_NEG: "real_neg (real_neg x) = x"
by (import hollight REAL_NEG_NEG)
-lemma REAL_MUL_RNEG: "ALL (x::hollight.real) y::hollight.real.
- real_mul x (real_neg y) = real_neg (real_mul x y)"
+lemma REAL_MUL_RNEG: "real_mul x (real_neg y) = real_neg (real_mul x y)"
by (import hollight REAL_MUL_RNEG)
-lemma REAL_MUL_LNEG: "ALL (x::hollight.real) y::hollight.real.
- real_mul (real_neg x) y = real_neg (real_mul x y)"
+lemma REAL_MUL_LNEG: "real_mul (real_neg x) y = real_neg (real_mul x y)"
by (import hollight REAL_MUL_LNEG)
-lemma REAL_NEG_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_neg (real_add x y) = real_add (real_neg x) (real_neg y)"
+lemma REAL_NEG_ADD: "real_neg (real_add x y) = real_add (real_neg x) (real_neg y)"
by (import hollight REAL_NEG_ADD)
-lemma REAL_ADD_RID: "ALL x::hollight.real. real_add x (real_of_num 0) = x"
+lemma REAL_ADD_RID: "real_add x (real_of_num 0) = x"
by (import hollight REAL_ADD_RID)
lemma REAL_NEG_0: "real_neg (real_of_num 0) = real_of_num 0"
by (import hollight REAL_NEG_0)
-lemma REAL_LE_LNEG: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_neg x) y = real_le (real_of_num 0) (real_add x y)"
+lemma REAL_LE_LNEG: "real_le (real_neg x) y = real_le (real_of_num 0) (real_add x y)"
by (import hollight REAL_LE_LNEG)
-lemma REAL_LE_NEG2: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_neg x) (real_neg y) = real_le y x"
+lemma REAL_LE_NEG2: "real_le (real_neg x) (real_neg y) = real_le y x"
by (import hollight REAL_LE_NEG2)
-lemma REAL_LE_RNEG: "ALL (x::hollight.real) y::hollight.real.
- real_le x (real_neg y) = real_le (real_add x y) (real_of_num 0)"
+lemma REAL_LE_RNEG: "real_le x (real_neg y) = real_le (real_add x y) (real_of_num 0)"
by (import hollight REAL_LE_RNEG)
-lemma REAL_OF_NUM_POW: "ALL (x::nat) n::nat. real_pow (real_of_num x) n = real_of_num (EXP x n)"
+lemma REAL_OF_NUM_POW: "real_pow (real_of_num x) n = real_of_num (x ^ n)"
by (import hollight REAL_OF_NUM_POW)
-lemma REAL_POW_NEG: "ALL (x::hollight.real) n::nat.
- real_pow (real_neg x) n =
- COND (EVEN n) (real_pow x n) (real_neg (real_pow x n))"
+lemma REAL_POW_NEG: "real_pow (real_neg x) n =
+(if even n then real_pow x n else real_neg (real_pow x n))"
by (import hollight REAL_POW_NEG)
-lemma REAL_ABS_NUM: "ALL x::nat. real_abs (real_of_num x) = real_of_num x"
+lemma REAL_ABS_NUM: "real_abs (real_of_num x) = real_of_num x"
by (import hollight REAL_ABS_NUM)
-lemma REAL_ABS_NEG: "ALL x::hollight.real. real_abs (real_neg x) = real_abs x"
+lemma REAL_ABS_NEG: "real_abs (real_neg x) = real_abs x"
by (import hollight REAL_ABS_NEG)
-lemma REAL_LTE_TOTAL: "ALL (x::hollight.real) xa::hollight.real. real_lt x xa | real_le xa x"
+lemma REAL_LTE_TOTAL: "real_lt x xa | real_le xa x"
by (import hollight REAL_LTE_TOTAL)
-lemma REAL_LET_TOTAL: "ALL (x::hollight.real) xa::hollight.real. real_le x xa | real_lt xa x"
+lemma REAL_LET_TOTAL: "real_le x xa | real_lt xa x"
by (import hollight REAL_LET_TOTAL)
-lemma REAL_LT_IMP_LE: "ALL (x::hollight.real) y::hollight.real. real_lt x y --> real_le x y"
+lemma REAL_LT_IMP_LE: "real_lt x y ==> real_le x y"
by (import hollight REAL_LT_IMP_LE)
-lemma REAL_LTE_TRANS: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt x y & real_le y z --> real_lt x z"
+lemma REAL_LTE_TRANS: "real_lt x y & real_le y z ==> real_lt x z"
by (import hollight REAL_LTE_TRANS)
-lemma REAL_LET_TRANS: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le x y & real_lt y z --> real_lt x z"
+lemma REAL_LET_TRANS: "real_le x y & real_lt y z ==> real_lt x z"
by (import hollight REAL_LET_TRANS)
-lemma REAL_LT_TRANS: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt x y & real_lt y z --> real_lt x z"
+lemma REAL_LT_TRANS: "real_lt x y & real_lt y z ==> real_lt x z"
by (import hollight REAL_LT_TRANS)
-lemma REAL_LE_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_of_num 0) x & real_le (real_of_num 0) y -->
- real_le (real_of_num 0) (real_add x y)"
+lemma REAL_LE_ADD: "real_le (real_of_num 0) x & real_le (real_of_num 0) y
+==> real_le (real_of_num 0) (real_add x y)"
by (import hollight REAL_LE_ADD)
-lemma REAL_LTE_ANTISYM: "ALL (x::hollight.real) y::hollight.real. ~ (real_lt x y & real_le y x)"
+lemma REAL_LTE_ANTISYM: "~ (real_lt x y & real_le y x)"
by (import hollight REAL_LTE_ANTISYM)
-lemma REAL_SUB_LE: "ALL (x::hollight.real) xa::hollight.real.
- real_le (real_of_num 0) (real_sub x xa) = real_le xa x"
+lemma REAL_SUB_LE: "real_le (real_of_num 0) (real_sub x xa) = real_le xa x"
by (import hollight REAL_SUB_LE)
-lemma REAL_NEG_SUB: "ALL (x::hollight.real) xa::hollight.real.
- real_neg (real_sub x xa) = real_sub xa x"
+lemma REAL_NEG_SUB: "real_neg (real_sub x xa) = real_sub xa x"
by (import hollight REAL_NEG_SUB)
-lemma REAL_LE_LT: "ALL (x::hollight.real) xa::hollight.real.
- real_le x xa = (real_lt x xa | x = xa)"
+lemma REAL_LE_LT: "real_le x xa = (real_lt x xa | x = xa)"
by (import hollight REAL_LE_LT)
-lemma REAL_SUB_LT: "ALL (x::hollight.real) xa::hollight.real.
- real_lt (real_of_num 0) (real_sub x xa) = real_lt xa x"
+lemma REAL_SUB_LT: "real_lt (real_of_num 0) (real_sub x xa) = real_lt xa x"
by (import hollight REAL_SUB_LT)
-lemma REAL_NOT_LT: "ALL (x::hollight.real) xa::hollight.real. (~ real_lt x xa) = real_le xa x"
+lemma REAL_NOT_LT: "(~ real_lt x xa) = real_le xa x"
by (import hollight REAL_NOT_LT)
-lemma REAL_SUB_0: "ALL (x::hollight.real) y::hollight.real.
- (real_sub x y = real_of_num 0) = (x = y)"
+lemma REAL_SUB_0: "(real_sub x y = real_of_num 0) = (x = y)"
by (import hollight REAL_SUB_0)
-lemma REAL_LT_LE: "ALL (x::hollight.real) y::hollight.real.
- real_lt x y = (real_le x y & x ~= y)"
+lemma REAL_LT_LE: "real_lt x y = (real_le x y & x ~= y)"
by (import hollight REAL_LT_LE)
-lemma REAL_LT_REFL: "ALL x::hollight.real. ~ real_lt x x"
+lemma REAL_LT_REFL: "~ real_lt x x"
by (import hollight REAL_LT_REFL)
-lemma REAL_LTE_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num 0) x & real_le (real_of_num 0) y -->
- real_lt (real_of_num 0) (real_add x y)"
+lemma REAL_LTE_ADD: "real_lt (real_of_num 0) x & real_le (real_of_num 0) y
+==> real_lt (real_of_num 0) (real_add x y)"
by (import hollight REAL_LTE_ADD)
-lemma REAL_LET_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_of_num 0) x & real_lt (real_of_num 0) y -->
- real_lt (real_of_num 0) (real_add x y)"
+lemma REAL_LET_ADD: "real_le (real_of_num 0) x & real_lt (real_of_num 0) y
+==> real_lt (real_of_num 0) (real_add x y)"
by (import hollight REAL_LET_ADD)
-lemma REAL_LT_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
- real_lt (real_of_num 0) (real_add x y)"
+lemma REAL_LT_ADD: "real_lt (real_of_num 0) x & real_lt (real_of_num 0) y
+==> real_lt (real_of_num 0) (real_add x y)"
by (import hollight REAL_LT_ADD)
-lemma REAL_ENTIRE: "ALL (x::hollight.real) y::hollight.real.
- (real_mul x y = real_of_num 0) = (x = real_of_num 0 | y = real_of_num 0)"
+lemma REAL_ENTIRE: "(real_mul x y = real_of_num 0) = (x = real_of_num 0 | y = real_of_num 0)"
by (import hollight REAL_ENTIRE)
-lemma REAL_LE_NEGTOTAL: "ALL x::hollight.real.
- real_le (real_of_num 0) x | real_le (real_of_num 0) (real_neg x)"
+lemma REAL_LE_NEGTOTAL: "real_le (real_of_num 0) x | real_le (real_of_num 0) (real_neg x)"
by (import hollight REAL_LE_NEGTOTAL)
-lemma REAL_LE_SQUARE: "ALL x::hollight.real. real_le (real_of_num 0) (real_mul x x)"
+lemma REAL_LE_SQUARE: "real_le (real_of_num 0) (real_mul x x)"
by (import hollight REAL_LE_SQUARE)
-lemma REAL_MUL_RID: "ALL x::hollight.real. real_mul x (real_of_num (NUMERAL_BIT1 0)) = x"
+lemma REAL_MUL_RID: "real_mul x (real_of_num 1) = x"
by (import hollight REAL_MUL_RID)
-lemma REAL_POW_2: "ALL x::hollight.real.
- real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = real_mul x x"
+lemma REAL_POW_2: "real_pow x 2 = real_mul x x"
by (import hollight REAL_POW_2)
-lemma REAL_POLY_CLAUSES: "(ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_add x (real_add y z) = real_add (real_add x y) z) &
-(ALL (x::hollight.real) y::hollight.real. real_add x y = real_add y x) &
-(ALL x::hollight.real. real_add (real_of_num 0) x = x) &
-(ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_mul x (real_mul y z) = real_mul (real_mul x y) z) &
-(ALL (x::hollight.real) y::hollight.real. real_mul x y = real_mul y x) &
-(ALL x::hollight.real. real_mul (real_of_num (NUMERAL_BIT1 0)) x = x) &
-(ALL x::hollight.real. real_mul (real_of_num 0) x = real_of_num 0) &
-(ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
+lemma REAL_POLY_CLAUSES: "(ALL x y z. real_add x (real_add y z) = real_add (real_add x y) z) &
+(ALL x y. real_add x y = real_add y x) &
+(ALL x. real_add (real_of_num 0) x = x) &
+(ALL x y z. real_mul x (real_mul y z) = real_mul (real_mul x y) z) &
+(ALL x y. real_mul x y = real_mul y x) &
+(ALL x. real_mul (real_of_num 1) x = x) &
+(ALL x. real_mul (real_of_num 0) x = real_of_num 0) &
+(ALL x xa xb.
real_mul x (real_add xa xb) =
real_add (real_mul x xa) (real_mul x xb)) &
-(ALL x::hollight.real. real_pow x 0 = real_of_num (NUMERAL_BIT1 0)) &
-(ALL (x::hollight.real) xa::nat.
- real_pow x (Suc xa) = real_mul x (real_pow x xa))"
+(ALL x. real_pow x 0 = real_of_num 1) &
+(ALL x xa. real_pow x (Suc xa) = real_mul x (real_pow x xa))"
by (import hollight REAL_POLY_CLAUSES)
-lemma REAL_POLY_NEG_CLAUSES: "(ALL x::hollight.real.
- real_neg x = real_mul (real_neg (real_of_num (NUMERAL_BIT1 0))) x) &
-(ALL (x::hollight.real) xa::hollight.real.
- real_sub x xa =
- real_add x (real_mul (real_neg (real_of_num (NUMERAL_BIT1 0))) xa))"
+lemma REAL_POLY_NEG_CLAUSES: "(ALL x. real_neg x = real_mul (real_neg (real_of_num 1)) x) &
+(ALL x xa.
+ real_sub x xa = real_add x (real_mul (real_neg (real_of_num 1)) xa))"
by (import hollight REAL_POLY_NEG_CLAUSES)
-lemma REAL_POS: "ALL x::nat. real_le (real_of_num 0) (real_of_num x)"
+lemma REAL_POS: "real_le (real_of_num 0) (real_of_num x)"
by (import hollight REAL_POS)
-lemma REAL_OF_NUM_LT: "ALL (x::nat) xa::nat. real_lt (real_of_num x) (real_of_num xa) = < x xa"
+lemma REAL_OF_NUM_LT: "real_lt (real_of_num x) (real_of_num xa) = (x < xa)"
by (import hollight REAL_OF_NUM_LT)
-lemma REAL_OF_NUM_GE: "ALL (x::nat) xa::nat.
- hollight.real_ge (real_of_num x) (real_of_num xa) = >= x xa"
+lemma REAL_OF_NUM_GE: "real_ge (real_of_num x) (real_of_num xa) = (xa <= x)"
by (import hollight REAL_OF_NUM_GE)
-lemma REAL_OF_NUM_GT: "ALL (x::nat) xa::nat.
- hollight.real_gt (real_of_num x) (real_of_num xa) = > x xa"
+lemma REAL_OF_NUM_GT: "real_gt (real_of_num x) (real_of_num xa) = (xa < x)"
by (import hollight REAL_OF_NUM_GT)
-lemma REAL_OF_NUM_SUC: "ALL x::nat.
- real_add (real_of_num x) (real_of_num (NUMERAL_BIT1 0)) =
- real_of_num (Suc x)"
+lemma REAL_OF_NUM_MAX: "real_max (real_of_num x) (real_of_num xa) = real_of_num (max x xa)"
+ by (import hollight REAL_OF_NUM_MAX)
+
+lemma REAL_OF_NUM_MIN: "real_min (real_of_num x) (real_of_num xa) = real_of_num (min x xa)"
+ by (import hollight REAL_OF_NUM_MIN)
+
+lemma REAL_OF_NUM_SUC: "real_add (real_of_num x) (real_of_num 1) = real_of_num (Suc x)"
by (import hollight REAL_OF_NUM_SUC)
-lemma REAL_OF_NUM_SUB: "ALL (m::nat) n::nat.
- <= m n --> real_sub (real_of_num n) (real_of_num m) = real_of_num (n - m)"
+lemma REAL_OF_NUM_SUB: "m <= n ==> real_sub (real_of_num n) (real_of_num m) = real_of_num (n - m)"
by (import hollight REAL_OF_NUM_SUB)
-lemma REAL_MUL_AC: "real_mul (m::hollight.real) (n::hollight.real) = real_mul n m &
-real_mul (real_mul m n) (p::hollight.real) = real_mul m (real_mul n p) &
+lemma REAL_MUL_AC: "real_mul m n = real_mul n m &
+real_mul (real_mul m n) p = real_mul m (real_mul n p) &
real_mul m (real_mul n p) = real_mul n (real_mul m p)"
by (import hollight REAL_MUL_AC)
-lemma REAL_ADD_RDISTRIB: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_mul (real_add x y) z = real_add (real_mul x z) (real_mul y z)"
+lemma REAL_ADD_RDISTRIB: "real_mul (real_add x y) z = real_add (real_mul x z) (real_mul y z)"
by (import hollight REAL_ADD_RDISTRIB)
-lemma REAL_LT_LADD_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt y z --> real_lt (real_add x y) (real_add x z)"
+lemma REAL_LT_LADD_IMP: "real_lt y z ==> real_lt (real_add x y) (real_add x z)"
by (import hollight REAL_LT_LADD_IMP)
-lemma REAL_LT_MUL: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
- real_lt (real_of_num 0) (real_mul x y)"
+lemma REAL_LT_MUL: "real_lt (real_of_num 0) x & real_lt (real_of_num 0) y
+==> real_lt (real_of_num 0) (real_mul x y)"
by (import hollight REAL_LT_MUL)
-lemma REAL_EQ_ADD_LCANCEL_0: "ALL (x::hollight.real) y::hollight.real.
- (real_add x y = x) = (y = real_of_num 0)"
+lemma REAL_EQ_ADD_LCANCEL_0: "(real_add x y = x) = (y = real_of_num 0)"
by (import hollight REAL_EQ_ADD_LCANCEL_0)
-lemma REAL_EQ_ADD_RCANCEL_0: "ALL (x::hollight.real) y::hollight.real.
- (real_add x y = y) = (x = real_of_num 0)"
+lemma REAL_EQ_ADD_RCANCEL_0: "(real_add x y = y) = (x = real_of_num 0)"
by (import hollight REAL_EQ_ADD_RCANCEL_0)
-lemma REAL_LNEG_UNIQ: "ALL (x::hollight.real) y::hollight.real.
- (real_add x y = real_of_num 0) = (x = real_neg y)"
+lemma REAL_LNEG_UNIQ: "(real_add x y = real_of_num 0) = (x = real_neg y)"
by (import hollight REAL_LNEG_UNIQ)
-lemma REAL_RNEG_UNIQ: "ALL (x::hollight.real) y::hollight.real.
- (real_add x y = real_of_num 0) = (y = real_neg x)"
+lemma REAL_RNEG_UNIQ: "(real_add x y = real_of_num 0) = (y = real_neg x)"
by (import hollight REAL_RNEG_UNIQ)
-lemma REAL_NEG_LMUL: "ALL (x::hollight.real) y::hollight.real.
- real_neg (real_mul x y) = real_mul (real_neg x) y"
+lemma REAL_NEG_LMUL: "real_neg (real_mul x y) = real_mul (real_neg x) y"
by (import hollight REAL_NEG_LMUL)
-lemma REAL_NEG_RMUL: "ALL (x::hollight.real) y::hollight.real.
- real_neg (real_mul x y) = real_mul x (real_neg y)"
+lemma REAL_NEG_RMUL: "real_neg (real_mul x y) = real_mul x (real_neg y)"
by (import hollight REAL_NEG_RMUL)
-lemma REAL_NEGNEG: "ALL x::hollight.real. real_neg (real_neg x) = x"
+lemma REAL_NEGNEG: "real_neg (real_neg x) = x"
by (import hollight REAL_NEGNEG)
-lemma REAL_NEG_MUL2: "ALL (x::hollight.real) y::hollight.real.
- real_mul (real_neg x) (real_neg y) = real_mul x y"
+lemma REAL_NEG_MUL2: "real_mul (real_neg x) (real_neg y) = real_mul x y"
by (import hollight REAL_NEG_MUL2)
-lemma REAL_LT_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_add x y) (real_add x z) = real_lt y z"
+lemma REAL_LT_LADD: "real_lt (real_add x y) (real_add x z) = real_lt y z"
by (import hollight REAL_LT_LADD)
-lemma REAL_LT_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_add x z) (real_add y z) = real_lt x y"
+lemma REAL_LT_RADD: "real_lt (real_add x z) (real_add y z) = real_lt x y"
by (import hollight REAL_LT_RADD)
-lemma REAL_LT_ANTISYM: "ALL (x::hollight.real) y::hollight.real. ~ (real_lt x y & real_lt y x)"
+lemma REAL_LT_ANTISYM: "~ (real_lt x y & real_lt y x)"
by (import hollight REAL_LT_ANTISYM)
-lemma REAL_LT_GT: "ALL (x::hollight.real) y::hollight.real. real_lt x y --> ~ real_lt y x"
+lemma REAL_LT_GT: "real_lt x y ==> ~ real_lt y x"
by (import hollight REAL_LT_GT)
-lemma REAL_NOT_EQ: "ALL (x::hollight.real) y::hollight.real.
- (x ~= y) = (real_lt x y | real_lt y x)"
+lemma REAL_NOT_EQ: "(x ~= y) = (real_lt x y | real_lt y x)"
by (import hollight REAL_NOT_EQ)
-lemma REAL_LE_TOTAL: "ALL (x::hollight.real) y::hollight.real. real_le x y | real_le y x"
- by (import hollight REAL_LE_TOTAL)
-
-lemma REAL_LE_REFL: "ALL x::hollight.real. real_le x x"
- by (import hollight REAL_LE_REFL)
-
-lemma REAL_LE_ANTISYM: "ALL (x::hollight.real) y::hollight.real.
- (real_le x y & real_le y x) = (x = y)"
- by (import hollight REAL_LE_ANTISYM)
-
-lemma REAL_LET_ANTISYM: "ALL (x::hollight.real) y::hollight.real. ~ (real_le x y & real_lt y x)"
+lemma REAL_LET_ANTISYM: "~ (real_le x y & real_lt y x)"
by (import hollight REAL_LET_ANTISYM)
-lemma REAL_NEG_LT0: "ALL x::hollight.real.
- real_lt (real_neg x) (real_of_num 0) = real_lt (real_of_num 0) x"
+lemma REAL_NEG_LT0: "real_lt (real_neg x) (real_of_num 0) = real_lt (real_of_num 0) x"
by (import hollight REAL_NEG_LT0)
-lemma REAL_NEG_GT0: "ALL x::hollight.real.
- real_lt (real_of_num 0) (real_neg x) = real_lt x (real_of_num 0)"
+lemma REAL_NEG_GT0: "real_lt (real_of_num 0) (real_neg x) = real_lt x (real_of_num 0)"
by (import hollight REAL_NEG_GT0)
-lemma REAL_NEG_LE0: "ALL x::hollight.real.
- real_le (real_neg x) (real_of_num 0) = real_le (real_of_num 0) x"
+lemma REAL_NEG_LE0: "real_le (real_neg x) (real_of_num 0) = real_le (real_of_num 0) x"
by (import hollight REAL_NEG_LE0)
-lemma REAL_NEG_GE0: "ALL x::hollight.real.
- real_le (real_of_num 0) (real_neg x) = real_le x (real_of_num 0)"
+lemma REAL_NEG_GE0: "real_le (real_of_num 0) (real_neg x) = real_le x (real_of_num 0)"
by (import hollight REAL_NEG_GE0)
-lemma REAL_LT_TOTAL: "ALL (x::hollight.real) y::hollight.real. x = y | real_lt x y | real_lt y x"
+lemma REAL_LT_TOTAL: "x = y | real_lt x y | real_lt y x"
by (import hollight REAL_LT_TOTAL)
-lemma REAL_LT_NEGTOTAL: "ALL x::hollight.real.
- x = real_of_num 0 |
- real_lt (real_of_num 0) x | real_lt (real_of_num 0) (real_neg x)"
+lemma REAL_LT_NEGTOTAL: "x = real_of_num 0 |
+real_lt (real_of_num 0) x | real_lt (real_of_num 0) (real_neg x)"
by (import hollight REAL_LT_NEGTOTAL)
-lemma REAL_LE_01: "real_le (real_of_num 0) (real_of_num (NUMERAL_BIT1 0))"
+lemma REAL_LE_01: "real_le (real_of_num 0) (real_of_num 1)"
by (import hollight REAL_LE_01)
-lemma REAL_LT_01: "real_lt (real_of_num 0) (real_of_num (NUMERAL_BIT1 0))"
+lemma REAL_LT_01: "real_lt (real_of_num 0) (real_of_num 1)"
by (import hollight REAL_LT_01)
-lemma REAL_LE_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_add x y) (real_add x z) = real_le y z"
+lemma REAL_LE_LADD: "real_le (real_add x y) (real_add x z) = real_le y z"
by (import hollight REAL_LE_LADD)
-lemma REAL_LE_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_add x z) (real_add y z) = real_le x y"
+lemma REAL_LE_RADD: "real_le (real_add x z) (real_add y z) = real_le x y"
by (import hollight REAL_LE_RADD)
-lemma REAL_LT_ADD2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_lt w x & real_lt y z --> real_lt (real_add w y) (real_add x z)"
+lemma REAL_LT_ADD2: "real_lt w x & real_lt y z ==> real_lt (real_add w y) (real_add x z)"
by (import hollight REAL_LT_ADD2)
-lemma REAL_LE_ADD2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_le w x & real_le y z --> real_le (real_add w y) (real_add x z)"
+lemma REAL_LE_ADD2: "real_le w x & real_le y z ==> real_le (real_add w y) (real_add x z)"
by (import hollight REAL_LE_ADD2)
-lemma REAL_LT_LNEG: "ALL (x::hollight.real) xa::hollight.real.
- real_lt (real_neg x) xa = real_lt (real_of_num 0) (real_add x xa)"
+lemma REAL_LT_LNEG: "real_lt (real_neg x) xa = real_lt (real_of_num 0) (real_add x xa)"
by (import hollight REAL_LT_LNEG)
-lemma REAL_LT_RNEG: "ALL (x::hollight.real) xa::hollight.real.
- real_lt x (real_neg xa) = real_lt (real_add x xa) (real_of_num 0)"
+lemma REAL_LT_RNEG: "real_lt x (real_neg xa) = real_lt (real_add x xa) (real_of_num 0)"
by (import hollight REAL_LT_RNEG)
-lemma REAL_LT_ADDNEG: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt y (real_add x (real_neg z)) = real_lt (real_add y z) x"
+lemma REAL_LT_ADDNEG: "real_lt y (real_add x (real_neg z)) = real_lt (real_add y z) x"
by (import hollight REAL_LT_ADDNEG)
-lemma REAL_LT_ADDNEG2: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_add x (real_neg y)) z = real_lt x (real_add z y)"
+lemma REAL_LT_ADDNEG2: "real_lt (real_add x (real_neg y)) z = real_lt x (real_add z y)"
by (import hollight REAL_LT_ADDNEG2)
-lemma REAL_LT_ADD1: "ALL (x::hollight.real) y::hollight.real.
- real_le x y --> real_lt x (real_add y (real_of_num (NUMERAL_BIT1 0)))"
+lemma REAL_LT_ADD1: "real_le x y ==> real_lt x (real_add y (real_of_num 1))"
by (import hollight REAL_LT_ADD1)
-lemma REAL_SUB_ADD: "ALL (x::hollight.real) y::hollight.real. real_add (real_sub x y) y = x"
+lemma REAL_SUB_ADD: "real_add (real_sub x y) y = x"
by (import hollight REAL_SUB_ADD)
-lemma REAL_SUB_ADD2: "ALL (x::hollight.real) y::hollight.real. real_add y (real_sub x y) = x"
+lemma REAL_SUB_ADD2: "real_add y (real_sub x y) = x"
by (import hollight REAL_SUB_ADD2)
-lemma REAL_SUB_REFL: "ALL x::hollight.real. real_sub x x = real_of_num 0"
+lemma REAL_SUB_REFL: "real_sub x x = real_of_num 0"
by (import hollight REAL_SUB_REFL)
-lemma REAL_LE_DOUBLE: "ALL x::hollight.real.
- real_le (real_of_num 0) (real_add x x) = real_le (real_of_num 0) x"
+lemma REAL_LE_DOUBLE: "real_le (real_of_num 0) (real_add x x) = real_le (real_of_num 0) x"
by (import hollight REAL_LE_DOUBLE)
-lemma REAL_LE_NEGL: "ALL x::hollight.real. real_le (real_neg x) x = real_le (real_of_num 0) x"
+lemma REAL_LE_NEGL: "real_le (real_neg x) x = real_le (real_of_num 0) x"
by (import hollight REAL_LE_NEGL)
-lemma REAL_LE_NEGR: "ALL x::hollight.real. real_le x (real_neg x) = real_le x (real_of_num 0)"
+lemma REAL_LE_NEGR: "real_le x (real_neg x) = real_le x (real_of_num 0)"
by (import hollight REAL_LE_NEGR)
-lemma REAL_NEG_EQ_0: "ALL x::hollight.real. (real_neg x = real_of_num 0) = (x = real_of_num 0)"
+lemma REAL_NEG_EQ_0: "(real_neg x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight REAL_NEG_EQ_0)
-lemma REAL_ADD_SUB: "ALL (x::hollight.real) y::hollight.real. real_sub (real_add x y) x = y"
+lemma REAL_ADD_SUB: "real_sub (real_add x y) x = y"
by (import hollight REAL_ADD_SUB)
-lemma REAL_NEG_EQ: "ALL (x::hollight.real) y::hollight.real. (real_neg x = y) = (x = real_neg y)"
+lemma REAL_NEG_EQ: "(real_neg x = y) = (x = real_neg y)"
by (import hollight REAL_NEG_EQ)
-lemma REAL_NEG_MINUS1: "ALL x::hollight.real.
- real_neg x = real_mul (real_neg (real_of_num (NUMERAL_BIT1 0))) x"
+lemma REAL_NEG_MINUS1: "real_neg x = real_mul (real_neg (real_of_num 1)) x"
by (import hollight REAL_NEG_MINUS1)
-lemma REAL_LT_IMP_NE: "ALL (x::hollight.real) y::hollight.real. real_lt x y --> x ~= y"
+lemma REAL_LT_IMP_NE: "real_lt x y ==> x ~= y"
by (import hollight REAL_LT_IMP_NE)
-lemma REAL_LE_ADDR: "ALL (x::hollight.real) y::hollight.real.
- real_le x (real_add x y) = real_le (real_of_num 0) y"
+lemma REAL_LE_ADDR: "real_le x (real_add x y) = real_le (real_of_num 0) y"
by (import hollight REAL_LE_ADDR)
-lemma REAL_LE_ADDL: "ALL (x::hollight.real) y::hollight.real.
- real_le y (real_add x y) = real_le (real_of_num 0) x"
+lemma REAL_LE_ADDL: "real_le y (real_add x y) = real_le (real_of_num 0) x"
by (import hollight REAL_LE_ADDL)
-lemma REAL_LT_ADDR: "ALL (x::hollight.real) y::hollight.real.
- real_lt x (real_add x y) = real_lt (real_of_num 0) y"
+lemma REAL_LT_ADDR: "real_lt x (real_add x y) = real_lt (real_of_num 0) y"
by (import hollight REAL_LT_ADDR)
-lemma REAL_LT_ADDL: "ALL (x::hollight.real) y::hollight.real.
- real_lt y (real_add x y) = real_lt (real_of_num 0) x"
+lemma REAL_LT_ADDL: "real_lt y (real_add x y) = real_lt (real_of_num 0) x"
by (import hollight REAL_LT_ADDL)
-lemma REAL_SUB_SUB: "ALL (x::hollight.real) y::hollight.real.
- real_sub (real_sub x y) x = real_neg y"
+lemma REAL_SUB_SUB: "real_sub (real_sub x y) x = real_neg y"
by (import hollight REAL_SUB_SUB)
-lemma REAL_LT_ADD_SUB: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_add x y) z = real_lt x (real_sub z y)"
+lemma REAL_LT_ADD_SUB: "real_lt (real_add x y) z = real_lt x (real_sub z y)"
by (import hollight REAL_LT_ADD_SUB)
-lemma REAL_LT_SUB_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_sub x y) z = real_lt x (real_add z y)"
+lemma REAL_LT_SUB_RADD: "real_lt (real_sub x y) z = real_lt x (real_add z y)"
by (import hollight REAL_LT_SUB_RADD)
-lemma REAL_LT_SUB_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt x (real_sub y z) = real_lt (real_add x z) y"
+lemma REAL_LT_SUB_LADD: "real_lt x (real_sub y z) = real_lt (real_add x z) y"
by (import hollight REAL_LT_SUB_LADD)
-lemma REAL_LE_SUB_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le x (real_sub y z) = real_le (real_add x z) y"
+lemma REAL_LE_SUB_LADD: "real_le x (real_sub y z) = real_le (real_add x z) y"
by (import hollight REAL_LE_SUB_LADD)
-lemma REAL_LE_SUB_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_sub x y) z = real_le x (real_add z y)"
+lemma REAL_LE_SUB_RADD: "real_le (real_sub x y) z = real_le x (real_add z y)"
by (import hollight REAL_LE_SUB_RADD)
-lemma REAL_LT_NEG: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_neg x) (real_neg y) = real_lt y x"
+lemma REAL_LT_NEG: "real_lt (real_neg x) (real_neg y) = real_lt y x"
by (import hollight REAL_LT_NEG)
-lemma REAL_LE_NEG: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_neg x) (real_neg y) = real_le y x"
+lemma REAL_LE_NEG: "real_le (real_neg x) (real_neg y) = real_le y x"
by (import hollight REAL_LE_NEG)
-lemma REAL_ADD2_SUB2: "ALL (a::hollight.real) (b::hollight.real) (c::hollight.real)
- d::hollight.real.
- real_sub (real_add a b) (real_add c d) =
- real_add (real_sub a c) (real_sub b d)"
+lemma REAL_ADD2_SUB2: "real_sub (real_add a b) (real_add c d) =
+real_add (real_sub a c) (real_sub b d)"
by (import hollight REAL_ADD2_SUB2)
-lemma REAL_SUB_LZERO: "ALL x::hollight.real. real_sub (real_of_num 0) x = real_neg x"
+lemma REAL_SUB_LZERO: "real_sub (real_of_num 0) x = real_neg x"
by (import hollight REAL_SUB_LZERO)
-lemma REAL_SUB_RZERO: "ALL x::hollight.real. real_sub x (real_of_num 0) = x"
+lemma REAL_SUB_RZERO: "real_sub x (real_of_num 0) = x"
by (import hollight REAL_SUB_RZERO)
-lemma REAL_LET_ADD2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_le w x & real_lt y z --> real_lt (real_add w y) (real_add x z)"
+lemma REAL_LET_ADD2: "real_le w x & real_lt y z ==> real_lt (real_add w y) (real_add x z)"
by (import hollight REAL_LET_ADD2)
-lemma REAL_LTE_ADD2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_lt w x & real_le y z --> real_lt (real_add w y) (real_add x z)"
+lemma REAL_LTE_ADD2: "real_lt w x & real_le y z ==> real_lt (real_add w y) (real_add x z)"
by (import hollight REAL_LTE_ADD2)
-lemma REAL_SUB_LNEG: "ALL (x::hollight.real) y::hollight.real.
- real_sub (real_neg x) y = real_neg (real_add x y)"
+lemma REAL_SUB_LNEG: "real_sub (real_neg x) y = real_neg (real_add x y)"
by (import hollight REAL_SUB_LNEG)
-lemma REAL_SUB_RNEG: "ALL (x::hollight.real) y::hollight.real.
- real_sub x (real_neg y) = real_add x y"
+lemma REAL_SUB_RNEG: "real_sub x (real_neg y) = real_add x y"
by (import hollight REAL_SUB_RNEG)
-lemma REAL_SUB_NEG2: "ALL (x::hollight.real) y::hollight.real.
- real_sub (real_neg x) (real_neg y) = real_sub y x"
+lemma REAL_SUB_NEG2: "real_sub (real_neg x) (real_neg y) = real_sub y x"
by (import hollight REAL_SUB_NEG2)
-lemma REAL_SUB_TRIANGLE: "ALL (a::hollight.real) (b::hollight.real) c::hollight.real.
- real_add (real_sub a b) (real_sub b c) = real_sub a c"
+lemma REAL_SUB_TRIANGLE: "real_add (real_sub a b) (real_sub b c) = real_sub a c"
by (import hollight REAL_SUB_TRIANGLE)
-lemma REAL_EQ_SUB_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- (x = real_sub y z) = (real_add x z = y)"
+lemma REAL_EQ_SUB_LADD: "(x = real_sub y z) = (real_add x z = y)"
by (import hollight REAL_EQ_SUB_LADD)
-lemma REAL_EQ_SUB_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- (real_sub x y = z) = (x = real_add z y)"
+lemma REAL_EQ_SUB_RADD: "(real_sub x y = z) = (x = real_add z y)"
by (import hollight REAL_EQ_SUB_RADD)
-lemma REAL_SUB_SUB2: "ALL (x::hollight.real) y::hollight.real. real_sub x (real_sub x y) = y"
+lemma REAL_SUB_SUB2: "real_sub x (real_sub x y) = y"
by (import hollight REAL_SUB_SUB2)
-lemma REAL_ADD_SUB2: "ALL (x::hollight.real) y::hollight.real.
- real_sub x (real_add x y) = real_neg y"
+lemma REAL_ADD_SUB2: "real_sub x (real_add x y) = real_neg y"
by (import hollight REAL_ADD_SUB2)
-lemma REAL_EQ_IMP_LE: "ALL (x::hollight.real) y::hollight.real. x = y --> real_le x y"
+lemma REAL_EQ_IMP_LE: "x = y ==> real_le x y"
by (import hollight REAL_EQ_IMP_LE)
-lemma REAL_POS_NZ: "ALL x::hollight.real. real_lt (real_of_num 0) x --> x ~= real_of_num 0"
+lemma REAL_POS_NZ: "real_lt (real_of_num 0) x ==> x ~= real_of_num 0"
by (import hollight REAL_POS_NZ)
-lemma REAL_DIFFSQ: "ALL (x::hollight.real) y::hollight.real.
- real_mul (real_add x y) (real_sub x y) =
- real_sub (real_mul x x) (real_mul y y)"
+lemma REAL_DIFFSQ: "real_mul (real_add x y) (real_sub x y) =
+real_sub (real_mul x x) (real_mul y y)"
by (import hollight REAL_DIFFSQ)
-lemma REAL_EQ_NEG2: "ALL (x::hollight.real) y::hollight.real. (real_neg x = real_neg y) = (x = y)"
+lemma REAL_EQ_NEG2: "(real_neg x = real_neg y) = (x = y)"
by (import hollight REAL_EQ_NEG2)
-lemma REAL_LT_NEG2: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_neg x) (real_neg y) = real_lt y x"
+lemma REAL_LT_NEG2: "real_lt (real_neg x) (real_neg y) = real_lt y x"
by (import hollight REAL_LT_NEG2)
-lemma REAL_SUB_LDISTRIB: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_mul x (real_sub y z) = real_sub (real_mul x y) (real_mul x z)"
+lemma REAL_SUB_LDISTRIB: "real_mul x (real_sub y z) = real_sub (real_mul x y) (real_mul x z)"
by (import hollight REAL_SUB_LDISTRIB)
-lemma REAL_SUB_RDISTRIB: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_mul (real_sub x y) z = real_sub (real_mul x z) (real_mul y z)"
+lemma REAL_SUB_RDISTRIB: "real_mul (real_sub x y) z = real_sub (real_mul x z) (real_mul y z)"
by (import hollight REAL_SUB_RDISTRIB)
-lemma REAL_ABS_ZERO: "ALL x::hollight.real. (real_abs x = real_of_num 0) = (x = real_of_num 0)"
+lemma REAL_ABS_ZERO: "(real_abs x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight REAL_ABS_ZERO)
lemma REAL_ABS_0: "real_abs (real_of_num 0) = real_of_num 0"
by (import hollight REAL_ABS_0)
-lemma REAL_ABS_1: "real_abs (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
+lemma REAL_ABS_1: "real_abs (real_of_num 1) = real_of_num 1"
by (import hollight REAL_ABS_1)
-lemma REAL_ABS_TRIANGLE: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_abs (real_add x y)) (real_add (real_abs x) (real_abs y))"
+lemma REAL_ABS_TRIANGLE: "real_le (real_abs (real_add x y)) (real_add (real_abs x) (real_abs y))"
by (import hollight REAL_ABS_TRIANGLE)
-lemma REAL_ABS_TRIANGLE_LE: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_add (real_abs x) (real_abs (real_sub y x))) z -->
- real_le (real_abs y) z"
+lemma REAL_ABS_TRIANGLE_LE: "real_le (real_add (real_abs x) (real_abs (real_sub y x))) z
+==> real_le (real_abs y) z"
by (import hollight REAL_ABS_TRIANGLE_LE)
-lemma REAL_ABS_TRIANGLE_LT: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_add (real_abs x) (real_abs (real_sub y x))) z -->
- real_lt (real_abs y) z"
+lemma REAL_ABS_TRIANGLE_LT: "real_lt (real_add (real_abs x) (real_abs (real_sub y x))) z
+==> real_lt (real_abs y) z"
by (import hollight REAL_ABS_TRIANGLE_LT)
-lemma REAL_ABS_POS: "ALL x::hollight.real. real_le (real_of_num 0) (real_abs x)"
+lemma REAL_ABS_POS: "real_le (real_of_num 0) (real_abs x)"
by (import hollight REAL_ABS_POS)
-lemma REAL_ABS_SUB: "ALL (x::hollight.real) y::hollight.real.
- real_abs (real_sub x y) = real_abs (real_sub y x)"
+lemma REAL_ABS_SUB: "real_abs (real_sub x y) = real_abs (real_sub y x)"
by (import hollight REAL_ABS_SUB)
-lemma REAL_ABS_NZ: "ALL x::hollight.real.
- (x ~= real_of_num 0) = real_lt (real_of_num 0) (real_abs x)"
+lemma REAL_ABS_NZ: "(x ~= real_of_num 0) = real_lt (real_of_num 0) (real_abs x)"
by (import hollight REAL_ABS_NZ)
-lemma REAL_ABS_ABS: "ALL x::hollight.real. real_abs (real_abs x) = real_abs x"
+lemma REAL_ABS_ABS: "real_abs (real_abs x) = real_abs x"
by (import hollight REAL_ABS_ABS)
-lemma REAL_ABS_LE: "ALL x::hollight.real. real_le x (real_abs x)"
+lemma REAL_ABS_LE: "real_le x (real_abs x)"
by (import hollight REAL_ABS_LE)
-lemma REAL_ABS_REFL: "ALL x::hollight.real. (real_abs x = x) = real_le (real_of_num 0) x"
+lemma REAL_ABS_REFL: "(real_abs x = x) = real_le (real_of_num 0) x"
by (import hollight REAL_ABS_REFL)
-lemma REAL_ABS_BETWEEN: "ALL (x::hollight.real) (y::hollight.real) d::hollight.real.
- (real_lt (real_of_num 0) d &
- real_lt (real_sub x d) y & real_lt y (real_add x d)) =
- real_lt (real_abs (real_sub y x)) d"
+lemma REAL_ABS_BETWEEN: "(real_lt (real_of_num 0) d &
+ real_lt (real_sub x d) y & real_lt y (real_add x d)) =
+real_lt (real_abs (real_sub y x)) d"
by (import hollight REAL_ABS_BETWEEN)
-lemma REAL_ABS_BOUND: "ALL (x::hollight.real) (y::hollight.real) d::hollight.real.
- real_lt (real_abs (real_sub x y)) d --> real_lt y (real_add x d)"
+lemma REAL_ABS_BOUND: "real_lt (real_abs (real_sub x y)) d ==> real_lt y (real_add x d)"
by (import hollight REAL_ABS_BOUND)
-lemma REAL_ABS_STILLNZ: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_abs (real_sub x y)) (real_abs y) --> x ~= real_of_num 0"
+lemma REAL_ABS_STILLNZ: "real_lt (real_abs (real_sub x y)) (real_abs y) ==> x ~= real_of_num 0"
by (import hollight REAL_ABS_STILLNZ)
-lemma REAL_ABS_CASES: "ALL x::hollight.real.
- x = real_of_num 0 | real_lt (real_of_num 0) (real_abs x)"
+lemma REAL_ABS_CASES: "x = real_of_num 0 | real_lt (real_of_num 0) (real_abs x)"
by (import hollight REAL_ABS_CASES)
-lemma REAL_ABS_BETWEEN1: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt x z & real_lt (real_abs (real_sub y x)) (real_sub z x) -->
- real_lt y z"
+lemma REAL_ABS_BETWEEN1: "real_lt x z & real_lt (real_abs (real_sub y x)) (real_sub z x)
+==> real_lt y z"
by (import hollight REAL_ABS_BETWEEN1)
-lemma REAL_ABS_SIGN: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_abs (real_sub x y)) y --> real_lt (real_of_num 0) x"
+lemma REAL_ABS_SIGN: "real_lt (real_abs (real_sub x y)) y ==> real_lt (real_of_num 0) x"
by (import hollight REAL_ABS_SIGN)
-lemma REAL_ABS_SIGN2: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_abs (real_sub x y)) (real_neg y) -->
- real_lt x (real_of_num 0)"
+lemma REAL_ABS_SIGN2: "real_lt (real_abs (real_sub x y)) (real_neg y) ==> real_lt x (real_of_num 0)"
by (import hollight REAL_ABS_SIGN2)
-lemma REAL_ABS_CIRCLE: "ALL (x::hollight.real) (y::hollight.real) h::hollight.real.
- real_lt (real_abs h) (real_sub (real_abs y) (real_abs x)) -->
- real_lt (real_abs (real_add x h)) (real_abs y)"
+lemma REAL_ABS_CIRCLE: "real_lt (real_abs h) (real_sub (real_abs y) (real_abs x))
+==> real_lt (real_abs (real_add x h)) (real_abs y)"
by (import hollight REAL_ABS_CIRCLE)
-lemma REAL_SUB_ABS: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_sub (real_abs x) (real_abs y)) (real_abs (real_sub x y))"
+lemma REAL_SUB_ABS: "real_le (real_sub (real_abs x) (real_abs y)) (real_abs (real_sub x y))"
by (import hollight REAL_SUB_ABS)
-lemma REAL_ABS_SUB_ABS: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_abs (real_sub (real_abs x) (real_abs y)))
- (real_abs (real_sub x y))"
+lemma REAL_ABS_SUB_ABS: "real_le (real_abs (real_sub (real_abs x) (real_abs y)))
+ (real_abs (real_sub x y))"
by (import hollight REAL_ABS_SUB_ABS)
-lemma REAL_ABS_BETWEEN2: "ALL (x0::hollight.real) (x::hollight.real) (y0::hollight.real)
- y::hollight.real.
- real_lt x0 y0 &
- real_lt
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
- (real_abs (real_sub x x0)))
- (real_sub y0 x0) &
- real_lt
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
- (real_abs (real_sub y y0)))
- (real_sub y0 x0) -->
- real_lt x y"
+lemma REAL_ABS_BETWEEN2: "real_lt x0 y0 &
+real_lt (real_mul (real_of_num 2) (real_abs (real_sub x x0)))
+ (real_sub y0 x0) &
+real_lt (real_mul (real_of_num 2) (real_abs (real_sub y y0)))
+ (real_sub y0 x0)
+==> real_lt x y"
by (import hollight REAL_ABS_BETWEEN2)
-lemma REAL_ABS_BOUNDS: "ALL (x::hollight.real) k::hollight.real.
- real_le (real_abs x) k = (real_le (real_neg k) x & real_le x k)"
+lemma REAL_ABS_BOUNDS: "real_le (real_abs x) k = (real_le (real_neg k) x & real_le x k)"
by (import hollight REAL_ABS_BOUNDS)
-lemma REAL_MIN_MAX: "ALL (x::hollight.real) y::hollight.real.
- real_min x y = real_neg (real_max (real_neg x) (real_neg y))"
+lemma REAL_BOUNDS_LE: "(real_le (real_neg k) x & real_le x k) = real_le (real_abs x) k"
+ by (import hollight REAL_BOUNDS_LE)
+
+lemma REAL_BOUNDS_LT: "(real_lt (real_neg k) x & real_lt x k) = real_lt (real_abs x) k"
+ by (import hollight REAL_BOUNDS_LT)
+
+lemma REAL_MIN_MAX: "real_min x y = real_neg (real_max (real_neg x) (real_neg y))"
by (import hollight REAL_MIN_MAX)
-lemma REAL_MAX_MIN: "ALL (x::hollight.real) y::hollight.real.
- real_max x y = real_neg (real_min (real_neg x) (real_neg y))"
+lemma REAL_MAX_MIN: "real_max x y = real_neg (real_min (real_neg x) (real_neg y))"
by (import hollight REAL_MAX_MIN)
-lemma REAL_MAX_MAX: "ALL (x::hollight.real) y::hollight.real.
- real_le x (real_max x y) & real_le y (real_max x y)"
+lemma REAL_MAX_MAX: "real_le x (real_max x y) & real_le y (real_max x y)"
by (import hollight REAL_MAX_MAX)
-lemma REAL_MIN_MIN: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_min x y) x & real_le (real_min x y) y"
+lemma REAL_MIN_MIN: "real_le (real_min x y) x & real_le (real_min x y) y"
by (import hollight REAL_MIN_MIN)
-lemma REAL_MAX_SYM: "ALL (x::hollight.real) y::hollight.real. real_max x y = real_max y x"
+lemma REAL_MAX_SYM: "real_max x y = real_max y x"
by (import hollight REAL_MAX_SYM)
-lemma REAL_MIN_SYM: "ALL (x::hollight.real) y::hollight.real. real_min x y = real_min y x"
+lemma REAL_MIN_SYM: "real_min x y = real_min y x"
by (import hollight REAL_MIN_SYM)
-lemma REAL_LE_MAX: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le z (real_max x y) = (real_le z x | real_le z y)"
+lemma REAL_LE_MAX: "real_le z (real_max x y) = (real_le z x | real_le z y)"
by (import hollight REAL_LE_MAX)
-lemma REAL_LE_MIN: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le z (real_min x y) = (real_le z x & real_le z y)"
+lemma REAL_LE_MIN: "real_le z (real_min x y) = (real_le z x & real_le z y)"
by (import hollight REAL_LE_MIN)
-lemma REAL_LT_MAX: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt z (real_max x y) = (real_lt z x | real_lt z y)"
+lemma REAL_LT_MAX: "real_lt z (real_max x y) = (real_lt z x | real_lt z y)"
by (import hollight REAL_LT_MAX)
-lemma REAL_LT_MIN: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt z (real_min x y) = (real_lt z x & real_lt z y)"
+lemma REAL_LT_MIN: "real_lt z (real_min x y) = (real_lt z x & real_lt z y)"
by (import hollight REAL_LT_MIN)
-lemma REAL_MAX_LE: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_max x y) z = (real_le x z & real_le y z)"
+lemma REAL_MAX_LE: "real_le (real_max x y) z = (real_le x z & real_le y z)"
by (import hollight REAL_MAX_LE)
-lemma REAL_MIN_LE: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le (real_min x y) z = (real_le x z | real_le y z)"
+lemma REAL_MIN_LE: "real_le (real_min x y) z = (real_le x z | real_le y z)"
by (import hollight REAL_MIN_LE)
-lemma REAL_MAX_LT: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_max x y) z = (real_lt x z & real_lt y z)"
+lemma REAL_MAX_LT: "real_lt (real_max x y) z = (real_lt x z & real_lt y z)"
by (import hollight REAL_MAX_LT)
-lemma REAL_MIN_LT: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_min x y) z = (real_lt x z | real_lt y z)"
+lemma REAL_MIN_LT: "real_lt (real_min x y) z = (real_lt x z | real_lt y z)"
by (import hollight REAL_MIN_LT)
-lemma REAL_MAX_ASSOC: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_max x (real_max y z) = real_max (real_max x y) z"
+lemma REAL_MAX_ASSOC: "real_max x (real_max y z) = real_max (real_max x y) z"
by (import hollight REAL_MAX_ASSOC)
-lemma REAL_MIN_ASSOC: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_min x (real_min y z) = real_min (real_min x y) z"
+lemma REAL_MIN_ASSOC: "real_min x (real_min y z) = real_min (real_min x y) z"
by (import hollight REAL_MIN_ASSOC)
-lemma REAL_MAX_ACI: "real_max (x::hollight.real) (y::hollight.real) = real_max y x &
-real_max (real_max x y) (z::hollight.real) = real_max x (real_max y z) &
+lemma REAL_MAX_ACI: "real_max x y = real_max y x &
+real_max (real_max x y) z = real_max x (real_max y z) &
real_max x (real_max y z) = real_max y (real_max x z) &
real_max x x = x & real_max x (real_max x y) = real_max x y"
by (import hollight REAL_MAX_ACI)
-lemma REAL_MIN_ACI: "real_min (x::hollight.real) (y::hollight.real) = real_min y x &
-real_min (real_min x y) (z::hollight.real) = real_min x (real_min y z) &
+lemma REAL_MIN_ACI: "real_min x y = real_min y x &
+real_min (real_min x y) z = real_min x (real_min y z) &
real_min x (real_min y z) = real_min y (real_min x z) &
real_min x x = x & real_min x (real_min x y) = real_min x y"
by (import hollight REAL_MIN_ACI)
-lemma REAL_ABS_MUL: "ALL (x::hollight.real) y::hollight.real.
- real_abs (real_mul x y) = real_mul (real_abs x) (real_abs y)"
+lemma REAL_ABS_MUL: "real_abs (real_mul x y) = real_mul (real_abs x) (real_abs y)"
by (import hollight REAL_ABS_MUL)
-lemma REAL_POW_LE: "ALL (x::hollight.real) n::nat.
- real_le (real_of_num 0) x --> real_le (real_of_num 0) (real_pow x n)"
+lemma REAL_POW_LE: "real_le (real_of_num 0) x ==> real_le (real_of_num 0) (real_pow x n)"
by (import hollight REAL_POW_LE)
-lemma REAL_POW_LT: "ALL (x::hollight.real) n::nat.
- real_lt (real_of_num 0) x --> real_lt (real_of_num 0) (real_pow x n)"
+lemma REAL_POW_LT: "real_lt (real_of_num 0) x ==> real_lt (real_of_num 0) (real_pow x n)"
by (import hollight REAL_POW_LT)
-lemma REAL_ABS_POW: "ALL (x::hollight.real) n::nat.
- real_abs (real_pow x n) = real_pow (real_abs x) n"
+lemma REAL_ABS_POW: "real_abs (real_pow x n) = real_pow (real_abs x) n"
by (import hollight REAL_ABS_POW)
-lemma REAL_LE_LMUL: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_le (real_of_num 0) x & real_le xa xb -->
- real_le (real_mul x xa) (real_mul x xb)"
+lemma REAL_LE_LMUL: "real_le (real_of_num 0) x & real_le xa xb
+==> real_le (real_mul x xa) (real_mul x xb)"
by (import hollight REAL_LE_LMUL)
-lemma REAL_LE_RMUL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_le x y & real_le (real_of_num 0) z -->
- real_le (real_mul x z) (real_mul y z)"
+lemma REAL_LE_RMUL: "real_le x y & real_le (real_of_num 0) z
+==> real_le (real_mul x z) (real_mul y z)"
by (import hollight REAL_LE_RMUL)
-lemma REAL_LT_LMUL: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) x & real_lt xa xb -->
- real_lt (real_mul x xa) (real_mul x xb)"
+lemma REAL_LT_LMUL: "real_lt (real_of_num 0) x & real_lt xa xb
+==> real_lt (real_mul x xa) (real_mul x xb)"
by (import hollight REAL_LT_LMUL)
-lemma REAL_LT_RMUL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt x y & real_lt (real_of_num 0) z -->
- real_lt (real_mul x z) (real_mul y z)"
+lemma REAL_LT_RMUL: "real_lt x y & real_lt (real_of_num 0) z
+==> real_lt (real_mul x z) (real_mul y z)"
by (import hollight REAL_LT_RMUL)
-lemma REAL_EQ_MUL_LCANCEL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- (real_mul x y = real_mul x z) = (x = real_of_num 0 | y = z)"
+lemma REAL_EQ_MUL_LCANCEL: "(real_mul x y = real_mul x z) = (x = real_of_num 0 | y = z)"
by (import hollight REAL_EQ_MUL_LCANCEL)
-lemma REAL_EQ_MUL_RCANCEL: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- (real_mul x xb = real_mul xa xb) = (x = xa | xb = real_of_num 0)"
+lemma REAL_EQ_MUL_RCANCEL: "(real_mul x xb = real_mul xa xb) = (x = xa | xb = real_of_num 0)"
by (import hollight REAL_EQ_MUL_RCANCEL)
-lemma REAL_MUL_LINV_UNIQ: "ALL (x::hollight.real) y::hollight.real.
- real_mul x y = real_of_num (NUMERAL_BIT1 0) --> real_inv y = x"
+lemma REAL_MUL_LINV_UNIQ: "real_mul x y = real_of_num 1 ==> real_inv y = x"
by (import hollight REAL_MUL_LINV_UNIQ)
-lemma REAL_MUL_RINV_UNIQ: "ALL (x::hollight.real) xa::hollight.real.
- real_mul x xa = real_of_num (NUMERAL_BIT1 0) --> real_inv x = xa"
+lemma REAL_MUL_RINV_UNIQ: "real_mul x xa = real_of_num 1 ==> real_inv x = xa"
by (import hollight REAL_MUL_RINV_UNIQ)
-lemma REAL_INV_INV: "ALL x::hollight.real. real_inv (real_inv x) = x"
+lemma REAL_INV_INV: "real_inv (real_inv x) = x"
by (import hollight REAL_INV_INV)
-lemma REAL_INV_EQ_0: "ALL x::hollight.real. (real_inv x = real_of_num 0) = (x = real_of_num 0)"
+lemma REAL_EQ_INV2: "(real_inv x = real_inv y) = (x = y)"
+ by (import hollight REAL_EQ_INV2)
+
+lemma REAL_INV_EQ_0: "(real_inv x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight REAL_INV_EQ_0)
-lemma REAL_LT_INV: "ALL x::hollight.real.
- real_lt (real_of_num 0) x --> real_lt (real_of_num 0) (real_inv x)"
+lemma REAL_LT_INV: "real_lt (real_of_num 0) x ==> real_lt (real_of_num 0) (real_inv x)"
by (import hollight REAL_LT_INV)
-lemma REAL_LT_INV_EQ: "ALL x::hollight.real.
- real_lt (real_of_num 0) (real_inv x) = real_lt (real_of_num 0) x"
+lemma REAL_LT_INV_EQ: "real_lt (real_of_num 0) (real_inv x) = real_lt (real_of_num 0) x"
by (import hollight REAL_LT_INV_EQ)
-lemma REAL_INV_NEG: "ALL x::hollight.real. real_inv (real_neg x) = real_neg (real_inv x)"
+lemma REAL_INV_NEG: "real_inv (real_neg x) = real_neg (real_inv x)"
by (import hollight REAL_INV_NEG)
-lemma REAL_LE_INV_EQ: "ALL x::hollight.real.
- real_le (real_of_num 0) (real_inv x) = real_le (real_of_num 0) x"
+lemma REAL_LE_INV_EQ: "real_le (real_of_num 0) (real_inv x) = real_le (real_of_num 0) x"
by (import hollight REAL_LE_INV_EQ)
-lemma REAL_LE_INV: "ALL x::hollight.real.
- real_le (real_of_num 0) x --> real_le (real_of_num 0) (real_inv x)"
+lemma REAL_LE_INV: "real_le (real_of_num 0) x ==> real_le (real_of_num 0) (real_inv x)"
by (import hollight REAL_LE_INV)
-lemma REAL_MUL_RINV: "ALL x::hollight.real.
- x ~= real_of_num 0 -->
- real_mul x (real_inv x) = real_of_num (NUMERAL_BIT1 0)"
+lemma REAL_MUL_RINV: "x ~= real_of_num 0 ==> real_mul x (real_inv x) = real_of_num 1"
by (import hollight REAL_MUL_RINV)
-lemma REAL_INV_1: "real_inv (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
+lemma REAL_INV_1: "real_inv (real_of_num 1) = real_of_num 1"
by (import hollight REAL_INV_1)
-lemma REAL_DIV_1: "ALL x::hollight.real. real_div x (real_of_num (NUMERAL_BIT1 0)) = x"
+lemma REAL_INV_EQ_1: "(real_inv x = real_of_num 1) = (x = real_of_num 1)"
+ by (import hollight REAL_INV_EQ_1)
+
+lemma REAL_DIV_1: "real_div x (real_of_num 1) = x"
by (import hollight REAL_DIV_1)
-lemma REAL_DIV_REFL: "ALL x::hollight.real.
- x ~= real_of_num 0 --> real_div x x = real_of_num (NUMERAL_BIT1 0)"
+lemma REAL_DIV_REFL: "x ~= real_of_num 0 ==> real_div x x = real_of_num 1"
by (import hollight REAL_DIV_REFL)
-lemma REAL_DIV_RMUL: "ALL (x::hollight.real) xa::hollight.real.
- xa ~= real_of_num 0 --> real_mul (real_div x xa) xa = x"
+lemma REAL_DIV_RMUL: "xa ~= real_of_num 0 ==> real_mul (real_div x xa) xa = x"
by (import hollight REAL_DIV_RMUL)
-lemma REAL_DIV_LMUL: "ALL (x::hollight.real) xa::hollight.real.
- xa ~= real_of_num 0 --> real_mul xa (real_div x xa) = x"
+lemma REAL_DIV_LMUL: "xa ~= real_of_num 0 ==> real_mul xa (real_div x xa) = x"
by (import hollight REAL_DIV_LMUL)
-lemma REAL_ABS_INV: "ALL x::hollight.real. real_abs (real_inv x) = real_inv (real_abs x)"
+lemma REAL_ABS_INV: "real_abs (real_inv x) = real_inv (real_abs x)"
by (import hollight REAL_ABS_INV)
-lemma REAL_ABS_DIV: "ALL (x::hollight.real) xa::hollight.real.
- real_abs (real_div x xa) = real_div (real_abs x) (real_abs xa)"
+lemma REAL_ABS_DIV: "real_abs (real_div x xa) = real_div (real_abs x) (real_abs xa)"
by (import hollight REAL_ABS_DIV)
-lemma REAL_INV_MUL: "ALL (x::hollight.real) y::hollight.real.
- real_inv (real_mul x y) = real_mul (real_inv x) (real_inv y)"
+lemma REAL_INV_MUL: "real_inv (real_mul x y) = real_mul (real_inv x) (real_inv y)"
by (import hollight REAL_INV_MUL)
-lemma REAL_INV_DIV: "ALL (x::hollight.real) xa::hollight.real.
- real_inv (real_div x xa) = real_div xa x"
+lemma REAL_INV_DIV: "real_inv (real_div x xa) = real_div xa x"
by (import hollight REAL_INV_DIV)
-lemma REAL_POW_MUL: "ALL (x::hollight.real) (y::hollight.real) n::nat.
- real_pow (real_mul x y) n = real_mul (real_pow x n) (real_pow y n)"
+lemma REAL_POW_MUL: "real_pow (real_mul x y) n = real_mul (real_pow x n) (real_pow y n)"
by (import hollight REAL_POW_MUL)
-lemma REAL_POW_INV: "ALL (x::hollight.real) n::nat.
- real_pow (real_inv x) n = real_inv (real_pow x n)"
+lemma REAL_POW_INV: "real_pow (real_inv x) n = real_inv (real_pow x n)"
by (import hollight REAL_POW_INV)
-lemma REAL_POW_DIV: "ALL (x::hollight.real) (xa::hollight.real) xb::nat.
- real_pow (real_div x xa) xb = real_div (real_pow x xb) (real_pow xa xb)"
+lemma REAL_INV_POW: "real_inv (real_pow x xa) = real_pow (real_inv x) xa"
+ by (import hollight REAL_INV_POW)
+
+lemma REAL_POW_DIV: "real_pow (real_div x xa) xb = real_div (real_pow x xb) (real_pow xa xb)"
by (import hollight REAL_POW_DIV)
-lemma REAL_POW_ADD: "ALL (x::hollight.real) (m::nat) n::nat.
- real_pow x (m + n) = real_mul (real_pow x m) (real_pow x n)"
+lemma REAL_POW_ADD: "real_pow x (m + n) = real_mul (real_pow x m) (real_pow x n)"
by (import hollight REAL_POW_ADD)
-lemma REAL_POW_NZ: "ALL (x::hollight.real) n::nat.
- x ~= real_of_num 0 --> real_pow x n ~= real_of_num 0"
+lemma REAL_POW_NZ: "x ~= real_of_num 0 ==> real_pow x n ~= real_of_num 0"
by (import hollight REAL_POW_NZ)
-lemma REAL_POW_SUB: "ALL (x::hollight.real) (m::nat) n::nat.
- x ~= real_of_num 0 & <= m n -->
- real_pow x (n - m) = real_div (real_pow x n) (real_pow x m)"
+lemma REAL_POW_SUB: "x ~= real_of_num 0 & m <= n
+==> real_pow x (n - m) = real_div (real_pow x n) (real_pow x m)"
by (import hollight REAL_POW_SUB)
-lemma REAL_LT_IMP_NZ: "ALL x::hollight.real. real_lt (real_of_num 0) x --> x ~= real_of_num 0"
+lemma REAL_LT_IMP_NZ: "real_lt (real_of_num 0) x ==> x ~= real_of_num 0"
by (import hollight REAL_LT_IMP_NZ)
-lemma REAL_LT_LCANCEL_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) x & real_lt (real_mul x y) (real_mul x z) -->
- real_lt y z"
+lemma REAL_LT_LCANCEL_IMP: "real_lt (real_of_num 0) x & real_lt (real_mul x y) (real_mul x z)
+==> real_lt y z"
by (import hollight REAL_LT_LCANCEL_IMP)
-lemma REAL_LT_RCANCEL_IMP: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb & real_lt (real_mul x xb) (real_mul xa xb) -->
- real_lt x xa"
+lemma REAL_LT_RCANCEL_IMP: "real_lt (real_of_num 0) xb & real_lt (real_mul x xb) (real_mul xa xb)
+==> real_lt x xa"
by (import hollight REAL_LT_RCANCEL_IMP)
-lemma REAL_LE_LCANCEL_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) x & real_le (real_mul x y) (real_mul x z) -->
- real_le y z"
+lemma REAL_LE_LCANCEL_IMP: "real_lt (real_of_num 0) x & real_le (real_mul x y) (real_mul x z)
+==> real_le y z"
by (import hollight REAL_LE_LCANCEL_IMP)
-lemma REAL_LE_RCANCEL_IMP: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb & real_le (real_mul x xb) (real_mul xa xb) -->
- real_le x xa"
+lemma REAL_LE_RCANCEL_IMP: "real_lt (real_of_num 0) xb & real_le (real_mul x xb) (real_mul xa xb)
+==> real_le x xa"
by (import hollight REAL_LE_RCANCEL_IMP)
-lemma REAL_LE_RMUL_EQ: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) z -->
- real_le (real_mul x z) (real_mul y z) = real_le x y"
+lemma REAL_LE_RMUL_EQ: "real_lt (real_of_num 0) z
+==> real_le (real_mul x z) (real_mul y z) = real_le x y"
by (import hollight REAL_LE_RMUL_EQ)
-lemma REAL_LE_LMUL_EQ: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) z -->
- real_le (real_mul z x) (real_mul z y) = real_le x y"
+lemma REAL_LE_LMUL_EQ: "real_lt (real_of_num 0) z
+==> real_le (real_mul z x) (real_mul z y) = real_le x y"
by (import hollight REAL_LE_LMUL_EQ)
-lemma REAL_LT_RMUL_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_lt (real_mul x xb) (real_mul xa xb) = real_lt x xa"
+lemma REAL_LT_RMUL_EQ: "real_lt (real_of_num 0) xb
+==> real_lt (real_mul x xb) (real_mul xa xb) = real_lt x xa"
by (import hollight REAL_LT_RMUL_EQ)
-lemma REAL_LT_LMUL_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_lt (real_mul xb x) (real_mul xb xa) = real_lt x xa"
+lemma REAL_LT_LMUL_EQ: "real_lt (real_of_num 0) xb
+==> real_lt (real_mul xb x) (real_mul xb xa) = real_lt x xa"
by (import hollight REAL_LT_LMUL_EQ)
-lemma REAL_LE_RDIV_EQ: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) z -->
- real_le x (real_div y z) = real_le (real_mul x z) y"
+lemma REAL_LE_MUL_EQ: "(ALL x y.
+ real_lt (real_of_num 0) x -->
+ real_le (real_of_num 0) (real_mul x y) = real_le (real_of_num 0) y) &
+(ALL x y.
+ real_lt (real_of_num 0) y -->
+ real_le (real_of_num 0) (real_mul x y) = real_le (real_of_num 0) x)"
+ by (import hollight REAL_LE_MUL_EQ)
+
+lemma REAL_LT_MUL_EQ: "(ALL x y.
+ real_lt (real_of_num 0) x -->
+ real_lt (real_of_num 0) (real_mul x y) = real_lt (real_of_num 0) y) &
+(ALL x y.
+ real_lt (real_of_num 0) y -->
+ real_lt (real_of_num 0) (real_mul x y) = real_lt (real_of_num 0) x)"
+ by (import hollight REAL_LT_MUL_EQ)
+
+lemma REAL_MUL_POS_LT: "real_lt (real_of_num 0) (real_mul x y) =
+(real_lt (real_of_num 0) x & real_lt (real_of_num 0) y |
+ real_lt x (real_of_num 0) & real_lt y (real_of_num 0))"
+ by (import hollight REAL_MUL_POS_LT)
+
+lemma REAL_MUL_POS_LE: "real_le (real_of_num 0) (real_mul x xa) =
+(x = real_of_num 0 |
+ xa = real_of_num 0 |
+ real_lt (real_of_num 0) x & real_lt (real_of_num 0) xa |
+ real_lt x (real_of_num 0) & real_lt xa (real_of_num 0))"
+ by (import hollight REAL_MUL_POS_LE)
+
+lemma REAL_LE_RDIV_EQ: "real_lt (real_of_num 0) z
+==> real_le x (real_div y z) = real_le (real_mul x z) y"
by (import hollight REAL_LE_RDIV_EQ)
-lemma REAL_LE_LDIV_EQ: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- real_lt (real_of_num 0) z -->
- real_le (real_div x z) y = real_le x (real_mul y z)"
+lemma REAL_LE_LDIV_EQ: "real_lt (real_of_num 0) z
+==> real_le (real_div x z) y = real_le x (real_mul y z)"
by (import hollight REAL_LE_LDIV_EQ)
-lemma REAL_LT_RDIV_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_lt x (real_div xa xb) = real_lt (real_mul x xb) xa"
+lemma REAL_LT_RDIV_EQ: "real_lt (real_of_num 0) xb
+==> real_lt x (real_div xa xb) = real_lt (real_mul x xb) xa"
by (import hollight REAL_LT_RDIV_EQ)
-lemma REAL_LT_LDIV_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_lt (real_div x xb) xa = real_lt x (real_mul xa xb)"
+lemma REAL_LT_LDIV_EQ: "real_lt (real_of_num 0) xb
+==> real_lt (real_div x xb) xa = real_lt x (real_mul xa xb)"
by (import hollight REAL_LT_LDIV_EQ)
-lemma REAL_EQ_RDIV_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- (x = real_div xa xb) = (real_mul x xb = xa)"
+lemma REAL_EQ_RDIV_EQ: "real_lt (real_of_num 0) xb ==> (x = real_div xa xb) = (real_mul x xb = xa)"
by (import hollight REAL_EQ_RDIV_EQ)
-lemma REAL_EQ_LDIV_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- (real_div x xb = xa) = (x = real_mul xa xb)"
+lemma REAL_EQ_LDIV_EQ: "real_lt (real_of_num 0) xb ==> (real_div x xb = xa) = (x = real_mul xa xb)"
by (import hollight REAL_EQ_LDIV_EQ)
-lemma REAL_LT_DIV2_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_lt (real_div x xb) (real_div xa xb) = real_lt x xa"
+lemma REAL_LT_DIV2_EQ: "real_lt (real_of_num 0) xb
+==> real_lt (real_div x xb) (real_div xa xb) = real_lt x xa"
by (import hollight REAL_LT_DIV2_EQ)
-lemma REAL_LE_DIV2_EQ: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- real_lt (real_of_num 0) xb -->
- real_le (real_div x xb) (real_div xa xb) = real_le x xa"
+lemma REAL_LE_DIV2_EQ: "real_lt (real_of_num 0) xb
+==> real_le (real_div x xb) (real_div xa xb) = real_le x xa"
by (import hollight REAL_LE_DIV2_EQ)
-lemma REAL_MUL_2: "ALL x::hollight.real.
- real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x = real_add x x"
+lemma REAL_MUL_2: "real_mul (real_of_num 2) x = real_add x x"
by (import hollight REAL_MUL_2)
-lemma REAL_POW_EQ_0: "ALL (x::hollight.real) n::nat.
- (real_pow x n = real_of_num 0) = (x = real_of_num 0 & n ~= 0)"
+lemma REAL_POW_EQ_0: "(real_pow x n = real_of_num 0) = (x = real_of_num 0 & n ~= 0)"
by (import hollight REAL_POW_EQ_0)
-lemma REAL_LE_MUL2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_le (real_of_num 0) w &
- real_le w x & real_le (real_of_num 0) y & real_le y z -->
- real_le (real_mul w y) (real_mul x z)"
+lemma REAL_LE_MUL2: "real_le (real_of_num 0) w &
+real_le w x & real_le (real_of_num 0) y & real_le y z
+==> real_le (real_mul w y) (real_mul x z)"
by (import hollight REAL_LE_MUL2)
-lemma REAL_LT_MUL2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
- z::hollight.real.
- real_le (real_of_num 0) w &
- real_lt w x & real_le (real_of_num 0) y & real_lt y z -->
- real_lt (real_mul w y) (real_mul x z)"
+lemma REAL_LT_MUL2: "real_le (real_of_num 0) w &
+real_lt w x & real_le (real_of_num 0) y & real_lt y z
+==> real_lt (real_mul w y) (real_mul x z)"
by (import hollight REAL_LT_MUL2)
-lemma REAL_LT_SQUARE: "ALL x::hollight.real.
- real_lt (real_of_num 0) (real_mul x x) = (x ~= real_of_num 0)"
+lemma REAL_LT_SQUARE: "real_lt (real_of_num 0) (real_mul x x) = (x ~= real_of_num 0)"
by (import hollight REAL_LT_SQUARE)
-lemma REAL_INV_LE_1: "ALL x::hollight.real.
- real_le (real_of_num (NUMERAL_BIT1 0)) x -->
- real_le (real_inv x) (real_of_num (NUMERAL_BIT1 0))"
+lemma REAL_POW_1: "real_pow x 1 = x"
+ by (import hollight REAL_POW_1)
+
+lemma REAL_POW_ONE: "real_pow (real_of_num 1) n = real_of_num 1"
+ by (import hollight REAL_POW_ONE)
+
+lemma REAL_LT_INV2: "real_lt (real_of_num 0) x & real_lt x y
+==> real_lt (real_inv y) (real_inv x)"
+ by (import hollight REAL_LT_INV2)
+
+lemma REAL_LE_INV2: "real_lt (real_of_num 0) x & real_le x y
+==> real_le (real_inv y) (real_inv x)"
+ by (import hollight REAL_LE_INV2)
+
+lemma REAL_LT_LINV: "real_lt (real_of_num 0) y & real_lt (real_inv y) x
+==> real_lt (real_inv x) y"
+ by (import hollight REAL_LT_LINV)
+
+lemma REAL_LT_RINV: "real_lt (real_of_num 0) x & real_lt x (real_inv y)
+==> real_lt y (real_inv x)"
+ by (import hollight REAL_LT_RINV)
+
+lemma REAL_LE_LINV: "real_lt (real_of_num 0) y & real_le (real_inv y) x
+==> real_le (real_inv x) y"
+ by (import hollight REAL_LE_LINV)
+
+lemma REAL_LE_RINV: "real_lt (real_of_num 0) x & real_le x (real_inv y)
+==> real_le y (real_inv x)"
+ by (import hollight REAL_LE_RINV)
+
+lemma REAL_INV_LE_1: "real_le (real_of_num 1) x ==> real_le (real_inv x) (real_of_num 1)"
by (import hollight REAL_INV_LE_1)
-lemma REAL_POW_LE_1: "ALL (n::nat) x::hollight.real.
- real_le (real_of_num (NUMERAL_BIT1 0)) x -->
- real_le (real_of_num (NUMERAL_BIT1 0)) (real_pow x n)"
- by (import hollight REAL_POW_LE_1)
-
-lemma REAL_POW_1_LE: "ALL (n::nat) x::hollight.real.
- real_le (real_of_num 0) x & real_le x (real_of_num (NUMERAL_BIT1 0)) -->
- real_le (real_pow x n) (real_of_num (NUMERAL_BIT1 0))"
- by (import hollight REAL_POW_1_LE)
-
-lemma REAL_POW_1: "ALL x::hollight.real. real_pow x (NUMERAL_BIT1 0) = x"
- by (import hollight REAL_POW_1)
-
-lemma REAL_POW_ONE: "ALL n::nat.
- real_pow (real_of_num (NUMERAL_BIT1 0)) n = real_of_num (NUMERAL_BIT1 0)"
- by (import hollight REAL_POW_ONE)
-
-lemma REAL_LT_INV2: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num 0) x & real_lt x y -->
- real_lt (real_inv y) (real_inv x)"
- by (import hollight REAL_LT_INV2)
-
-lemma REAL_LE_INV2: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num 0) x & real_le x y -->
- real_le (real_inv y) (real_inv x)"
- by (import hollight REAL_LE_INV2)
-
-lemma REAL_INV_1_LE: "ALL x::hollight.real.
- real_lt (real_of_num 0) x & real_le x (real_of_num (NUMERAL_BIT1 0)) -->
- real_le (real_of_num (NUMERAL_BIT1 0)) (real_inv x)"
+lemma REAL_INV_1_LE: "real_lt (real_of_num 0) x & real_le x (real_of_num 1)
+==> real_le (real_of_num 1) (real_inv x)"
by (import hollight REAL_INV_1_LE)
-lemma REAL_SUB_INV: "ALL (x::hollight.real) xa::hollight.real.
- x ~= real_of_num 0 & xa ~= real_of_num 0 -->
- real_sub (real_inv x) (real_inv xa) =
- real_div (real_sub xa x) (real_mul x xa)"
+lemma REAL_INV_LT_1: "real_lt (real_of_num 1) x ==> real_lt (real_inv x) (real_of_num 1)"
+ by (import hollight REAL_INV_LT_1)
+
+lemma REAL_INV_1_LT: "real_lt (real_of_num 0) x & real_lt x (real_of_num 1)
+==> real_lt (real_of_num 1) (real_inv x)"
+ by (import hollight REAL_INV_1_LT)
+
+lemma REAL_SUB_INV: "x ~= real_of_num 0 & xa ~= real_of_num 0
+==> real_sub (real_inv x) (real_inv xa) =
+ real_div (real_sub xa x) (real_mul x xa)"
by (import hollight REAL_SUB_INV)
-lemma REAL_DOWN: "ALL d::hollight.real.
- real_lt (real_of_num 0) d -->
- (EX x::hollight.real. real_lt (real_of_num 0) x & real_lt x d)"
+lemma REAL_DOWN: "real_lt (real_of_num 0) d ==> EX x. real_lt (real_of_num 0) x & real_lt x d"
by (import hollight REAL_DOWN)
-lemma REAL_DOWN2: "ALL (d1::hollight.real) d2::hollight.real.
- real_lt (real_of_num 0) d1 & real_lt (real_of_num 0) d2 -->
- (EX e::hollight.real.
- real_lt (real_of_num 0) e & real_lt e d1 & real_lt e d2)"
+lemma REAL_DOWN2: "real_lt (real_of_num 0) d1 & real_lt (real_of_num 0) d2
+==> EX e. real_lt (real_of_num 0) e & real_lt e d1 & real_lt e d2"
by (import hollight REAL_DOWN2)
-lemma REAL_POW_LE2: "ALL (n::nat) (x::hollight.real) y::hollight.real.
- real_le (real_of_num 0) x & real_le x y -->
- real_le (real_pow x n) (real_pow y n)"
+lemma REAL_POW_LE2: "real_le (real_of_num 0) x & real_le x y
+==> real_le (real_pow x n) (real_pow y n)"
by (import hollight REAL_POW_LE2)
-lemma REAL_POW_MONO: "ALL (m::nat) (n::nat) x::hollight.real.
- real_le (real_of_num (NUMERAL_BIT1 0)) x & <= m n -->
- real_le (real_pow x m) (real_pow x n)"
+lemma REAL_POW_LE_1: "real_le (real_of_num 1) x ==> real_le (real_of_num 1) (real_pow x n)"
+ by (import hollight REAL_POW_LE_1)
+
+lemma REAL_POW_1_LE: "real_le (real_of_num 0) x & real_le x (real_of_num 1)
+==> real_le (real_pow x n) (real_of_num 1)"
+ by (import hollight REAL_POW_1_LE)
+
+lemma REAL_POW_MONO: "real_le (real_of_num 1) x & m <= n ==> real_le (real_pow x m) (real_pow x n)"
by (import hollight REAL_POW_MONO)
-lemma REAL_POW_LT2: "ALL (n::nat) (x::hollight.real) y::hollight.real.
- n ~= 0 & real_le (real_of_num 0) x & real_lt x y -->
- real_lt (real_pow x n) (real_pow y n)"
+lemma REAL_POW_LT2: "n ~= 0 & real_le (real_of_num 0) x & real_lt x y
+==> real_lt (real_pow x n) (real_pow y n)"
by (import hollight REAL_POW_LT2)
-lemma REAL_POW_MONO_LT: "ALL (m::nat) (n::nat) x::hollight.real.
- real_lt (real_of_num (NUMERAL_BIT1 0)) x & < m n -->
- real_lt (real_pow x m) (real_pow x n)"
+lemma REAL_POW_LT_1: "n ~= 0 & real_lt (real_of_num 1) x
+==> real_lt (real_of_num 1) (real_pow x n)"
+ by (import hollight REAL_POW_LT_1)
+
+lemma REAL_POW_1_LT: "n ~= 0 & real_le (real_of_num 0) x & real_lt x (real_of_num 1)
+==> real_lt (real_pow x n) (real_of_num 1)"
+ by (import hollight REAL_POW_1_LT)
+
+lemma REAL_POW_MONO_LT: "real_lt (real_of_num 1) x & m < n ==> real_lt (real_pow x m) (real_pow x n)"
by (import hollight REAL_POW_MONO_LT)
-lemma REAL_POW_POW: "ALL (x::hollight.real) (m::nat) n::nat.
- real_pow (real_pow x m) n = real_pow x (m * n)"
+lemma REAL_POW_POW: "real_pow (real_pow x m) n = real_pow x (m * n)"
by (import hollight REAL_POW_POW)
-lemma REAL_EQ_RCANCEL_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- z ~= real_of_num 0 & real_mul x z = real_mul y z --> x = y"
+lemma REAL_EQ_RCANCEL_IMP: "z ~= real_of_num 0 & real_mul x z = real_mul y z ==> x = y"
by (import hollight REAL_EQ_RCANCEL_IMP)
-lemma REAL_EQ_LCANCEL_IMP: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
- xb ~= real_of_num 0 & real_mul xb x = real_mul xb xa --> x = xa"
+lemma REAL_EQ_LCANCEL_IMP: "xb ~= real_of_num 0 & real_mul xb x = real_mul xb xa ==> x = xa"
by (import hollight REAL_EQ_LCANCEL_IMP)
-lemma REAL_LT_DIV: "ALL (x::hollight.real) xa::hollight.real.
- real_lt (real_of_num 0) x & real_lt (real_of_num 0) xa -->
- real_lt (real_of_num 0) (real_div x xa)"
+lemma REAL_LT_DIV: "real_lt (real_of_num 0) x & real_lt (real_of_num 0) xa
+==> real_lt (real_of_num 0) (real_div x xa)"
by (import hollight REAL_LT_DIV)
-lemma REAL_LE_DIV: "ALL (x::hollight.real) xa::hollight.real.
- real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
- real_le (real_of_num 0) (real_div x xa)"
+lemma REAL_LE_DIV: "real_le (real_of_num 0) x & real_le (real_of_num 0) xa
+==> real_le (real_of_num 0) (real_div x xa)"
by (import hollight REAL_LE_DIV)
-lemma REAL_DIV_POW2: "ALL (x::hollight.real) (m::nat) n::nat.
- x ~= real_of_num 0 -->
- real_div (real_pow x m) (real_pow x n) =
- COND (<= n m) (real_pow x (m - n)) (real_inv (real_pow x (n - m)))"
+lemma REAL_DIV_POW2: "x ~= real_of_num 0
+==> real_div (real_pow x m) (real_pow x n) =
+ (if n <= m then real_pow x (m - n) else real_inv (real_pow x (n - m)))"
by (import hollight REAL_DIV_POW2)
-lemma REAL_DIV_POW2_ALT: "ALL (x::hollight.real) (m::nat) n::nat.
- x ~= real_of_num 0 -->
- real_div (real_pow x m) (real_pow x n) =
- COND (< n m) (real_pow x (m - n)) (real_inv (real_pow x (n - m)))"
+lemma REAL_DIV_POW2_ALT: "x ~= real_of_num 0
+==> real_div (real_pow x m) (real_pow x n) =
+ (if n < m then real_pow x (m - n) else real_inv (real_pow x (n - m)))"
by (import hollight REAL_DIV_POW2_ALT)
-lemma REAL_LT_POW2: "ALL x::nat.
- real_lt (real_of_num 0)
- (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x)"
+lemma REAL_LT_POW2: "real_lt (real_of_num 0) (real_pow (real_of_num 2) x)"
by (import hollight REAL_LT_POW2)
-lemma REAL_LE_POW2: "ALL n::nat.
- real_le (real_of_num (NUMERAL_BIT1 0))
- (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) n)"
+lemma REAL_LE_POW2: "real_le (real_of_num 1) (real_pow (real_of_num 2) n)"
by (import hollight REAL_LE_POW2)
-lemma REAL_POW2_ABS: "ALL x::hollight.real.
- real_pow (real_abs x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)) =
- real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))"
+lemma REAL_POW2_ABS: "real_pow (real_abs x) 2 = real_pow x 2"
by (import hollight REAL_POW2_ABS)
-lemma REAL_LE_SQUARE_ABS: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_abs x) (real_abs y) =
- real_le (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
- (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
+lemma REAL_LE_SQUARE_ABS: "real_le (real_abs x) (real_abs y) = real_le (real_pow x 2) (real_pow y 2)"
by (import hollight REAL_LE_SQUARE_ABS)
-lemma REAL_SOS_EQ_0: "ALL (x::hollight.real) y::hollight.real.
- (real_add (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
- (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
- real_of_num 0) =
- (x = real_of_num 0 & y = real_of_num 0)"
+lemma REAL_LT_SQUARE_ABS: "real_lt (real_abs x) (real_abs xa) = real_lt (real_pow x 2) (real_pow xa 2)"
+ by (import hollight REAL_LT_SQUARE_ABS)
+
+lemma REAL_EQ_SQUARE_ABS: "(real_abs x = real_abs xa) = (real_pow x 2 = real_pow xa 2)"
+ by (import hollight REAL_EQ_SQUARE_ABS)
+
+lemma REAL_LE_POW_2: "real_le (real_of_num 0) (real_pow x 2)"
+ by (import hollight REAL_LE_POW_2)
+
+lemma REAL_SOS_EQ_0: "(real_add (real_pow x 2) (real_pow y 2) = real_of_num 0) =
+(x = real_of_num 0 & y = real_of_num 0)"
by (import hollight REAL_SOS_EQ_0)
-lemma REAL_WLOG_LE: "(ALL (x::hollight.real) y::hollight.real.
- (P::hollight.real => hollight.real => bool) x y = P y x) &
-(ALL (x::hollight.real) y::hollight.real. real_le x y --> P x y) -->
-(ALL x::hollight.real. All (P x))"
+lemma REAL_POW_ZERO: "real_pow (real_of_num 0) n =
+(if n = 0 then real_of_num 1 else real_of_num 0)"
+ by (import hollight REAL_POW_ZERO)
+
+lemma REAL_POW_MONO_INV: "real_le (real_of_num 0) x & real_le x (real_of_num 1) & n <= m
+==> real_le (real_pow x m) (real_pow x n)"
+ by (import hollight REAL_POW_MONO_INV)
+
+lemma REAL_POW_LE2_REV: "n ~= 0 & real_le (real_of_num 0) y & real_le (real_pow x n) (real_pow y n)
+==> real_le x y"
+ by (import hollight REAL_POW_LE2_REV)
+
+lemma REAL_POW_LT2_REV: "real_le (real_of_num 0) y & real_lt (real_pow x n) (real_pow y n)
+==> real_lt x y"
+ by (import hollight REAL_POW_LT2_REV)
+
+lemma REAL_POW_EQ: "x ~= 0 &
+real_le (real_of_num 0) xa &
+real_le (real_of_num 0) xb & real_pow xa x = real_pow xb x
+==> xa = xb"
+ by (import hollight REAL_POW_EQ)
+
+lemma REAL_POW_EQ_ABS: "n ~= 0 & real_pow x n = real_pow y n ==> real_abs x = real_abs y"
+ by (import hollight REAL_POW_EQ_ABS)
+
+lemma REAL_POW_EQ_1_IMP: "n ~= 0 & real_pow x n = real_of_num 1 ==> real_abs x = real_of_num 1"
+ by (import hollight REAL_POW_EQ_1_IMP)
+
+lemma REAL_POW_EQ_1: "(real_pow x n = real_of_num 1) =
+(real_abs x = real_of_num 1 & (real_lt x (real_of_num 0) --> even n) |
+ n = 0)"
+ by (import hollight REAL_POW_EQ_1)
+
+lemma REAL_POW_LT2_ODD: "real_lt x y & odd n ==> real_lt (real_pow x n) (real_pow y n)"
+ by (import hollight REAL_POW_LT2_ODD)
+
+lemma REAL_POW_LE2_ODD: "real_le xa xb & odd x ==> real_le (real_pow xa x) (real_pow xb x)"
+ by (import hollight REAL_POW_LE2_ODD)
+
+lemma REAL_POW_LT2_ODD_EQ: "odd n ==> real_lt (real_pow x n) (real_pow y n) = real_lt x y"
+ by (import hollight REAL_POW_LT2_ODD_EQ)
+
+lemma REAL_POW_LE2_ODD_EQ: "odd n ==> real_le (real_pow x n) (real_pow y n) = real_le x y"
+ by (import hollight REAL_POW_LE2_ODD_EQ)
+
+lemma REAL_POW_EQ_ODD_EQ: "odd x ==> (real_pow xa x = real_pow xb x) = (xa = xb)"
+ by (import hollight REAL_POW_EQ_ODD_EQ)
+
+lemma REAL_POW_EQ_ODD: "odd n & real_pow x n = real_pow y n ==> x = y"
+ by (import hollight REAL_POW_EQ_ODD)
+
+lemma REAL_POW_EQ_EQ: "(real_pow x n = real_pow y n) =
+(if even n then n = 0 | real_abs x = real_abs y else x = y)"
+ by (import hollight REAL_POW_EQ_EQ)
+
+definition
+ real_sgn :: "hollight.real => hollight.real" where
+ "real_sgn ==
+%u. if real_lt (real_of_num 0) u then real_of_num 1
+ else if real_lt u (real_of_num 0) then real_neg (real_of_num 1)
+ else real_of_num 0"
+
+lemma DEF_real_sgn: "real_sgn =
+(%u. if real_lt (real_of_num 0) u then real_of_num 1
+ else if real_lt u (real_of_num 0) then real_neg (real_of_num 1)
+ else real_of_num 0)"
+ by (import hollight DEF_real_sgn)
+
+lemma REAL_SGN_0: "real_sgn (real_of_num 0) = real_of_num 0"
+ by (import hollight REAL_SGN_0)
+
+lemma REAL_SGN_NEG: "real_sgn (real_neg x) = real_neg (real_sgn x)"
+ by (import hollight REAL_SGN_NEG)
+
+lemma REAL_SGN_ABS: "real_mul (real_sgn x) (real_abs x) = x"
+ by (import hollight REAL_SGN_ABS)
+
+lemma REAL_ABS_SGN: "real_abs (real_sgn x) = real_sgn (real_abs x)"
+ by (import hollight REAL_ABS_SGN)
+
+lemma REAL_SGN: "real_sgn x = real_div x (real_abs x)"
+ by (import hollight REAL_SGN)
+
+lemma REAL_SGN_MUL: "real_sgn (real_mul x xa) = real_mul (real_sgn x) (real_sgn xa)"
+ by (import hollight REAL_SGN_MUL)
+
+lemma REAL_SGN_INV: "real_sgn (real_inv x) = real_sgn x"
+ by (import hollight REAL_SGN_INV)
+
+lemma REAL_SGN_DIV: "real_sgn (real_div x xa) = real_div (real_sgn x) (real_sgn xa)"
+ by (import hollight REAL_SGN_DIV)
+
+lemma REAL_WLOG_LE: "(ALL x y. P x y = P y x) & (ALL x y. real_le x y --> P x y) ==> P x xa"
by (import hollight REAL_WLOG_LE)
-lemma REAL_WLOG_LT: "(ALL x::hollight.real. (P::hollight.real => hollight.real => bool) x x) &
-(ALL (x::hollight.real) y::hollight.real. P x y = P y x) &
-(ALL (x::hollight.real) y::hollight.real. real_lt x y --> P x y) -->
-(ALL x::hollight.real. All (P x))"
+lemma REAL_WLOG_LT: "(ALL x. P x x) & (ALL x y. P x y = P y x) & (ALL x y. real_lt x y --> P x y)
+==> P x xa"
by (import hollight REAL_WLOG_LT)
-definition mod_real :: "hollight.real => hollight.real => hollight.real => bool" where
- "mod_real ==
-%(u::hollight.real) (ua::hollight.real) ub::hollight.real.
- EX q::hollight.real. real_sub ua ub = real_mul q u"
-
-lemma DEF_mod_real: "mod_real =
-(%(u::hollight.real) (ua::hollight.real) ub::hollight.real.
- EX q::hollight.real. real_sub ua ub = real_mul q u)"
- by (import hollight DEF_mod_real)
-
-definition DECIMAL :: "nat => nat => hollight.real" where
- "DECIMAL == %(u::nat) ua::nat. real_div (real_of_num u) (real_of_num ua)"
-
-lemma DEF_DECIMAL: "DECIMAL = (%(u::nat) ua::nat. real_div (real_of_num u) (real_of_num ua))"
+definition
+ DECIMAL :: "nat => nat => hollight.real" where
+ "DECIMAL == %u ua. real_div (real_of_num u) (real_of_num ua)"
+
+lemma DEF_DECIMAL: "DECIMAL = (%u ua. real_div (real_of_num u) (real_of_num ua))"
by (import hollight DEF_DECIMAL)
-lemma RAT_LEMMA1: "(y1::hollight.real) ~= real_of_num 0 &
-(y2::hollight.real) ~= real_of_num 0 -->
-real_add (real_div (x1::hollight.real) y1)
- (real_div (x2::hollight.real) y2) =
-real_mul (real_add (real_mul x1 y2) (real_mul x2 y1))
- (real_mul (real_inv y1) (real_inv y2))"
+lemma RAT_LEMMA1: "y1 ~= real_of_num 0 & y2 ~= real_of_num 0
+==> real_add (real_div x1 y1) (real_div x2 y2) =
+ real_mul (real_add (real_mul x1 y2) (real_mul x2 y1))
+ (real_mul (real_inv y1) (real_inv y2))"
by (import hollight RAT_LEMMA1)
-lemma RAT_LEMMA2: "real_lt (real_of_num 0) (y1::hollight.real) &
-real_lt (real_of_num 0) (y2::hollight.real) -->
-real_add (real_div (x1::hollight.real) y1)
- (real_div (x2::hollight.real) y2) =
-real_mul (real_add (real_mul x1 y2) (real_mul x2 y1))
- (real_mul (real_inv y1) (real_inv y2))"
+lemma RAT_LEMMA2: "real_lt (real_of_num 0) y1 & real_lt (real_of_num 0) y2
+==> real_add (real_div x1 y1) (real_div x2 y2) =
+ real_mul (real_add (real_mul x1 y2) (real_mul x2 y1))
+ (real_mul (real_inv y1) (real_inv y2))"
by (import hollight RAT_LEMMA2)
-lemma RAT_LEMMA3: "real_lt (real_of_num 0) (y1::hollight.real) &
-real_lt (real_of_num 0) (y2::hollight.real) -->
-real_sub (real_div (x1::hollight.real) y1)
- (real_div (x2::hollight.real) y2) =
-real_mul (real_sub (real_mul x1 y2) (real_mul x2 y1))
- (real_mul (real_inv y1) (real_inv y2))"
+lemma RAT_LEMMA3: "real_lt (real_of_num 0) y1 & real_lt (real_of_num 0) y2
+==> real_sub (real_div x1 y1) (real_div x2 y2) =
+ real_mul (real_sub (real_mul x1 y2) (real_mul x2 y1))
+ (real_mul (real_inv y1) (real_inv y2))"
by (import hollight RAT_LEMMA3)
-lemma RAT_LEMMA4: "real_lt (real_of_num 0) (y1::hollight.real) &
-real_lt (real_of_num 0) (y2::hollight.real) -->
-real_le (real_div (x1::hollight.real) y1)
- (real_div (x2::hollight.real) y2) =
-real_le (real_mul x1 y2) (real_mul x2 y1)"
+lemma RAT_LEMMA4: "real_lt (real_of_num 0) y1 & real_lt (real_of_num 0) y2
+==> real_le (real_div x1 y1) (real_div x2 y2) =
+ real_le (real_mul x1 y2) (real_mul x2 y1)"
by (import hollight RAT_LEMMA4)
-lemma RAT_LEMMA5: "real_lt (real_of_num 0) (y1::hollight.real) &
-real_lt (real_of_num 0) (y2::hollight.real) -->
-(real_div (x1::hollight.real) y1 = real_div (x2::hollight.real) y2) =
-(real_mul x1 y2 = real_mul x2 y1)"
+lemma RAT_LEMMA5: "real_lt (real_of_num 0) y1 & real_lt (real_of_num 0) y2
+==> (real_div x1 y1 = real_div x2 y2) = (real_mul x1 y2 = real_mul x2 y1)"
by (import hollight RAT_LEMMA5)
-definition is_int :: "hollight.real => bool" where
- "is_int ==
-%u::hollight.real.
- EX n::nat. u = real_of_num n | u = real_neg (real_of_num n)"
-
-lemma DEF_is_int: "is_int =
-(%u::hollight.real.
- EX n::nat. u = real_of_num n | u = real_neg (real_of_num n))"
- by (import hollight DEF_is_int)
-
-typedef (open) int = "Collect is_int" morphisms "dest_int" "mk_int"
- apply (rule light_ex_imp_nonempty[where t="real_of_num (NUMERAL 0)"])
+lemma REAL_INTEGRAL: "(ALL x. real_mul (real_of_num 0) x = real_of_num 0) &
+(ALL x y z. (real_add x y = real_add x z) = (y = z)) &
+(ALL w x y z.
+ (real_add (real_mul w y) (real_mul x z) =
+ real_add (real_mul w z) (real_mul x y)) =
+ (w = x | y = z))"
+ by (import hollight REAL_INTEGRAL)
+
+definition
+ integer :: "hollight.real => bool" where
+ "integer == %u. EX n. real_abs u = real_of_num n"
+
+lemma DEF_integer: "integer = (%u. EX n. real_abs u = real_of_num n)"
+ by (import hollight DEF_integer)
+
+lemma is_int: "integer x = (EX n. x = real_of_num n | x = real_neg (real_of_num n))"
+ by (import hollight is_int)
+
+typedef (open) int = "Collect integer" morphisms "real_of_int" "int_of_real"
+ apply (rule light_ex_imp_nonempty[where t="Eps integer"])
by (import hollight TYDEF_int)
syntax
- dest_int :: _
+ real_of_int :: _
syntax
- mk_int :: _
+ int_of_real :: _
lemmas "TYDEF_int_@intern" = typedef_hol2hollight
[where a="a :: hollight.int" and r=r ,
OF type_definition_int]
-lemma dest_int_rep: "ALL x::hollight.int.
- EX n::nat.
- dest_int x = real_of_num n | dest_int x = real_neg (real_of_num n)"
+lemma dest_int_rep: "EX n. hollight.real_of_int x = real_of_num n |
+ hollight.real_of_int x = real_neg (real_of_num n)"
by (import hollight dest_int_rep)
-definition int_le :: "hollight.int => hollight.int => bool" where
- "int_le ==
-%(u::hollight.int) ua::hollight.int. real_le (dest_int u) (dest_int ua)"
-
-lemma DEF_int_le: "int_le =
-(%(u::hollight.int) ua::hollight.int. real_le (dest_int u) (dest_int ua))"
+definition
+ int_le :: "hollight.int => hollight.int => bool" where
+ "int_le == %u ua. real_le (hollight.real_of_int u) (hollight.real_of_int ua)"
+
+lemma DEF_int_le: "int_le = (%u ua. real_le (hollight.real_of_int u) (hollight.real_of_int ua))"
by (import hollight DEF_int_le)
-definition int_lt :: "hollight.int => hollight.int => bool" where
- "int_lt ==
-%(u::hollight.int) ua::hollight.int. real_lt (dest_int u) (dest_int ua)"
-
-lemma DEF_int_lt: "int_lt =
-(%(u::hollight.int) ua::hollight.int. real_lt (dest_int u) (dest_int ua))"
+definition
+ int_lt :: "hollight.int => hollight.int => bool" where
+ "int_lt == %u ua. real_lt (hollight.real_of_int u) (hollight.real_of_int ua)"
+
+lemma DEF_int_lt: "int_lt = (%u ua. real_lt (hollight.real_of_int u) (hollight.real_of_int ua))"
by (import hollight DEF_int_lt)
-definition int_ge :: "hollight.int => hollight.int => bool" where
- "int_ge ==
-%(u::hollight.int) ua::hollight.int.
- hollight.real_ge (dest_int u) (dest_int ua)"
-
-lemma DEF_int_ge: "int_ge =
-(%(u::hollight.int) ua::hollight.int.
- hollight.real_ge (dest_int u) (dest_int ua))"
+definition
+ int_ge :: "hollight.int => hollight.int => bool" where
+ "int_ge == %u ua. real_ge (hollight.real_of_int u) (hollight.real_of_int ua)"
+
+lemma DEF_int_ge: "int_ge = (%u ua. real_ge (hollight.real_of_int u) (hollight.real_of_int ua))"
by (import hollight DEF_int_ge)
-definition int_gt :: "hollight.int => hollight.int => bool" where
- "int_gt ==
-%(u::hollight.int) ua::hollight.int.
- hollight.real_gt (dest_int u) (dest_int ua)"
-
-lemma DEF_int_gt: "int_gt =
-(%(u::hollight.int) ua::hollight.int.
- hollight.real_gt (dest_int u) (dest_int ua))"
+definition
+ int_gt :: "hollight.int => hollight.int => bool" where
+ "int_gt == %u ua. real_gt (hollight.real_of_int u) (hollight.real_of_int ua)"
+
+lemma DEF_int_gt: "int_gt = (%u ua. real_gt (hollight.real_of_int u) (hollight.real_of_int ua))"
by (import hollight DEF_int_gt)
-definition int_of_num :: "nat => hollight.int" where
- "int_of_num == %u::nat. mk_int (real_of_num u)"
-
-lemma DEF_int_of_num: "int_of_num = (%u::nat. mk_int (real_of_num u))"
+definition
+ int_of_num :: "nat => hollight.int" where
+ "int_of_num == %u. int_of_real (real_of_num u)"
+
+lemma DEF_int_of_num: "int_of_num = (%u. int_of_real (real_of_num u))"
by (import hollight DEF_int_of_num)
-lemma int_of_num_th: "ALL x::nat. dest_int (int_of_num x) = real_of_num x"
+lemma int_of_num_th: "hollight.real_of_int (int_of_num x) = real_of_num x"
by (import hollight int_of_num_th)
-definition int_neg :: "hollight.int => hollight.int" where
- "int_neg == %u::hollight.int. mk_int (real_neg (dest_int u))"
-
-lemma DEF_int_neg: "int_neg = (%u::hollight.int. mk_int (real_neg (dest_int u)))"
+definition
+ int_neg :: "hollight.int => hollight.int" where
+ "int_neg == %u. int_of_real (real_neg (hollight.real_of_int u))"
+
+lemma DEF_int_neg: "int_neg = (%u. int_of_real (real_neg (hollight.real_of_int u)))"
by (import hollight DEF_int_neg)
-lemma int_neg_th: "ALL x::hollight.int. dest_int (int_neg x) = real_neg (dest_int x)"
+lemma int_neg_th: "hollight.real_of_int (int_neg x) = real_neg (hollight.real_of_int x)"
by (import hollight int_neg_th)
-definition int_add :: "hollight.int => hollight.int => hollight.int" where
+definition
+ int_add :: "hollight.int => hollight.int => hollight.int" where
"int_add ==
-%(u::hollight.int) ua::hollight.int.
- mk_int (real_add (dest_int u) (dest_int ua))"
+%u ua.
+ int_of_real (real_add (hollight.real_of_int u) (hollight.real_of_int ua))"
lemma DEF_int_add: "int_add =
-(%(u::hollight.int) ua::hollight.int.
- mk_int (real_add (dest_int u) (dest_int ua)))"
+(%u ua.
+ int_of_real
+ (real_add (hollight.real_of_int u) (hollight.real_of_int ua)))"
by (import hollight DEF_int_add)
-lemma int_add_th: "ALL (x::hollight.int) xa::hollight.int.
- dest_int (int_add x xa) = real_add (dest_int x) (dest_int xa)"
+lemma int_add_th: "hollight.real_of_int (int_add x xa) =
+real_add (hollight.real_of_int x) (hollight.real_of_int xa)"
by (import hollight int_add_th)
-definition int_sub :: "hollight.int => hollight.int => hollight.int" where
+definition
+ int_sub :: "hollight.int => hollight.int => hollight.int" where
"int_sub ==
-%(u::hollight.int) ua::hollight.int.
- mk_int (real_sub (dest_int u) (dest_int ua))"
+%u ua.
+ int_of_real (real_sub (hollight.real_of_int u) (hollight.real_of_int ua))"
lemma DEF_int_sub: "int_sub =
-(%(u::hollight.int) ua::hollight.int.
- mk_int (real_sub (dest_int u) (dest_int ua)))"
+(%u ua.
+ int_of_real
+ (real_sub (hollight.real_of_int u) (hollight.real_of_int ua)))"
by (import hollight DEF_int_sub)
-lemma int_sub_th: "ALL (x::hollight.int) xa::hollight.int.
- dest_int (int_sub x xa) = real_sub (dest_int x) (dest_int xa)"
+lemma int_sub_th: "hollight.real_of_int (int_sub x xa) =
+real_sub (hollight.real_of_int x) (hollight.real_of_int xa)"
by (import hollight int_sub_th)
-definition int_mul :: "hollight.int => hollight.int => hollight.int" where
+definition
+ int_mul :: "hollight.int => hollight.int => hollight.int" where
"int_mul ==
-%(u::hollight.int) ua::hollight.int.
- mk_int (real_mul (dest_int u) (dest_int ua))"
+%u ua.
+ int_of_real (real_mul (hollight.real_of_int u) (hollight.real_of_int ua))"
lemma DEF_int_mul: "int_mul =
-(%(u::hollight.int) ua::hollight.int.
- mk_int (real_mul (dest_int u) (dest_int ua)))"
+(%u ua.
+ int_of_real
+ (real_mul (hollight.real_of_int u) (hollight.real_of_int ua)))"
by (import hollight DEF_int_mul)
-lemma int_mul_th: "ALL (x::hollight.int) y::hollight.int.
- dest_int (int_mul x y) = real_mul (dest_int x) (dest_int y)"
+lemma int_mul_th: "hollight.real_of_int (int_mul x y) =
+real_mul (hollight.real_of_int x) (hollight.real_of_int y)"
by (import hollight int_mul_th)
-definition int_abs :: "hollight.int => hollight.int" where
- "int_abs == %u::hollight.int. mk_int (real_abs (dest_int u))"
-
-lemma DEF_int_abs: "int_abs = (%u::hollight.int. mk_int (real_abs (dest_int u)))"
+definition
+ int_abs :: "hollight.int => hollight.int" where
+ "int_abs == %u. int_of_real (real_abs (hollight.real_of_int u))"
+
+lemma DEF_int_abs: "int_abs = (%u. int_of_real (real_abs (hollight.real_of_int u)))"
by (import hollight DEF_int_abs)
-lemma int_abs_th: "ALL x::hollight.int. dest_int (int_abs x) = real_abs (dest_int x)"
+lemma int_abs_th: "hollight.real_of_int (int_abs x) = real_abs (hollight.real_of_int x)"
by (import hollight int_abs_th)
-definition int_max :: "hollight.int => hollight.int => hollight.int" where
+definition
+ int_sgn :: "hollight.int => hollight.int" where
+ "int_sgn == %u. int_of_real (real_sgn (hollight.real_of_int u))"
+
+lemma DEF_int_sgn: "int_sgn = (%u. int_of_real (real_sgn (hollight.real_of_int u)))"
+ by (import hollight DEF_int_sgn)
+
+lemma int_sgn_th: "hollight.real_of_int (int_sgn x) = real_sgn (hollight.real_of_int x)"
+ by (import hollight int_sgn_th)
+
+definition
+ int_max :: "hollight.int => hollight.int => hollight.int" where
"int_max ==
-%(u::hollight.int) ua::hollight.int.
- mk_int (real_max (dest_int u) (dest_int ua))"
+%u ua.
+ int_of_real (real_max (hollight.real_of_int u) (hollight.real_of_int ua))"
lemma DEF_int_max: "int_max =
-(%(u::hollight.int) ua::hollight.int.
- mk_int (real_max (dest_int u) (dest_int ua)))"
+(%u ua.
+ int_of_real
+ (real_max (hollight.real_of_int u) (hollight.real_of_int ua)))"
by (import hollight DEF_int_max)
-lemma int_max_th: "ALL (x::hollight.int) y::hollight.int.
- dest_int (int_max x y) = real_max (dest_int x) (dest_int y)"
+lemma int_max_th: "hollight.real_of_int (int_max x y) =
+real_max (hollight.real_of_int x) (hollight.real_of_int y)"
by (import hollight int_max_th)
-definition int_min :: "hollight.int => hollight.int => hollight.int" where
+definition
+ int_min :: "hollight.int => hollight.int => hollight.int" where
"int_min ==
-%(u::hollight.int) ua::hollight.int.
- mk_int (real_min (dest_int u) (dest_int ua))"
+%u ua.
+ int_of_real (real_min (hollight.real_of_int u) (hollight.real_of_int ua))"
lemma DEF_int_min: "int_min =
-(%(u::hollight.int) ua::hollight.int.
- mk_int (real_min (dest_int u) (dest_int ua)))"
+(%u ua.
+ int_of_real
+ (real_min (hollight.real_of_int u) (hollight.real_of_int ua)))"
by (import hollight DEF_int_min)
-lemma int_min_th: "ALL (x::hollight.int) y::hollight.int.
- dest_int (int_min x y) = real_min (dest_int x) (dest_int y)"
+lemma int_min_th: "hollight.real_of_int (int_min x y) =
+real_min (hollight.real_of_int x) (hollight.real_of_int y)"
by (import hollight int_min_th)
-definition int_pow :: "hollight.int => nat => hollight.int" where
- "int_pow == %(u::hollight.int) ua::nat. mk_int (real_pow (dest_int u) ua)"
-
-lemma DEF_int_pow: "int_pow = (%(u::hollight.int) ua::nat. mk_int (real_pow (dest_int u) ua))"
+definition
+ int_pow :: "hollight.int => nat => hollight.int" where
+ "int_pow == %u ua. int_of_real (real_pow (hollight.real_of_int u) ua)"
+
+lemma DEF_int_pow: "int_pow = (%u ua. int_of_real (real_pow (hollight.real_of_int u) ua))"
by (import hollight DEF_int_pow)
-lemma int_pow_th: "ALL (x::hollight.int) xa::nat.
- dest_int (int_pow x xa) = real_pow (dest_int x) xa"
+lemma int_pow_th: "hollight.real_of_int (int_pow x xa) = real_pow (hollight.real_of_int x) xa"
by (import hollight int_pow_th)
-lemma INT_IMAGE: "ALL x::hollight.int.
- (EX n::nat. x = int_of_num n) | (EX n::nat. x = int_neg (int_of_num n))"
+lemma INT_IMAGE: "(EX n. x = int_of_num n) | (EX n. x = int_neg (int_of_num n))"
by (import hollight INT_IMAGE)
-lemma INT_LT_DISCRETE: "ALL (x::hollight.int) y::hollight.int.
- int_lt x y = int_le (int_add x (int_of_num (NUMERAL_BIT1 0))) y"
+lemma INT_LT_DISCRETE: "int_lt x y = int_le (int_add x (int_of_num 1)) y"
by (import hollight INT_LT_DISCRETE)
-lemma INT_GT_DISCRETE: "ALL (x::hollight.int) xa::hollight.int.
- int_gt x xa = int_ge x (int_add xa (int_of_num (NUMERAL_BIT1 0)))"
+lemma INT_GT_DISCRETE: "int_gt x xa = int_ge x (int_add xa (int_of_num 1))"
by (import hollight INT_GT_DISCRETE)
-lemma INT_FORALL_POS: "(ALL n::nat. (P::hollight.int => bool) (int_of_num n)) =
-(ALL i::hollight.int. int_le (int_of_num 0) i --> P i)"
+lemma INT_FORALL_POS: "(ALL n. P (int_of_num n)) = (ALL i. int_le (int_of_num 0) i --> P i)"
by (import hollight INT_FORALL_POS)
-lemma INT_ABS_MUL_1: "ALL (x::hollight.int) y::hollight.int.
- (int_abs (int_mul x y) = int_of_num (NUMERAL_BIT1 0)) =
- (int_abs x = int_of_num (NUMERAL_BIT1 0) &
- int_abs y = int_of_num (NUMERAL_BIT1 0))"
+lemma INT_EXISTS_POS: "(EX n. P (int_of_num n)) = (EX i. int_le (int_of_num 0) i & P i)"
+ by (import hollight INT_EXISTS_POS)
+
+lemma INT_FORALL_ABS: "(ALL n. x (int_of_num n)) = (ALL xa. x (int_abs xa))"
+ by (import hollight INT_FORALL_ABS)
+
+lemma INT_EXISTS_ABS: "(EX n. P (int_of_num n)) = (EX x. P (int_abs x))"
+ by (import hollight INT_EXISTS_ABS)
+
+lemma INT_ABS_MUL_1: "(int_abs (int_mul x y) = int_of_num 1) =
+(int_abs x = int_of_num 1 & int_abs y = int_of_num 1)"
by (import hollight INT_ABS_MUL_1)
-lemma INT_POW: "int_pow (x::hollight.int) 0 = int_of_num (NUMERAL_BIT1 0) &
-(ALL xa::nat. int_pow x (Suc xa) = int_mul x (int_pow x xa))"
+lemma INT_WOP: "(EX x. int_le (int_of_num 0) x & P x) =
+(EX x. int_le (int_of_num 0) x &
+ P x & (ALL y. int_le (int_of_num 0) y & P y --> int_le x y))"
+ by (import hollight INT_WOP)
+
+lemma INT_POW: "int_pow x 0 = int_of_num 1 &
+(ALL xa. int_pow x (Suc xa) = int_mul x (int_pow x xa))"
by (import hollight INT_POW)
-lemma INT_ABS: "ALL x::hollight.int.
- int_abs x = COND (int_le (int_of_num 0) x) x (int_neg x)"
+lemma INT_ABS: "int_abs x = (if int_le (int_of_num 0) x then x else int_neg x)"
by (import hollight INT_ABS)
-lemma INT_GE: "ALL (x::hollight.int) xa::hollight.int. int_ge x xa = int_le xa x"
+lemma INT_GE: "int_ge x xa = int_le xa x"
by (import hollight INT_GE)
-lemma INT_GT: "ALL (x::hollight.int) xa::hollight.int. int_gt x xa = int_lt xa x"
+lemma INT_GT: "int_gt x xa = int_lt xa x"
by (import hollight INT_GT)
-lemma INT_LT: "ALL (x::hollight.int) xa::hollight.int. int_lt x xa = (~ int_le xa x)"
+lemma INT_LT: "int_lt x xa = (~ int_le xa x)"
by (import hollight INT_LT)
-lemma INT_ARCH: "ALL (x::hollight.int) d::hollight.int.
- d ~= int_of_num 0 --> (EX c::hollight.int. int_lt x (int_mul c d))"
+lemma INT_ARCH: "d ~= int_of_num 0 ==> EX c. int_lt x (int_mul c d)"
by (import hollight INT_ARCH)
-definition mod_int :: "hollight.int => hollight.int => hollight.int => bool" where
- "mod_int ==
-%(u::hollight.int) (ua::hollight.int) ub::hollight.int.
- EX q::hollight.int. int_sub ua ub = int_mul q u"
-
-lemma DEF_mod_int: "mod_int =
-(%(u::hollight.int) (ua::hollight.int) ub::hollight.int.
- EX q::hollight.int. int_sub ua ub = int_mul q u)"
- by (import hollight DEF_mod_int)
-
-definition IN :: "'A => ('A => bool) => bool" where
- "IN == %(u::'A::type) ua::'A::type => bool. ua u"
-
-lemma DEF_IN: "IN = (%(u::'A::type) ua::'A::type => bool. ua u)"
- by (import hollight DEF_IN)
-
-lemma EXTENSION: "ALL (x::'A::type => bool) xa::'A::type => bool.
- (x = xa) = (ALL xb::'A::type. IN xb x = IN xb xa)"
- by (import hollight EXTENSION)
-
-definition GSPEC :: "('A => bool) => 'A => bool" where
- "GSPEC == %u::'A::type => bool. u"
-
-lemma DEF_GSPEC: "GSPEC = (%u::'A::type => bool. u)"
- by (import hollight DEF_GSPEC)
-
-definition SETSPEC :: "'q_37056 => bool => 'q_37056 => bool" where
- "SETSPEC == %(u::'q_37056::type) (ua::bool) ub::'q_37056::type. ua & u = ub"
-
-lemma DEF_SETSPEC: "SETSPEC = (%(u::'q_37056::type) (ua::bool) ub::'q_37056::type. ua & u = ub)"
- by (import hollight DEF_SETSPEC)
-
-lemma IN_ELIM_THM: "(ALL (P::(bool => 'q_37089::type => bool) => bool) x::'q_37089::type.
- IN x (GSPEC (%v::'q_37089::type. P (SETSPEC v))) =
- P (%(p::bool) t::'q_37089::type. p & x = t)) &
-(ALL (p::'q_37120::type => bool) x::'q_37120::type.
- IN x
- (GSPEC (%v::'q_37120::type. EX y::'q_37120::type. SETSPEC v (p y) y)) =
- p x) &
-(ALL (P::(bool => 'q_37148::type => bool) => bool) x::'q_37148::type.
- GSPEC (%v::'q_37148::type. P (SETSPEC v)) x =
- P (%(p::bool) t::'q_37148::type. p & x = t)) &
-(ALL (p::'q_37177::type => bool) x::'q_37177::type.
- GSPEC (%v::'q_37177::type. EX y::'q_37177::type. SETSPEC v (p y) y) x =
- p x) &
-(ALL (p::'q_37194::type => bool) x::'q_37194::type. IN x p = p x)"
- by (import hollight IN_ELIM_THM)
-
-definition EMPTY :: "'A => bool" where
- "EMPTY == %x::'A::type. False"
-
-lemma DEF_EMPTY: "EMPTY = (%x::'A::type. False)"
- by (import hollight DEF_EMPTY)
-
-definition INSERT :: "'A => ('A => bool) => 'A => bool" where
- "INSERT == %(u::'A::type) (ua::'A::type => bool) y::'A::type. IN y ua | y = u"
-
-lemma DEF_INSERT: "INSERT =
-(%(u::'A::type) (ua::'A::type => bool) y::'A::type. IN y ua | y = u)"
- by (import hollight DEF_INSERT)
+lemma INT_DIVMOD_EXIST_0: "EX x xa.
+ if n = int_of_num 0 then x = int_of_num 0 & xa = m
+ else int_le (int_of_num 0) xa &
+ int_lt xa (int_abs n) & m = int_add (int_mul x n) xa"
+ by (import hollight INT_DIVMOD_EXIST_0)
consts
- UNIV :: "'A => bool"
+ div :: "hollight.int => hollight.int => hollight.int" ("div")
defs
- UNIV_def: "hollight.UNIV == %x::'A::type. True"
-
-lemma DEF_UNIV: "hollight.UNIV = (%x::'A::type. True)"
- by (import hollight DEF_UNIV)
+ div_def: "div ==
+SOME q.
+ EX r. ALL m n.
+ if n = int_of_num 0 then q m n = int_of_num 0 & r m n = m
+ else int_le (int_of_num 0) (r m n) &
+ int_lt (r m n) (int_abs n) &
+ m = int_add (int_mul (q m n) n) (r m n)"
+
+lemma DEF_div: "div =
+(SOME q.
+ EX r. ALL m n.
+ if n = int_of_num 0 then q m n = int_of_num 0 & r m n = m
+ else int_le (int_of_num 0) (r m n) &
+ int_lt (r m n) (int_abs n) &
+ m = int_add (int_mul (q m n) n) (r m n))"
+ by (import hollight DEF_div)
+
+definition
+ rem :: "hollight.int => hollight.int => hollight.int" where
+ "rem ==
+SOME r.
+ ALL m n.
+ if n = int_of_num 0 then div m n = int_of_num 0 & r m n = m
+ else int_le (int_of_num 0) (r m n) &
+ int_lt (r m n) (int_abs n) &
+ m = int_add (int_mul (div m n) n) (r m n)"
+
+lemma DEF_rem: "rem =
+(SOME r.
+ ALL m n.
+ if n = int_of_num 0 then div m n = int_of_num 0 & r m n = m
+ else int_le (int_of_num 0) (r m n) &
+ int_lt (r m n) (int_abs n) &
+ m = int_add (int_mul (div m n) n) (r m n))"
+ by (import hollight DEF_rem)
+
+lemma INT_DIVISION: "n ~= int_of_num 0
+==> m = int_add (int_mul (div m n) n) (rem m n) &
+ int_le (int_of_num 0) (rem m n) & int_lt (rem m n) (int_abs n)"
+ by (import hollight INT_DIVISION)
+
+lemma sth: "(ALL x y z. int_add x (int_add y z) = int_add (int_add x y) z) &
+(ALL x y. int_add x y = int_add y x) &
+(ALL x. int_add (int_of_num 0) x = x) &
+(ALL x y z. int_mul x (int_mul y z) = int_mul (int_mul x y) z) &
+(ALL x y. int_mul x y = int_mul y x) &
+(ALL x. int_mul (int_of_num 1) x = x) &
+(ALL x. int_mul (int_of_num 0) x = int_of_num 0) &
+(ALL x y z. int_mul x (int_add y z) = int_add (int_mul x y) (int_mul x z)) &
+(ALL x. int_pow x 0 = int_of_num 1) &
+(ALL x xa. int_pow x (Suc xa) = int_mul x (int_pow x xa))"
+ by (import hollight sth)
+
+lemma INT_INTEGRAL: "(ALL x. int_mul (int_of_num 0) x = int_of_num 0) &
+(ALL x y z. (int_add x y = int_add x z) = (y = z)) &
+(ALL w x y z.
+ (int_add (int_mul w y) (int_mul x z) =
+ int_add (int_mul w z) (int_mul x y)) =
+ (w = x | y = z))"
+ by (import hollight INT_INTEGRAL)
+
+lemma INT_DIVMOD_UNIQ: "m = int_add (int_mul q n) r & int_le (int_of_num 0) r & int_lt r (int_abs n)
+==> div m n = q & rem m n = r"
+ by (import hollight INT_DIVMOD_UNIQ)
consts
- UNION :: "('A => bool) => ('A => bool) => 'A => bool"
+ eqeq :: "'A => 'A => ('A => 'A => bool) => bool"
+
+defs
+ eqeq_def: "hollight.eqeq == %(u::'A) (ua::'A) ub::'A => 'A => bool. ub u ua"
+
+lemma DEF__equal__equal_: "hollight.eqeq = (%(u::'A) (ua::'A) ub::'A => 'A => bool. ub u ua)"
+ by (import hollight DEF__equal__equal_)
+
+definition
+ real_mod :: "hollight.real => hollight.real => hollight.real => bool" where
+ "real_mod == %u ua ub. EX q. integer q & real_sub ua ub = real_mul q u"
+
+lemma DEF_real_mod: "real_mod = (%u ua ub. EX q. integer q & real_sub ua ub = real_mul q u)"
+ by (import hollight DEF_real_mod)
+
+definition
+ int_divides :: "hollight.int => hollight.int => bool" where
+ "int_divides == %u ua. EX x. ua = int_mul u x"
+
+lemma DEF_int_divides: "int_divides = (%u ua. EX x. ua = int_mul u x)"
+ by (import hollight DEF_int_divides)
+
+consts
+ int_mod :: "hollight.int => hollight.int => hollight.int => bool"
defs
- UNION_def: "hollight.UNION ==
-%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u | IN x ua) x)"
-
-lemma DEF_UNION: "hollight.UNION =
-(%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u | IN x ua) x))"
- by (import hollight DEF_UNION)
-
-definition UNIONS :: "(('A => bool) => bool) => 'A => bool" where
- "UNIONS ==
-%u::('A::type => bool) => bool.
- GSPEC
- (%ua::'A::type.
- EX x::'A::type.
- SETSPEC ua (EX ua::'A::type => bool. IN ua u & IN x ua) x)"
-
-lemma DEF_UNIONS: "UNIONS =
-(%u::('A::type => bool) => bool.
- GSPEC
- (%ua::'A::type.
- EX x::'A::type.
- SETSPEC ua (EX ua::'A::type => bool. IN ua u & IN x ua) x))"
- by (import hollight DEF_UNIONS)
+ int_mod_def: "hollight.int_mod == %u ua ub. int_divides u (int_sub ua ub)"
+
+lemma DEF_int_mod: "hollight.int_mod = (%u ua ub. int_divides u (int_sub ua ub))"
+ by (import hollight DEF_int_mod)
+
+lemma int_congruent: "hollight.eqeq x xa (hollight.int_mod xb) =
+(EX d. int_sub x xa = int_mul xb d)"
+ by (import hollight int_congruent)
consts
- INTER :: "('A => bool) => ('A => bool) => 'A => bool"
+ int_coprime :: "hollight.int * hollight.int => bool"
+
+defs
+ int_coprime_def: "hollight.int_coprime ==
+%u. EX x y. int_add (int_mul (fst u) x) (int_mul (snd u) y) = int_of_num 1"
+
+lemma DEF_int_coprime: "hollight.int_coprime =
+(%u. EX x y. int_add (int_mul (fst u) x) (int_mul (snd u) y) = int_of_num 1)"
+ by (import hollight DEF_int_coprime)
+
+lemma FORALL_UNCURRY: "All (P::('A => 'B => 'C) => bool) =
+(ALL f::'A * 'B => 'C. P (%(a::'A) b::'B. f (a, b)))"
+ by (import hollight FORALL_UNCURRY)
+
+lemma EXISTS_UNCURRY: "Ex (x::('A => 'B => 'C) => bool) =
+(EX f::'A * 'B => 'C. x (%(a::'A) b::'B. f (a, b)))"
+ by (import hollight EXISTS_UNCURRY)
+
+lemma WF_INT_MEASURE: "(ALL x::'A. int_le (int_of_num (0::nat)) ((m::'A => hollight.int) x)) &
+(ALL x::'A. (ALL y::'A. int_lt (m y) (m x) --> (P::'A => bool) y) --> P x)
+==> P (x::'A)"
+ by (import hollight WF_INT_MEASURE)
+
+lemma WF_INT_MEASURE_2: "(ALL (x::'A) y::'B.
+ int_le (int_of_num (0::nat)) ((m::'A => 'B => hollight.int) x y)) &
+(ALL (x::'A) y::'B.
+ (ALL (x'::'A) y'::'B.
+ int_lt (m x' y') (m x y) --> (P::'A => 'B => bool) x' y') -->
+ P x y)
+==> P (x::'A) (xa::'B)"
+ by (import hollight WF_INT_MEASURE_2)
+
+lemma INT_GCD_EXISTS: "EX d. int_divides d a &
+ int_divides d b & (EX x y. d = int_add (int_mul a x) (int_mul b y))"
+ by (import hollight INT_GCD_EXISTS)
+
+lemma INT_GCD_EXISTS_POS: "EX d. int_le (int_of_num 0) d &
+ int_divides d a &
+ int_divides d b & (EX x y. d = int_add (int_mul a x) (int_mul b y))"
+ by (import hollight INT_GCD_EXISTS_POS)
+
+consts
+ int_gcd :: "hollight.int * hollight.int => hollight.int"
defs
- INTER_def: "hollight.INTER ==
-%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u & IN x ua) x)"
-
-lemma DEF_INTER: "hollight.INTER =
-(%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u & IN x ua) x))"
- by (import hollight DEF_INTER)
-
-definition INTERS :: "(('A => bool) => bool) => 'A => bool" where
- "INTERS ==
-%u::('A::type => bool) => bool.
- GSPEC
- (%ua::'A::type.
- EX x::'A::type.
- SETSPEC ua (ALL ua::'A::type => bool. IN ua u --> IN x ua) x)"
-
-lemma DEF_INTERS: "INTERS =
-(%u::('A::type => bool) => bool.
- GSPEC
- (%ua::'A::type.
- EX x::'A::type.
- SETSPEC ua (ALL ua::'A::type => bool. IN ua u --> IN x ua) x))"
- by (import hollight DEF_INTERS)
-
-definition DIFF :: "('A => bool) => ('A => bool) => 'A => bool" where
- "DIFF ==
-%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u & ~ IN x ua) x)"
-
-lemma DEF_DIFF: "DIFF =
-(%(u::'A::type => bool) ua::'A::type => bool.
- GSPEC
- (%ub::'A::type. EX x::'A::type. SETSPEC ub (IN x u & ~ IN x ua) x))"
- by (import hollight DEF_DIFF)
-
-lemma INSERT: "INSERT (x::'A::type) (s::'A::type => bool) =
-GSPEC (%u::'A::type. EX y::'A::type. SETSPEC u (IN y s | y = x) y)"
+ int_gcd_def: "hollight.int_gcd ==
+SOME d.
+ ALL a b.
+ int_le (int_of_num 0) (d (a, b)) &
+ int_divides (d (a, b)) a &
+ int_divides (d (a, b)) b &
+ (EX x y. d (a, b) = int_add (int_mul a x) (int_mul b y))"
+
+lemma DEF_int_gcd: "hollight.int_gcd =
+(SOME d.
+ ALL a b.
+ int_le (int_of_num 0) (d (a, b)) &
+ int_divides (d (a, b)) a &
+ int_divides (d (a, b)) b &
+ (EX x y. d (a, b) = int_add (int_mul a x) (int_mul b y)))"
+ by (import hollight DEF_int_gcd)
+
+definition
+ num_of_int :: "hollight.int => nat" where
+ "num_of_int == %u. SOME n. int_of_num n = u"
+
+lemma DEF_num_of_int: "num_of_int = (%u. SOME n. int_of_num n = u)"
+ by (import hollight DEF_num_of_int)
+
+lemma NUM_OF_INT_OF_NUM: "num_of_int (int_of_num x) = x"
+ by (import hollight NUM_OF_INT_OF_NUM)
+
+lemma INT_OF_NUM_OF_INT: "int_le (int_of_num 0) x ==> int_of_num (num_of_int x) = x"
+ by (import hollight INT_OF_NUM_OF_INT)
+
+lemma NUM_OF_INT: "int_le (int_of_num 0) x = (int_of_num (num_of_int x) = x)"
+ by (import hollight NUM_OF_INT)
+
+definition
+ num_divides :: "nat => nat => bool" where
+ "num_divides == %u ua. int_divides (int_of_num u) (int_of_num ua)"
+
+lemma DEF_num_divides: "num_divides = (%u ua. int_divides (int_of_num u) (int_of_num ua))"
+ by (import hollight DEF_num_divides)
+
+definition
+ num_mod :: "nat => nat => nat => bool" where
+ "num_mod ==
+%u ua ub. hollight.int_mod (int_of_num u) (int_of_num ua) (int_of_num ub)"
+
+lemma DEF_num_mod: "num_mod =
+(%u ua ub. hollight.int_mod (int_of_num u) (int_of_num ua) (int_of_num ub))"
+ by (import hollight DEF_num_mod)
+
+lemma num_congruent: "hollight.eqeq x xa (num_mod xb) =
+hollight.eqeq (int_of_num x) (int_of_num xa)
+ (hollight.int_mod (int_of_num xb))"
+ by (import hollight num_congruent)
+
+definition
+ num_coprime :: "nat * nat => bool" where
+ "num_coprime ==
+%u. hollight.int_coprime (int_of_num (fst u), int_of_num (snd u))"
+
+lemma DEF_num_coprime: "num_coprime =
+(%u. hollight.int_coprime (int_of_num (fst u), int_of_num (snd u)))"
+ by (import hollight DEF_num_coprime)
+
+definition
+ num_gcd :: "nat * nat => nat" where
+ "num_gcd ==
+%u. num_of_int (hollight.int_gcd (int_of_num (fst u), int_of_num (snd u)))"
+
+lemma DEF_num_gcd: "num_gcd =
+(%u. num_of_int (hollight.int_gcd (int_of_num (fst u), int_of_num (snd u))))"
+ by (import hollight DEF_num_gcd)
+
+lemma NUM_GCD: "int_of_num (num_gcd (x, xa)) =
+hollight.int_gcd (int_of_num x, int_of_num xa)"
+ by (import hollight NUM_GCD)
+
+lemma IN_ELIM_THM: "(ALL (P::(bool => 'q_43295 => bool) => bool) x::'q_43295.
+ (x : {v::'q_43295. P (SETSPEC v)}) =
+ P (%(p::bool) t::'q_43295. p & x = t)) &
+(ALL (p::'q_43326 => bool) x::'q_43326.
+ (x : {v::'q_43326. EX y::'q_43326. p y & v = y}) = p x) &
+(ALL (P::(bool => 'q_43354 => bool) => bool) x::'q_43354.
+ {v::'q_43354. P (SETSPEC v)} x =
+ P (%(p::bool) t::'q_43354. p & x = t)) &
+(ALL (p::'q_43383 => bool) x::'q_43383.
+ {v::'q_43383. EX y::'q_43383. p y & v = y} x = p x) &
+(ALL (p::'q_43400 => bool) x::'q_43400. (x : p) = p x)"
+ by (import hollight IN_ELIM_THM)
+
+lemma INSERT: "insert (x::'A) (s::'A => bool) = {u::'A. EX y::'A. (y : s | y = x) & u = y}"
by (import hollight INSERT)
-definition DELETE :: "('A => bool) => 'A => 'A => bool" where
- "DELETE ==
-%(u::'A::type => bool) ua::'A::type.
- GSPEC (%ub::'A::type. EX y::'A::type. SETSPEC ub (IN y u & y ~= ua) y)"
-
-lemma DEF_DELETE: "DELETE =
-(%(u::'A::type => bool) ua::'A::type.
- GSPEC (%ub::'A::type. EX y::'A::type. SETSPEC ub (IN y u & y ~= ua) y))"
- by (import hollight DEF_DELETE)
-
-definition SUBSET :: "('A => bool) => ('A => bool) => bool" where
- "SUBSET ==
-%(u::'A::type => bool) ua::'A::type => bool.
- ALL x::'A::type. IN x u --> IN x ua"
-
-lemma DEF_SUBSET: "SUBSET =
-(%(u::'A::type => bool) ua::'A::type => bool.
- ALL x::'A::type. IN x u --> IN x ua)"
- by (import hollight DEF_SUBSET)
-
-definition PSUBSET :: "('A => bool) => ('A => bool) => bool" where
- "PSUBSET ==
-%(u::'A::type => bool) ua::'A::type => bool. SUBSET u ua & u ~= ua"
-
-lemma DEF_PSUBSET: "PSUBSET =
-(%(u::'A::type => bool) ua::'A::type => bool. SUBSET u ua & u ~= ua)"
- by (import hollight DEF_PSUBSET)
-
-definition DISJOINT :: "('A => bool) => ('A => bool) => bool" where
- "DISJOINT ==
-%(u::'A::type => bool) ua::'A::type => bool. hollight.INTER u ua = EMPTY"
-
-lemma DEF_DISJOINT: "DISJOINT =
-(%(u::'A::type => bool) ua::'A::type => bool. hollight.INTER u ua = EMPTY)"
- by (import hollight DEF_DISJOINT)
-
-definition SING :: "('A => bool) => bool" where
- "SING == %u::'A::type => bool. EX x::'A::type. u = INSERT x EMPTY"
-
-lemma DEF_SING: "SING = (%u::'A::type => bool. EX x::'A::type. u = INSERT x EMPTY)"
+definition
+ SING :: "('A => bool) => bool" where
+ "SING == %u::'A => bool. EX x::'A. u = {x}"
+
+lemma DEF_SING: "SING = (%u::'A => bool. EX x::'A. u = {x})"
by (import hollight DEF_SING)
-definition FINITE :: "('A => bool) => bool" where
- "FINITE ==
-%a::'A::type => bool.
- ALL FINITE'::('A::type => bool) => bool.
- (ALL a::'A::type => bool.
- a = EMPTY |
- (EX (x::'A::type) s::'A::type => bool.
- a = INSERT x s & FINITE' s) -->
- FINITE' a) -->
- FINITE' a"
-
-lemma DEF_FINITE: "FINITE =
-(%a::'A::type => bool.
- ALL FINITE'::('A::type => bool) => bool.
- (ALL a::'A::type => bool.
- a = EMPTY |
- (EX (x::'A::type) s::'A::type => bool.
- a = INSERT x s & FINITE' s) -->
- FINITE' a) -->
- FINITE' a)"
- by (import hollight DEF_FINITE)
-
-definition INFINITE :: "('A => bool) => bool" where
- "INFINITE == %u::'A::type => bool. ~ FINITE u"
-
-lemma DEF_INFINITE: "INFINITE = (%u::'A::type => bool. ~ FINITE u)"
- by (import hollight DEF_INFINITE)
-
-definition IMAGE :: "('A => 'B) => ('A => bool) => 'B => bool" where
- "IMAGE ==
-%(u::'A::type => 'B::type) ua::'A::type => bool.
- GSPEC
- (%ub::'B::type.
- EX y::'B::type. SETSPEC ub (EX x::'A::type. IN x ua & y = u x) y)"
-
-lemma DEF_IMAGE: "IMAGE =
-(%(u::'A::type => 'B::type) ua::'A::type => bool.
- GSPEC
- (%ub::'B::type.
- EX y::'B::type. SETSPEC ub (EX x::'A::type. IN x ua & y = u x) y))"
- by (import hollight DEF_IMAGE)
-
-definition INJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
+definition
+ INJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
"INJ ==
-%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- (ALL x::'A::type. IN x ua --> IN (u x) ub) &
- (ALL (x::'A::type) y::'A::type. IN x ua & IN y ua & u x = u y --> x = y)"
+%(u::'A => 'B) (ua::'A => bool) ub::'B => bool.
+ (ALL x::'A. x : ua --> u x : ub) &
+ (ALL (x::'A) y::'A. x : ua & y : ua & u x = u y --> x = y)"
lemma DEF_INJ: "INJ =
-(%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- (ALL x::'A::type. IN x ua --> IN (u x) ub) &
- (ALL (x::'A::type) y::'A::type.
- IN x ua & IN y ua & u x = u y --> x = y))"
+(%(u::'A => 'B) (ua::'A => bool) ub::'B => bool.
+ (ALL x::'A. x : ua --> u x : ub) &
+ (ALL (x::'A) y::'A. x : ua & y : ua & u x = u y --> x = y))"
by (import hollight DEF_INJ)
-definition SURJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
+definition
+ SURJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
"SURJ ==
-%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- (ALL x::'A::type. IN x ua --> IN (u x) ub) &
- (ALL x::'B::type. IN x ub --> (EX y::'A::type. IN y ua & u y = x))"
+%(u::'A => 'B) (ua::'A => bool) ub::'B => bool.
+ (ALL x::'A. x : ua --> u x : ub) &
+ (ALL x::'B. x : ub --> (EX y::'A. y : ua & u y = x))"
lemma DEF_SURJ: "SURJ =
-(%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- (ALL x::'A::type. IN x ua --> IN (u x) ub) &
- (ALL x::'B::type. IN x ub --> (EX y::'A::type. IN y ua & u y = x)))"
+(%(u::'A => 'B) (ua::'A => bool) ub::'B => bool.
+ (ALL x::'A. x : ua --> u x : ub) &
+ (ALL x::'B. x : ub --> (EX y::'A. y : ua & u y = x)))"
by (import hollight DEF_SURJ)
-definition BIJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
+definition
+ BIJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool" where
"BIJ ==
-%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- INJ u ua ub & SURJ u ua ub"
+%(u::'A => 'B) (ua::'A => bool) ub::'B => bool. INJ u ua ub & SURJ u ua ub"
lemma DEF_BIJ: "BIJ =
-(%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => bool.
- INJ u ua ub & SURJ u ua ub)"
+(%(u::'A => 'B) (ua::'A => bool) ub::'B => bool. INJ u ua ub & SURJ u ua ub)"
by (import hollight DEF_BIJ)
-definition CHOICE :: "('A => bool) => 'A" where
- "CHOICE == %u::'A::type => bool. SOME x::'A::type. IN x u"
-
-lemma DEF_CHOICE: "CHOICE = (%u::'A::type => bool. SOME x::'A::type. IN x u)"
- by (import hollight DEF_CHOICE)
-
-definition REST :: "('A => bool) => 'A => bool" where
- "REST == %u::'A::type => bool. DELETE u (CHOICE u)"
-
-lemma DEF_REST: "REST = (%u::'A::type => bool. DELETE u (CHOICE u))"
+definition
+ REST :: "('A => bool) => 'A => bool" where
+ "REST == %u::'A => bool. u - {Eps u}"
+
+lemma DEF_REST: "REST = (%u::'A => bool. u - {Eps u})"
by (import hollight DEF_REST)
-definition CARD_GE :: "('q_37693 => bool) => ('q_37690 => bool) => bool" where
- "CARD_GE ==
-%(u::'q_37693::type => bool) ua::'q_37690::type => bool.
- EX f::'q_37693::type => 'q_37690::type.
- ALL y::'q_37690::type.
- IN y ua --> (EX x::'q_37693::type. IN x u & y = f x)"
-
-lemma DEF_CARD_GE: "CARD_GE =
-(%(u::'q_37693::type => bool) ua::'q_37690::type => bool.
- EX f::'q_37693::type => 'q_37690::type.
- ALL y::'q_37690::type.
- IN y ua --> (EX x::'q_37693::type. IN x u & y = f x))"
- by (import hollight DEF_CARD_GE)
-
-definition CARD_LE :: "('q_37702 => bool) => ('q_37701 => bool) => bool" where
- "CARD_LE ==
-%(u::'q_37702::type => bool) ua::'q_37701::type => bool. CARD_GE ua u"
-
-lemma DEF_CARD_LE: "CARD_LE =
-(%(u::'q_37702::type => bool) ua::'q_37701::type => bool. CARD_GE ua u)"
- by (import hollight DEF_CARD_LE)
-
-definition CARD_EQ :: "('q_37712 => bool) => ('q_37713 => bool) => bool" where
- "CARD_EQ ==
-%(u::'q_37712::type => bool) ua::'q_37713::type => bool.
- CARD_LE u ua & CARD_LE ua u"
-
-lemma DEF_CARD_EQ: "CARD_EQ =
-(%(u::'q_37712::type => bool) ua::'q_37713::type => bool.
- CARD_LE u ua & CARD_LE ua u)"
- by (import hollight DEF_CARD_EQ)
-
-definition CARD_GT :: "('q_37727 => bool) => ('q_37728 => bool) => bool" where
- "CARD_GT ==
-%(u::'q_37727::type => bool) ua::'q_37728::type => bool.
- CARD_GE u ua & ~ CARD_GE ua u"
-
-lemma DEF_CARD_GT: "CARD_GT =
-(%(u::'q_37727::type => bool) ua::'q_37728::type => bool.
- CARD_GE u ua & ~ CARD_GE ua u)"
- by (import hollight DEF_CARD_GT)
-
-definition CARD_LT :: "('q_37743 => bool) => ('q_37744 => bool) => bool" where
- "CARD_LT ==
-%(u::'q_37743::type => bool) ua::'q_37744::type => bool.
- CARD_LE u ua & ~ CARD_LE ua u"
-
-lemma DEF_CARD_LT: "CARD_LT =
-(%(u::'q_37743::type => bool) ua::'q_37744::type => bool.
- CARD_LE u ua & ~ CARD_LE ua u)"
- by (import hollight DEF_CARD_LT)
-
-definition COUNTABLE :: "('q_37757 => bool) => bool" where
- "(op ==::(('q_37757::type => bool) => bool)
- => (('q_37757::type => bool) => bool) => prop)
- (COUNTABLE::('q_37757::type => bool) => bool)
- ((CARD_GE::(nat => bool) => ('q_37757::type => bool) => bool)
- (hollight.UNIV::nat => bool))"
-
-lemma DEF_COUNTABLE: "(op =::(('q_37757::type => bool) => bool)
- => (('q_37757::type => bool) => bool) => bool)
- (COUNTABLE::('q_37757::type => bool) => bool)
- ((CARD_GE::(nat => bool) => ('q_37757::type => bool) => bool)
- (hollight.UNIV::nat => bool))"
- by (import hollight DEF_COUNTABLE)
-
-lemma NOT_IN_EMPTY: "ALL x::'A::type. ~ IN x EMPTY"
+lemma NOT_IN_EMPTY: "(x::'A) ~: {}"
by (import hollight NOT_IN_EMPTY)
-lemma IN_UNIV: "ALL x::'A::type. IN x hollight.UNIV"
- by (import hollight IN_UNIV)
-
-lemma IN_UNION: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- IN xb (hollight.UNION x xa) = (IN xb x | IN xb xa)"
- by (import hollight IN_UNION)
-
-lemma IN_UNIONS: "ALL (x::('A::type => bool) => bool) xa::'A::type.
- IN xa (UNIONS x) = (EX t::'A::type => bool. IN t x & IN xa t)"
+lemma IN_UNIONS: "((xa::'A) : Union (x::('A => bool) => bool)) =
+(EX t::'A => bool. t : x & xa : t)"
by (import hollight IN_UNIONS)
-lemma IN_INTER: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- IN xb (hollight.INTER x xa) = (IN xb x & IN xb xa)"
- by (import hollight IN_INTER)
-
-lemma IN_INTERS: "ALL (x::('A::type => bool) => bool) xa::'A::type.
- IN xa (INTERS x) = (ALL t::'A::type => bool. IN t x --> IN xa t)"
+lemma IN_INTERS: "((xa::'A) : Inter (x::('A => bool) => bool)) =
+(ALL t::'A => bool. t : x --> xa : t)"
by (import hollight IN_INTERS)
-lemma IN_DIFF: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- IN xb (DIFF x xa) = (IN xb x & ~ IN xb xa)"
- by (import hollight IN_DIFF)
-
-lemma IN_INSERT: "ALL (x::'A::type) (xa::'A::type) xb::'A::type => bool.
- IN x (INSERT xa xb) = (x = xa | IN x xb)"
- by (import hollight IN_INSERT)
-
-lemma IN_DELETE: "ALL (x::'A::type => bool) (xa::'A::type) xb::'A::type.
- IN xa (DELETE x xb) = (IN xa x & xa ~= xb)"
+lemma IN_DELETE: "((xa::'A) : (x::'A => bool) - {xb::'A}) = (xa : x & xa ~= xb)"
by (import hollight IN_DELETE)
-lemma IN_SING: "ALL (x::'A::type) xa::'A::type. IN x (INSERT xa EMPTY) = (x = xa)"
- by (import hollight IN_SING)
-
-lemma IN_IMAGE: "ALL (x::'B::type) (xa::'A::type => bool) xb::'A::type => 'B::type.
- IN x (IMAGE xb xa) = (EX xc::'A::type. x = xb xc & IN xc xa)"
+lemma IN_IMAGE: "((x::'B) : (xb::'A => 'B) ` (xa::'A => bool)) =
+(EX xc::'A. x = xb xc & xc : xa)"
by (import hollight IN_IMAGE)
-lemma IN_REST: "ALL (x::'A::type) xa::'A::type => bool.
- IN x (REST xa) = (IN x xa & x ~= CHOICE xa)"
+lemma IN_REST: "((x::'A) : REST (xa::'A => bool)) = (x : xa & x ~= Eps xa)"
by (import hollight IN_REST)
-lemma CHOICE_DEF: "ALL x::'A::type => bool. x ~= EMPTY --> IN (CHOICE x) x"
+lemma FORALL_IN_INSERT: "(ALL xc::'q_44214.
+ xc : insert (xa::'q_44214) (xb::'q_44214 => bool) -->
+ (x::'q_44214 => bool) xc) =
+(x xa & (ALL xa::'q_44214. xa : xb --> x xa))"
+ by (import hollight FORALL_IN_INSERT)
+
+lemma EXISTS_IN_INSERT: "(EX xc::'q_44255.
+ xc : insert (xa::'q_44255) (xb::'q_44255 => bool) &
+ (x::'q_44255 => bool) xc) =
+(x xa | (EX xa::'q_44255. xa : xb & x xa))"
+ by (import hollight EXISTS_IN_INSERT)
+
+lemma CHOICE_DEF: "(x::'A => bool) ~= {} ==> Eps x : x"
by (import hollight CHOICE_DEF)
-lemma NOT_EQUAL_SETS: "ALL (x::'A::type => bool) xa::'A::type => bool.
- (x ~= xa) = (EX xb::'A::type. IN xb xa = (~ IN xb x))"
+lemma NOT_EQUAL_SETS: "((x::'A => bool) ~= (xa::'A => bool)) = (EX xb::'A. (xb : xa) = (xb ~: x))"
by (import hollight NOT_EQUAL_SETS)
-lemma MEMBER_NOT_EMPTY: "ALL x::'A::type => bool. (EX xa::'A::type. IN xa x) = (x ~= EMPTY)"
- by (import hollight MEMBER_NOT_EMPTY)
-
-lemma UNIV_NOT_EMPTY: "(Not::bool => bool)
- ((op =::('A::type => bool) => ('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool) (EMPTY::'A::type => bool))"
- by (import hollight UNIV_NOT_EMPTY)
-
-lemma EMPTY_NOT_UNIV: "(Not::bool => bool)
- ((op =::('A::type => bool) => ('A::type => bool) => bool)
- (EMPTY::'A::type => bool) (hollight.UNIV::'A::type => bool))"
+lemma EMPTY_NOT_UNIV: "(op ~=::('A::type => bool) => ('A::type => bool) => bool)
+ ({}::'A::type => bool) (UNIV::'A::type => bool)"
by (import hollight EMPTY_NOT_UNIV)
-lemma EQ_UNIV: "(ALL x::'A::type. IN x (s::'A::type => bool)) = (s = hollight.UNIV)"
+lemma EQ_UNIV: "(ALL x::'A. x : (s::'A => bool)) = (s = UNIV)"
by (import hollight EQ_UNIV)
-lemma SUBSET_TRANS: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- SUBSET x xa & SUBSET xa xb --> SUBSET x xb"
- by (import hollight SUBSET_TRANS)
-
-lemma SUBSET_REFL: "ALL x::'A::type => bool. SUBSET x x"
- by (import hollight SUBSET_REFL)
-
-lemma SUBSET_ANTISYM: "ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET x xa & SUBSET xa x --> x = xa"
- by (import hollight SUBSET_ANTISYM)
-
-lemma EMPTY_SUBSET: "(All::(('A::type => bool) => bool) => bool)
- ((SUBSET::('A::type => bool) => ('A::type => bool) => bool)
- (EMPTY::'A::type => bool))"
- by (import hollight EMPTY_SUBSET)
-
-lemma SUBSET_EMPTY: "ALL x::'A::type => bool. SUBSET x EMPTY = (x = EMPTY)"
- by (import hollight SUBSET_EMPTY)
-
-lemma SUBSET_UNIV: "ALL x::'A::type => bool. SUBSET x hollight.UNIV"
- by (import hollight SUBSET_UNIV)
-
-lemma UNIV_SUBSET: "ALL x::'A::type => bool. SUBSET hollight.UNIV x = (x = hollight.UNIV)"
+lemma UNIV_SUBSET: "(UNIV <= (x::'A => bool)) = (x = UNIV)"
by (import hollight UNIV_SUBSET)
-lemma PSUBSET_TRANS: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- PSUBSET x xa & PSUBSET xa xb --> PSUBSET x xb"
- by (import hollight PSUBSET_TRANS)
-
-lemma PSUBSET_SUBSET_TRANS: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- PSUBSET x xa & SUBSET xa xb --> PSUBSET x xb"
- by (import hollight PSUBSET_SUBSET_TRANS)
-
-lemma SUBSET_PSUBSET_TRANS: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- SUBSET x xa & PSUBSET xa xb --> PSUBSET x xb"
- by (import hollight SUBSET_PSUBSET_TRANS)
-
-lemma PSUBSET_IRREFL: "ALL x::'A::type => bool. ~ PSUBSET x x"
- by (import hollight PSUBSET_IRREFL)
-
-lemma NOT_PSUBSET_EMPTY: "ALL x::'A::type => bool. ~ PSUBSET x EMPTY"
- by (import hollight NOT_PSUBSET_EMPTY)
-
-lemma NOT_UNIV_PSUBSET: "ALL x::'A::type => bool. ~ PSUBSET hollight.UNIV x"
- by (import hollight NOT_UNIV_PSUBSET)
-
-lemma PSUBSET_UNIV: "ALL x::'A::type => bool.
- PSUBSET x hollight.UNIV = (EX xa::'A::type. ~ IN xa x)"
+lemma SING_SUBSET: "({xa::'q_44493} <= (x::'q_44493 => bool)) = (xa : x)"
+ by (import hollight SING_SUBSET)
+
+lemma PSUBSET_UNIV: "((x::'A => bool) < UNIV) = (EX xa::'A. xa ~: x)"
by (import hollight PSUBSET_UNIV)
-lemma UNION_ASSOC: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- hollight.UNION (hollight.UNION x xa) xb =
- hollight.UNION x (hollight.UNION xa xb)"
- by (import hollight UNION_ASSOC)
-
-lemma UNION_IDEMPOT: "ALL x::'A::type => bool. hollight.UNION x x = x"
- by (import hollight UNION_IDEMPOT)
-
-lemma UNION_COMM: "ALL (x::'A::type => bool) xa::'A::type => bool.
- hollight.UNION x xa = hollight.UNION xa x"
- by (import hollight UNION_COMM)
-
-lemma SUBSET_UNION: "(ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET x (hollight.UNION x xa)) &
-(ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET x (hollight.UNION xa x))"
+lemma PSUBSET_ALT: "((x::'A => bool) < (xa::'A => bool)) =
+(x <= xa & (EX a::'A. a : xa & a ~: x))"
+ by (import hollight PSUBSET_ALT)
+
+lemma SUBSET_UNION: "(ALL (x::'A => bool) xa::'A => bool. x <= x Un xa) &
+(ALL (x::'A => bool) xa::'A => bool. x <= xa Un x)"
by (import hollight SUBSET_UNION)
-lemma SUBSET_UNION_ABSORPTION: "ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET x xa = (hollight.UNION x xa = xa)"
- by (import hollight SUBSET_UNION_ABSORPTION)
-
-lemma UNION_EMPTY: "(ALL x::'A::type => bool. hollight.UNION EMPTY x = x) &
-(ALL x::'A::type => bool. hollight.UNION x EMPTY = x)"
+lemma UNION_EMPTY: "(ALL x::'A => bool. {} Un x = x) & (ALL x::'A => bool. x Un {} = x)"
by (import hollight UNION_EMPTY)
-lemma UNION_UNIV: "(ALL x::'A::type => bool. hollight.UNION hollight.UNIV x = hollight.UNIV) &
-(ALL x::'A::type => bool. hollight.UNION x hollight.UNIV = hollight.UNIV)"
+lemma UNION_UNIV: "(ALL x::'A => bool. UNIV Un x = UNIV) &
+(ALL x::'A => bool. x Un UNIV = UNIV)"
by (import hollight UNION_UNIV)
-lemma EMPTY_UNION: "ALL (x::'A::type => bool) xa::'A::type => bool.
- (hollight.UNION x xa = EMPTY) = (x = EMPTY & xa = EMPTY)"
- by (import hollight EMPTY_UNION)
-
-lemma UNION_SUBSET: "ALL (x::'q_38594::type => bool) (xa::'q_38594::type => bool)
- xb::'q_38594::type => bool.
- SUBSET (hollight.UNION x xa) xb = (SUBSET x xb & SUBSET xa xb)"
- by (import hollight UNION_SUBSET)
-
-lemma INTER_ASSOC: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- hollight.INTER (hollight.INTER x xa) xb =
- hollight.INTER x (hollight.INTER xa xb)"
- by (import hollight INTER_ASSOC)
-
-lemma INTER_IDEMPOT: "ALL x::'A::type => bool. hollight.INTER x x = x"
- by (import hollight INTER_IDEMPOT)
-
-lemma INTER_COMM: "ALL (x::'A::type => bool) xa::'A::type => bool.
- hollight.INTER x xa = hollight.INTER xa x"
- by (import hollight INTER_COMM)
-
-lemma INTER_SUBSET: "(ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET (hollight.INTER x xa) x) &
-(ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET (hollight.INTER xa x) x)"
+lemma INTER_SUBSET: "(ALL (x::'A => bool) xa::'A => bool. x Int xa <= x) &
+(ALL (x::'A => bool) xa::'A => bool. xa Int x <= x)"
by (import hollight INTER_SUBSET)
-lemma SUBSET_INTER_ABSORPTION: "ALL (x::'A::type => bool) xa::'A::type => bool.
- SUBSET x xa = (hollight.INTER x xa = x)"
- by (import hollight SUBSET_INTER_ABSORPTION)
-
-lemma INTER_EMPTY: "(ALL x::'A::type => bool. hollight.INTER EMPTY x = EMPTY) &
-(ALL x::'A::type => bool. hollight.INTER x EMPTY = EMPTY)"
+lemma INTER_EMPTY: "(ALL x::'A => bool. {} Int x = {}) & (ALL x::'A => bool. x Int {} = {})"
by (import hollight INTER_EMPTY)
-lemma INTER_UNIV: "(ALL x::'A::type => bool. hollight.INTER hollight.UNIV x = x) &
-(ALL x::'A::type => bool. hollight.INTER x hollight.UNIV = x)"
+lemma INTER_UNIV: "(ALL x::'A => bool. UNIV Int x = x) & (ALL x::'A => bool. x Int UNIV = x)"
by (import hollight INTER_UNIV)
-lemma UNION_OVER_INTER: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- hollight.INTER x (hollight.UNION xa xb) =
- hollight.UNION (hollight.INTER x xa) (hollight.INTER x xb)"
- by (import hollight UNION_OVER_INTER)
-
-lemma INTER_OVER_UNION: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- hollight.UNION x (hollight.INTER xa xb) =
- hollight.INTER (hollight.UNION x xa) (hollight.UNION x xb)"
- by (import hollight INTER_OVER_UNION)
-
-lemma IN_DISJOINT: "ALL (x::'A::type => bool) xa::'A::type => bool.
- DISJOINT x xa = (~ (EX xb::'A::type. IN xb x & IN xb xa))"
+lemma IN_DISJOINT: "((x::'A => bool) Int (xa::'A => bool) = {}) =
+(~ (EX xb::'A. xb : x & xb : xa))"
by (import hollight IN_DISJOINT)
-lemma DISJOINT_SYM: "ALL (x::'A::type => bool) xa::'A::type => bool.
- DISJOINT x xa = DISJOINT xa x"
+lemma DISJOINT_SYM: "((x::'A => bool) Int (xa::'A => bool) = {}) = (xa Int x = {})"
by (import hollight DISJOINT_SYM)
-lemma DISJOINT_EMPTY: "ALL x::'A::type => bool. DISJOINT EMPTY x & DISJOINT x EMPTY"
+lemma DISJOINT_EMPTY: "{} Int (x::'A => bool) = {} & x Int {} = {}"
by (import hollight DISJOINT_EMPTY)
-lemma DISJOINT_EMPTY_REFL: "ALL x::'A::type => bool. (x = EMPTY) = DISJOINT x x"
+lemma DISJOINT_EMPTY_REFL: "((x::'A => bool) = {}) = (x Int x = {})"
by (import hollight DISJOINT_EMPTY_REFL)
-lemma DISJOINT_UNION: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type => bool.
- DISJOINT (hollight.UNION x xa) xb = (DISJOINT x xb & DISJOINT xa xb)"
+lemma DISJOINT_UNION: "(((x::'A => bool) Un (xa::'A => bool)) Int (xb::'A => bool) = {}) =
+(x Int xb = {} & xa Int xb = {})"
by (import hollight DISJOINT_UNION)
-lemma DIFF_EMPTY: "ALL x::'A::type => bool. DIFF x EMPTY = x"
- by (import hollight DIFF_EMPTY)
-
-lemma EMPTY_DIFF: "ALL x::'A::type => bool. DIFF EMPTY x = EMPTY"
- by (import hollight EMPTY_DIFF)
-
-lemma DIFF_UNIV: "ALL x::'A::type => bool. DIFF x hollight.UNIV = EMPTY"
- by (import hollight DIFF_UNIV)
-
-lemma DIFF_DIFF: "ALL (x::'A::type => bool) xa::'A::type => bool.
- DIFF (DIFF x xa) xa = DIFF x xa"
- by (import hollight DIFF_DIFF)
-
-lemma DIFF_EQ_EMPTY: "ALL x::'A::type => bool. DIFF x x = EMPTY"
- by (import hollight DIFF_EQ_EMPTY)
-
-lemma SUBSET_DIFF: "ALL (x::'q_39012::type => bool) xa::'q_39012::type => bool.
- SUBSET (DIFF x xa) x"
- by (import hollight SUBSET_DIFF)
-
-lemma COMPONENT: "ALL (x::'A::type) s::'A::type => bool. IN x (INSERT x s)"
- by (import hollight COMPONENT)
-
-lemma DECOMPOSITION: "ALL (s::'A::type => bool) x::'A::type.
- IN x s = (EX t::'A::type => bool. s = INSERT x t & ~ IN x t)"
+lemma DECOMPOSITION: "((x::'A) : (s::'A => bool)) = (EX t::'A => bool. s = insert x t & x ~: t)"
by (import hollight DECOMPOSITION)
-lemma SET_CASES: "ALL s::'A::type => bool.
- s = EMPTY |
- (EX (x::'A::type) t::'A::type => bool. s = INSERT x t & ~ IN x t)"
+lemma SET_CASES: "(s::'A => bool) = {} | (EX (x::'A) t::'A => bool. s = insert x t & x ~: t)"
by (import hollight SET_CASES)
-lemma ABSORPTION: "ALL (x::'A::type) xa::'A::type => bool. IN x xa = (INSERT x xa = xa)"
+lemma ABSORPTION: "((x::'A) : (xa::'A => bool)) = (insert x xa = xa)"
by (import hollight ABSORPTION)
-lemma INSERT_INSERT: "ALL (x::'A::type) xa::'A::type => bool. INSERT x (INSERT x xa) = INSERT x xa"
- by (import hollight INSERT_INSERT)
-
-lemma INSERT_COMM: "ALL (x::'A::type) (xa::'A::type) xb::'A::type => bool.
- INSERT x (INSERT xa xb) = INSERT xa (INSERT x xb)"
- by (import hollight INSERT_COMM)
-
-lemma INSERT_UNIV: "ALL x::'A::type. INSERT x hollight.UNIV = hollight.UNIV"
+lemma INSERT_UNIV: "insert (x::'A) UNIV = UNIV"
by (import hollight INSERT_UNIV)
-lemma NOT_INSERT_EMPTY: "ALL (x::'A::type) xa::'A::type => bool. INSERT x xa ~= EMPTY"
- by (import hollight NOT_INSERT_EMPTY)
-
-lemma NOT_EMPTY_INSERT: "ALL (x::'A::type) xa::'A::type => bool. EMPTY ~= INSERT x xa"
- by (import hollight NOT_EMPTY_INSERT)
-
-lemma INSERT_UNION: "ALL (x::'A::type) (s::'A::type => bool) t::'A::type => bool.
- hollight.UNION (INSERT x s) t =
- COND (IN x t) (hollight.UNION s t) (INSERT x (hollight.UNION s t))"
+lemma INSERT_UNION: "insert (x::'A) (s::'A => bool) Un (t::'A => bool) =
+(if x : t then s Un t else insert x (s Un t))"
by (import hollight INSERT_UNION)
-lemma INSERT_UNION_EQ: "ALL (x::'A::type) (xa::'A::type => bool) xb::'A::type => bool.
- hollight.UNION (INSERT x xa) xb = INSERT x (hollight.UNION xa xb)"
- by (import hollight INSERT_UNION_EQ)
-
-lemma INSERT_INTER: "ALL (x::'A::type) (s::'A::type => bool) t::'A::type => bool.
- hollight.INTER (INSERT x s) t =
- COND (IN x t) (INSERT x (hollight.INTER s t)) (hollight.INTER s t)"
- by (import hollight INSERT_INTER)
-
-lemma DISJOINT_INSERT: "ALL (x::'A::type) (xa::'A::type => bool) xb::'A::type => bool.
- DISJOINT (INSERT x xa) xb = (DISJOINT xa xb & ~ IN x xb)"
+lemma DISJOINT_INSERT: "(insert (x::'A) (xa::'A => bool) Int (xb::'A => bool) = {}) =
+(xa Int xb = {} & x ~: xb)"
by (import hollight DISJOINT_INSERT)
-lemma INSERT_SUBSET: "ALL (x::'A::type) (xa::'A::type => bool) xb::'A::type => bool.
- SUBSET (INSERT x xa) xb = (IN x xb & SUBSET xa xb)"
- by (import hollight INSERT_SUBSET)
-
-lemma SUBSET_INSERT: "ALL (x::'A::type) xa::'A::type => bool.
- ~ IN x xa -->
- (ALL xb::'A::type => bool. SUBSET xa (INSERT x xb) = SUBSET xa xb)"
- by (import hollight SUBSET_INSERT)
-
-lemma INSERT_DIFF: "ALL (s::'A::type => bool) (t::'A::type => bool) x::'A::type.
- DIFF (INSERT x s) t = COND (IN x t) (DIFF s t) (INSERT x (DIFF s t))"
- by (import hollight INSERT_DIFF)
-
-lemma INSERT_AC: "INSERT (x::'q_39468::type)
- (INSERT (y::'q_39468::type) (s::'q_39468::type => bool)) =
-INSERT y (INSERT x s) &
-INSERT x (INSERT x s) = INSERT x s"
+lemma INSERT_AC: "insert (x::'q_45764) (insert (y::'q_45764) (s::'q_45764 => bool)) =
+insert y (insert x s) &
+insert x (insert x s) = insert x s"
by (import hollight INSERT_AC)
-lemma INTER_ACI: "hollight.INTER (p::'q_39535::type => bool) (q::'q_39535::type => bool) =
-hollight.INTER q p &
-hollight.INTER (hollight.INTER p q) (r::'q_39535::type => bool) =
-hollight.INTER p (hollight.INTER q r) &
-hollight.INTER p (hollight.INTER q r) =
-hollight.INTER q (hollight.INTER p r) &
-hollight.INTER p p = p &
-hollight.INTER p (hollight.INTER p q) = hollight.INTER p q"
+lemma INTER_ACI: "(p::'q_45831 => bool) Int (q::'q_45831 => bool) = q Int p &
+p Int q Int (r::'q_45831 => bool) = p Int (q Int r) &
+p Int (q Int r) = q Int (p Int r) & p Int p = p & p Int (p Int q) = p Int q"
by (import hollight INTER_ACI)
-lemma UNION_ACI: "hollight.UNION (p::'q_39601::type => bool) (q::'q_39601::type => bool) =
-hollight.UNION q p &
-hollight.UNION (hollight.UNION p q) (r::'q_39601::type => bool) =
-hollight.UNION p (hollight.UNION q r) &
-hollight.UNION p (hollight.UNION q r) =
-hollight.UNION q (hollight.UNION p r) &
-hollight.UNION p p = p &
-hollight.UNION p (hollight.UNION p q) = hollight.UNION p q"
+lemma UNION_ACI: "(p::'q_45897 => bool) Un (q::'q_45897 => bool) = q Un p &
+p Un q Un (r::'q_45897 => bool) = p Un (q Un r) &
+p Un (q Un r) = q Un (p Un r) & p Un p = p & p Un (p Un q) = p Un q"
by (import hollight UNION_ACI)
-lemma DELETE_NON_ELEMENT: "ALL (x::'A::type) xa::'A::type => bool. (~ IN x xa) = (DELETE xa x = xa)"
+lemma DELETE_NON_ELEMENT: "((x::'A) ~: (xa::'A => bool)) = (xa - {x} = xa)"
by (import hollight DELETE_NON_ELEMENT)
-lemma IN_DELETE_EQ: "ALL (s::'A::type => bool) (x::'A::type) x'::'A::type.
- (IN x s = IN x' s) = (IN x (DELETE s x') = IN x' (DELETE s x))"
+lemma IN_DELETE_EQ: "(((x::'A) : (s::'A => bool)) = ((x'::'A) : s)) =
+((x : s - {x'}) = (x' : s - {x}))"
by (import hollight IN_DELETE_EQ)
-lemma EMPTY_DELETE: "ALL x::'A::type. DELETE EMPTY x = EMPTY"
+lemma EMPTY_DELETE: "{} - {x::'A} = {}"
by (import hollight EMPTY_DELETE)
-lemma DELETE_DELETE: "ALL (x::'A::type) xa::'A::type => bool. DELETE (DELETE xa x) x = DELETE xa x"
+lemma DELETE_DELETE: "(xa::'A => bool) - {x::'A} - {x} = xa - {x}"
by (import hollight DELETE_DELETE)
-lemma DELETE_COMM: "ALL (x::'A::type) (xa::'A::type) xb::'A::type => bool.
- DELETE (DELETE xb x) xa = DELETE (DELETE xb xa) x"
+lemma DELETE_COMM: "(xb::'A => bool) - {x::'A} - {xa::'A} = xb - {xa} - {x}"
by (import hollight DELETE_COMM)
-lemma DELETE_SUBSET: "ALL (x::'A::type) xa::'A::type => bool. SUBSET (DELETE xa x) xa"
+lemma DELETE_SUBSET: "(xa::'A => bool) - {x::'A} <= xa"
by (import hollight DELETE_SUBSET)
-lemma SUBSET_DELETE: "ALL (x::'A::type) (xa::'A::type => bool) xb::'A::type => bool.
- SUBSET xa (DELETE xb x) = (~ IN x xa & SUBSET xa xb)"
+lemma SUBSET_DELETE: "((xa::'A => bool) <= (xb::'A => bool) - {x::'A}) = (x ~: xa & xa <= xb)"
by (import hollight SUBSET_DELETE)
-lemma SUBSET_INSERT_DELETE: "ALL (x::'A::type) (xa::'A::type => bool) xb::'A::type => bool.
- SUBSET xa (INSERT x xb) = SUBSET (DELETE xa x) xb"
+lemma SUBSET_INSERT_DELETE: "((xa::'A => bool) <= insert (x::'A) (xb::'A => bool)) = (xa - {x} <= xb)"
by (import hollight SUBSET_INSERT_DELETE)
-lemma DIFF_INSERT: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- DIFF x (INSERT xb xa) = DIFF (DELETE x xb) xa"
- by (import hollight DIFF_INSERT)
-
-lemma PSUBSET_INSERT_SUBSET: "ALL (x::'A::type => bool) xa::'A::type => bool.
- PSUBSET x xa = (EX xb::'A::type. ~ IN xb x & SUBSET (INSERT xb x) xa)"
+lemma PSUBSET_INSERT_SUBSET: "((x::'A => bool) < (xa::'A => bool)) =
+(EX xb::'A. xb ~: x & insert xb x <= xa)"
by (import hollight PSUBSET_INSERT_SUBSET)
-lemma PSUBSET_MEMBER: "ALL (x::'A::type => bool) xa::'A::type => bool.
- PSUBSET x xa = (SUBSET x xa & (EX y::'A::type. IN y xa & ~ IN y x))"
+lemma PSUBSET_MEMBER: "((x::'A => bool) < (xa::'A => bool)) =
+(x <= xa & (EX y::'A. y : xa & y ~: x))"
by (import hollight PSUBSET_MEMBER)
-lemma DELETE_INSERT: "ALL (x::'A::type) (y::'A::type) s::'A::type => bool.
- DELETE (INSERT x s) y = COND (x = y) (DELETE s y) (INSERT x (DELETE s y))"
+lemma DELETE_INSERT: "insert (x::'A) (s::'A => bool) - {y::'A} =
+(if x = y then s - {y} else insert x (s - {y}))"
by (import hollight DELETE_INSERT)
-lemma INSERT_DELETE: "ALL (x::'A::type) xa::'A::type => bool.
- IN x xa --> INSERT x (DELETE xa x) = xa"
- by (import hollight INSERT_DELETE)
-
-lemma DELETE_INTER: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- hollight.INTER (DELETE x xb) xa = DELETE (hollight.INTER x xa) xb"
+lemma DELETE_INTER: "((x::'A => bool) - {xb::'A}) Int (xa::'A => bool) = x Int xa - {xb}"
by (import hollight DELETE_INTER)
-lemma DISJOINT_DELETE_SYM: "ALL (x::'A::type => bool) (xa::'A::type => bool) xb::'A::type.
- DISJOINT (DELETE x xb) xa = DISJOINT (DELETE xa xb) x"
+lemma DISJOINT_DELETE_SYM: "(((x::'A => bool) - {xb::'A}) Int (xa::'A => bool) = {}) =
+((xa - {xb}) Int x = {})"
by (import hollight DISJOINT_DELETE_SYM)
-lemma UNIONS_0: "(op =::('q_40008::type => bool) => ('q_40008::type => bool) => bool)
- ((UNIONS::(('q_40008::type => bool) => bool) => 'q_40008::type => bool)
- (EMPTY::('q_40008::type => bool) => bool))
- (EMPTY::'q_40008::type => bool)"
- by (import hollight UNIONS_0)
-
-lemma UNIONS_1: "UNIONS (INSERT (s::'q_40014::type => bool) EMPTY) = s"
- by (import hollight UNIONS_1)
-
-lemma UNIONS_2: "UNIONS
- (INSERT (s::'q_40034::type => bool)
- (INSERT (t::'q_40034::type => bool) EMPTY)) =
-hollight.UNION s t"
- by (import hollight UNIONS_2)
-
-lemma UNIONS_INSERT: "UNIONS
- (INSERT (s::'q_40048::type => bool)
- (u::('q_40048::type => bool) => bool)) =
-hollight.UNION s (UNIONS u)"
- by (import hollight UNIONS_INSERT)
-
-lemma FORALL_IN_UNIONS: "ALL (x::'q_40090::type => bool) xa::('q_40090::type => bool) => bool.
- (ALL xb::'q_40090::type. IN xb (UNIONS xa) --> x xb) =
- (ALL (t::'q_40090::type => bool) xb::'q_40090::type.
- IN t xa & IN xb t --> x xb)"
+lemma FORALL_IN_UNIONS: "(ALL x::'q_46386.
+ x : Union (s::('q_46386 => bool) => bool) --> (P::'q_46386 => bool) x) =
+(ALL (t::'q_46386 => bool) x::'q_46386. t : s & x : t --> P x)"
by (import hollight FORALL_IN_UNIONS)
-lemma EMPTY_UNIONS: "ALL x::('q_40116::type => bool) => bool.
- (UNIONS x = EMPTY) =
- (ALL xa::'q_40116::type => bool. IN xa x --> xa = EMPTY)"
+lemma EXISTS_IN_UNIONS: "(EX x::'q_46428.
+ x : Union (s::('q_46428 => bool) => bool) & (P::'q_46428 => bool) x) =
+(EX (t::'q_46428 => bool) x::'q_46428. t : s & x : t & P x)"
+ by (import hollight EXISTS_IN_UNIONS)
+
+lemma EMPTY_UNIONS: "(Union (x::('q_46454 => bool) => bool) = {}) =
+(ALL xa::'q_46454 => bool. xa : x --> xa = {})"
by (import hollight EMPTY_UNIONS)
-lemma INTERS_0: "(op =::('q_40124::type => bool) => ('q_40124::type => bool) => bool)
- ((INTERS::(('q_40124::type => bool) => bool) => 'q_40124::type => bool)
- (EMPTY::('q_40124::type => bool) => bool))
- (hollight.UNIV::'q_40124::type => bool)"
- by (import hollight INTERS_0)
-
-lemma INTERS_1: "INTERS (INSERT (s::'q_40130::type => bool) EMPTY) = s"
- by (import hollight INTERS_1)
-
-lemma INTERS_2: "INTERS
- (INSERT (s::'q_40150::type => bool)
- (INSERT (t::'q_40150::type => bool) EMPTY)) =
-hollight.INTER s t"
- by (import hollight INTERS_2)
-
-lemma INTERS_INSERT: "INTERS
- (INSERT (s::'q_40164::type => bool)
- (u::('q_40164::type => bool) => bool)) =
-hollight.INTER s (INTERS u)"
- by (import hollight INTERS_INSERT)
-
-lemma IMAGE_CLAUSES: "IMAGE (f::'q_40190::type => 'q_40194::type) EMPTY = EMPTY &
-IMAGE f (INSERT (x::'q_40190::type) (s::'q_40190::type => bool)) =
-INSERT (f x) (IMAGE f s)"
+lemma INTER_UNIONS: "(ALL (x::('q_46493 => bool) => bool) xa::'q_46493 => bool.
+ Union x Int xa =
+ Union
+ {u::'q_46493 => bool.
+ EX xb::'q_46493 => bool. xb : x & u = xb Int xa}) &
+(ALL (x::('q_46529 => bool) => bool) xa::'q_46529 => bool.
+ xa Int Union x =
+ Union
+ {u::'q_46529 => bool. EX xb::'q_46529 => bool. xb : x & u = xa Int xb})"
+ by (import hollight INTER_UNIONS)
+
+lemma UNIONS_SUBSET: "(Union (x::('q_46545 => bool) => bool) <= (xa::'q_46545 => bool)) =
+(ALL xb::'q_46545 => bool. xb : x --> xb <= xa)"
+ by (import hollight UNIONS_SUBSET)
+
+lemma IMAGE_CLAUSES: "(f::'q_46676 => 'q_46680) ` {} = {} &
+f ` insert (x::'q_46676) (s::'q_46676 => bool) = insert (f x) (f ` s)"
by (import hollight IMAGE_CLAUSES)
-lemma IMAGE_UNION: "ALL (x::'q_40217::type => 'q_40228::type) (xa::'q_40217::type => bool)
- xb::'q_40217::type => bool.
- IMAGE x (hollight.UNION xa xb) = hollight.UNION (IMAGE x xa) (IMAGE x xb)"
- by (import hollight IMAGE_UNION)
-
-lemma IMAGE_o: "ALL (x::'q_40261::type => 'q_40257::type)
- (xa::'q_40252::type => 'q_40261::type) xb::'q_40252::type => bool.
- IMAGE (x o xa) xb = IMAGE x (IMAGE xa xb)"
- by (import hollight IMAGE_o)
-
-lemma IMAGE_SUBSET: "ALL (x::'q_40279::type => 'q_40290::type) (xa::'q_40279::type => bool)
- xb::'q_40279::type => bool.
- SUBSET xa xb --> SUBSET (IMAGE x xa) (IMAGE x xb)"
- by (import hollight IMAGE_SUBSET)
-
-lemma IMAGE_DIFF_INJ: "(ALL (x::'q_40321::type) y::'q_40321::type.
- (f::'q_40321::type => 'q_40332::type) x = f y --> x = y) -->
-IMAGE f (DIFF (s::'q_40321::type => bool) (t::'q_40321::type => bool)) =
-DIFF (IMAGE f s) (IMAGE f t)"
+lemma IMAGE_INTER_INJ: "(!!(xa::'q_46846) y::'q_46846.
+ (x::'q_46846 => 'q_46857) xa = x y ==> xa = y)
+==> x ` ((xa::'q_46846 => bool) Int (xb::'q_46846 => bool)) =
+ x ` xa Int x ` xb"
+ by (import hollight IMAGE_INTER_INJ)
+
+lemma IMAGE_DIFF_INJ: "(!!(xa::'q_46900) y::'q_46900.
+ (x::'q_46900 => 'q_46911) xa = x y ==> xa = y)
+==> x ` ((xa::'q_46900 => bool) - (xb::'q_46900 => bool)) = x ` xa - x ` xb"
by (import hollight IMAGE_DIFF_INJ)
-lemma IMAGE_DELETE_INJ: "(ALL x::'q_40367::type.
- (f::'q_40367::type => 'q_40366::type) x = f (a::'q_40367::type) -->
- x = a) -->
-IMAGE f (DELETE (s::'q_40367::type => bool) a) = DELETE (IMAGE f s) (f a)"
+lemma IMAGE_DELETE_INJ: "(!!xa::'q_46958.
+ (x::'q_46958 => 'q_46957) xa = x (xb::'q_46958) ==> xa = xb)
+==> x ` ((xa::'q_46958 => bool) - {xb}) = x ` xa - {x xb}"
by (import hollight IMAGE_DELETE_INJ)
-lemma IMAGE_EQ_EMPTY: "ALL (x::'q_40390::type => 'q_40386::type) xa::'q_40390::type => bool.
- (IMAGE x xa = EMPTY) = (xa = EMPTY)"
- by (import hollight IMAGE_EQ_EMPTY)
-
-lemma FORALL_IN_IMAGE: "ALL (x::'q_40426::type => 'q_40425::type) xa::'q_40426::type => bool.
- (ALL xb::'q_40425::type.
- IN xb (IMAGE x xa) --> (P::'q_40425::type => bool) xb) =
- (ALL xb::'q_40426::type. IN xb xa --> P (x xb))"
+lemma FORALL_IN_IMAGE: "(ALL xb::'q_47016.
+ xb : (x::'q_47017 => 'q_47016) ` (xa::'q_47017 => bool) -->
+ (P::'q_47016 => bool) xb) =
+(ALL xb::'q_47017. xb : xa --> P (x xb))"
by (import hollight FORALL_IN_IMAGE)
-lemma EXISTS_IN_IMAGE: "ALL (x::'q_40462::type => 'q_40461::type) xa::'q_40462::type => bool.
- (EX xb::'q_40461::type.
- IN xb (IMAGE x xa) & (P::'q_40461::type => bool) xb) =
- (EX xb::'q_40462::type. IN xb xa & P (x xb))"
+lemma EXISTS_IN_IMAGE: "(EX xb::'q_47052.
+ xb : (x::'q_47053 => 'q_47052) ` (xa::'q_47053 => bool) &
+ (P::'q_47052 => bool) xb) =
+(EX xb::'q_47053. xb : xa & P (x xb))"
by (import hollight EXISTS_IN_IMAGE)
-lemma SUBSET_IMAGE: "ALL (f::'A::type => 'B::type) (s::'B::type => bool) t::'A::type => bool.
- SUBSET s (IMAGE f t) =
- (EX x::'A::type => bool. SUBSET x t & s = IMAGE f x)"
- by (import hollight SUBSET_IMAGE)
-
-lemma IMAGE_CONST: "ALL (s::'q_40548::type => bool) c::'q_40553::type.
- IMAGE (%x::'q_40548::type. c) s = COND (s = EMPTY) EMPTY (INSERT c EMPTY)"
- by (import hollight IMAGE_CONST)
-
-lemma SIMPLE_IMAGE: "ALL (x::'q_40581::type => 'q_40585::type) xa::'q_40581::type => bool.
- GSPEC
- (%u::'q_40585::type.
- EX xb::'q_40581::type. SETSPEC u (IN xb xa) (x xb)) =
- IMAGE x xa"
+lemma FORALL_SUBSET_IMAGE: "(ALL xc<=(xa::'q_47140 => 'q_47156) ` (xb::'q_47140 => bool).
+ (x::('q_47156 => bool) => bool) xc) =
+(ALL t<=xb. x (xa ` t))"
+ by (import hollight FORALL_SUBSET_IMAGE)
+
+lemma EXISTS_SUBSET_IMAGE: "(EX xc<=(xa::'q_47183 => 'q_47199) ` (xb::'q_47183 => bool).
+ (x::('q_47199 => bool) => bool) xc) =
+(EX t<=xb. x (xa ` t))"
+ by (import hollight EXISTS_SUBSET_IMAGE)
+
+lemma SIMPLE_IMAGE: "{u::'q_47262.
+ EX xb::'q_47258.
+ xb : (xa::'q_47258 => bool) & u = (x::'q_47258 => 'q_47262) xb} =
+x ` xa"
by (import hollight SIMPLE_IMAGE)
-lemma EMPTY_GSPEC: "GSPEC (%u::'q_40602::type. Ex (SETSPEC u False)) = EMPTY"
+lemma SIMPLE_IMAGE_GEN: "{u::'q_47292.
+ EX xa::'q_47305.
+ (P::'q_47305 => bool) xa & u = (x::'q_47305 => 'q_47292) xa} =
+x ` {u::'q_47305. EX x::'q_47305. P x & u = x}"
+ by (import hollight SIMPLE_IMAGE_GEN)
+
+lemma IMAGE_UNIONS: "(x::'q_47323 => 'q_47332) ` Union (xa::('q_47323 => bool) => bool) =
+Union (op ` x ` xa)"
+ by (import hollight IMAGE_UNIONS)
+
+lemma SURJECTIVE_IMAGE_EQ: "(ALL y::'q_47396.
+ y : (xa::'q_47396 => bool) -->
+ (EX x::'q_47400. (f::'q_47400 => 'q_47396) x = y)) &
+(ALL xb::'q_47400. (f xb : xa) = (xb : (x::'q_47400 => bool)))
+==> f ` x = xa"
+ by (import hollight SURJECTIVE_IMAGE_EQ)
+
+lemma EMPTY_GSPEC: "{u::'q_47425. Ex (SETSPEC u False)} = {}"
by (import hollight EMPTY_GSPEC)
-lemma IN_ELIM_PAIR_THM: "ALL (x::'q_40648::type => 'q_40647::type => bool) (xa::'q_40648::type)
- xb::'q_40647::type.
- IN (xa, xb)
- (GSPEC
- (%xa::'q_40648::type * 'q_40647::type.
- EX (xb::'q_40648::type) y::'q_40647::type.
- SETSPEC xa (x xb y) (xb, y))) =
- x xa xb"
+lemma SING_GSPEC: "(ALL x::'q_47454. {u::'q_47454. EX xa::'q_47454. xa = x & u = xa} = {x}) &
+(ALL x::'q_47480. {u::'q_47480. EX xa::'q_47480. x = xa & u = xa} = {x})"
+ by (import hollight SING_GSPEC)
+
+lemma IN_ELIM_PAIR_THM: "((xa::'q_47526, xb::'q_47525)
+ : {xa::'q_47526 * 'q_47525.
+ EX (xb::'q_47526) y::'q_47525.
+ (x::'q_47526 => 'q_47525 => bool) xb y & xa = (xb, y)}) =
+x xa xb"
by (import hollight IN_ELIM_PAIR_THM)
-lemma FINITE_INDUCT_STRONG: "ALL P::('A::type => bool) => bool.
- P EMPTY &
- (ALL (x::'A::type) s::'A::type => bool.
- P s & ~ IN x s & FINITE s --> P (INSERT x s)) -->
- (ALL s::'A::type => bool. FINITE s --> P s)"
- by (import hollight FINITE_INDUCT_STRONG)
-
-lemma FINITE_SUBSET: "ALL (x::'A::type => bool) t::'A::type => bool.
- FINITE t & SUBSET x t --> FINITE x"
- by (import hollight FINITE_SUBSET)
-
-lemma FINITE_UNION_IMP: "ALL (x::'A::type => bool) xa::'A::type => bool.
- FINITE x & FINITE xa --> FINITE (hollight.UNION x xa)"
- by (import hollight FINITE_UNION_IMP)
-
-lemma FINITE_UNION: "ALL (s::'A::type => bool) t::'A::type => bool.
- FINITE (hollight.UNION s t) = (FINITE s & FINITE t)"
- by (import hollight FINITE_UNION)
-
-lemma FINITE_INTER: "ALL (s::'A::type => bool) t::'A::type => bool.
- FINITE s | FINITE t --> FINITE (hollight.INTER s t)"
- by (import hollight FINITE_INTER)
-
-lemma FINITE_INSERT: "ALL (s::'A::type => bool) x::'A::type. FINITE (INSERT x s) = FINITE s"
- by (import hollight FINITE_INSERT)
-
-lemma FINITE_DELETE_IMP: "ALL (s::'A::type => bool) x::'A::type. FINITE s --> FINITE (DELETE s x)"
+lemma SET_PAIR_THM: "{u::'q_47570 * 'q_47569.
+ EX p::'q_47570 * 'q_47569. (x::'q_47570 * 'q_47569 => bool) p & u = p} =
+{u::'q_47570 * 'q_47569.
+ EX (a::'q_47570) b::'q_47569. x (a, b) & u = (a, b)}"
+ by (import hollight SET_PAIR_THM)
+
+lemma FORALL_IN_GSPEC: "(ALL (P::'q_47618 => bool) f::'q_47618 => 'q_47739.
+ (ALL z::'q_47739.
+ z : {u::'q_47739. EX x::'q_47618. P x & u = f x} -->
+ (Q::'q_47739 => bool) z) =
+ (ALL x::'q_47618. P x --> Q (f x))) &
+(ALL (P::'q_47675 => 'q_47674 => bool) f::'q_47675 => 'q_47674 => 'q_47739.
+ (ALL z::'q_47739.
+ z : {u::'q_47739.
+ EX (x::'q_47675) y::'q_47674. P x y & u = f x y} -->
+ Q z) =
+ (ALL (x::'q_47675) y::'q_47674. P x y --> Q (f x y))) &
+(ALL (P::'q_47742 => 'q_47741 => 'q_47740 => bool)
+ f::'q_47742 => 'q_47741 => 'q_47740 => 'q_47739.
+ (ALL z::'q_47739.
+ z : {u::'q_47739.
+ EX (w::'q_47742) (x::'q_47741) y::'q_47740.
+ P w x y & u = f w x y} -->
+ Q z) =
+ (ALL (w::'q_47742) (x::'q_47741) y::'q_47740. P w x y --> Q (f w x y)))"
+ by (import hollight FORALL_IN_GSPEC)
+
+lemma EXISTS_IN_GSPEC: "(ALL (P::'q_47788 => bool) f::'q_47788 => 'q_47909.
+ (EX z::'q_47909.
+ z : {u::'q_47909. EX x::'q_47788. P x & u = f x} &
+ (Q::'q_47909 => bool) z) =
+ (EX x::'q_47788. P x & Q (f x))) &
+(ALL (P::'q_47845 => 'q_47844 => bool) f::'q_47845 => 'q_47844 => 'q_47909.
+ (EX z::'q_47909.
+ z : {u::'q_47909. EX (x::'q_47845) y::'q_47844. P x y & u = f x y} &
+ Q z) =
+ (EX (x::'q_47845) y::'q_47844. P x y & Q (f x y))) &
+(ALL (P::'q_47912 => 'q_47911 => 'q_47910 => bool)
+ f::'q_47912 => 'q_47911 => 'q_47910 => 'q_47909.
+ (EX z::'q_47909.
+ z : {u::'q_47909.
+ EX (w::'q_47912) (x::'q_47911) y::'q_47910.
+ P w x y & u = f w x y} &
+ Q z) =
+ (EX (w::'q_47912) (x::'q_47911) y::'q_47910. P w x y & Q (f w x y)))"
+ by (import hollight EXISTS_IN_GSPEC)
+
+lemma SET_PROVE_CASES: "(P::('A => bool) => bool) {} &
+(ALL (a::'A) s::'A => bool. a ~: s --> P (insert a s))
+==> P (x::'A => bool)"
+ by (import hollight SET_PROVE_CASES)
+
+lemma UNIONS_IMAGE: "Union ((f::'q_47989 => 'q_47973 => bool) ` (s::'q_47989 => bool)) =
+{u::'q_47973. EX y::'q_47973. (EX x::'q_47989. x : s & y : f x) & u = y}"
+ by (import hollight UNIONS_IMAGE)
+
+lemma INTERS_IMAGE: "Inter ((f::'q_48032 => 'q_48016 => bool) ` (s::'q_48032 => bool)) =
+{u::'q_48016. EX y::'q_48016. (ALL x::'q_48032. x : s --> y : f x) & u = y}"
+ by (import hollight INTERS_IMAGE)
+
+lemma UNIONS_GSPEC: "(ALL (P::'q_48085 => bool) f::'q_48085 => 'q_48071 => bool.
+ Union {u::'q_48071 => bool. EX x::'q_48085. P x & u = f x} =
+ {u::'q_48071.
+ EX a::'q_48071. (EX x::'q_48085. P x & a : f x) & u = a}) &
+(ALL (P::'q_48149 => 'q_48148 => bool)
+ f::'q_48149 => 'q_48148 => 'q_48129 => bool.
+ Union
+ {u::'q_48129 => bool.
+ EX (x::'q_48149) y::'q_48148. P x y & u = f x y} =
+ {u::'q_48129.
+ EX a::'q_48129.
+ (EX (x::'q_48149) y::'q_48148. P x y & a : f x y) & u = a}) &
+(ALL (P::'q_48223 => 'q_48222 => 'q_48221 => bool)
+ f::'q_48223 => 'q_48222 => 'q_48221 => 'q_48197 => bool.
+ Union
+ {u::'q_48197 => bool.
+ EX (x::'q_48223) (y::'q_48222) z::'q_48221. P x y z & u = f x y z} =
+ {u::'q_48197.
+ EX a::'q_48197.
+ (EX (x::'q_48223) (y::'q_48222) z::'q_48221.
+ P x y z & a : f x y z) &
+ u = a})"
+ by (import hollight UNIONS_GSPEC)
+
+lemma INTERS_GSPEC: "(ALL (P::'q_48276 => bool) f::'q_48276 => 'q_48262 => bool.
+ Inter {u::'q_48262 => bool. EX x::'q_48276. P x & u = f x} =
+ {u::'q_48262.
+ EX a::'q_48262. (ALL x::'q_48276. P x --> a : f x) & u = a}) &
+(ALL (P::'q_48340 => 'q_48339 => bool)
+ f::'q_48340 => 'q_48339 => 'q_48320 => bool.
+ Inter
+ {u::'q_48320 => bool.
+ EX (x::'q_48340) y::'q_48339. P x y & u = f x y} =
+ {u::'q_48320.
+ EX a::'q_48320.
+ (ALL (x::'q_48340) y::'q_48339. P x y --> a : f x y) & u = a}) &
+(ALL (P::'q_48414 => 'q_48413 => 'q_48412 => bool)
+ f::'q_48414 => 'q_48413 => 'q_48412 => 'q_48388 => bool.
+ Inter
+ {u::'q_48388 => bool.
+ EX (x::'q_48414) (y::'q_48413) z::'q_48412. P x y z & u = f x y z} =
+ {u::'q_48388.
+ EX a::'q_48388.
+ (ALL (x::'q_48414) (y::'q_48413) z::'q_48412.
+ P x y z --> a : f x y z) &
+ u = a})"
+ by (import hollight INTERS_GSPEC)
+
+lemma DIFF_INTERS: "(x::'q_48451 => bool) - Inter (xa::('q_48451 => bool) => bool) =
+Union {u::'q_48451 => bool. EX xb::'q_48451 => bool. xb : xa & u = x - xb}"
+ by (import hollight DIFF_INTERS)
+
+lemma INTERS_UNIONS: "Inter (x::('q_48486 => bool) => bool) =
+UNIV -
+Union {u::'q_48486 => bool. EX t::'q_48486 => bool. t : x & u = UNIV - t}"
+ by (import hollight INTERS_UNIONS)
+
+lemma UNIONS_INTERS: "Union (s::('q_48521 => bool) => bool) =
+UNIV -
+Inter {u::'q_48521 => bool. EX t::'q_48521 => bool. t : s & u = UNIV - t}"
+ by (import hollight UNIONS_INTERS)
+
+lemma FINITE_SING: "finite {x::'q_48799}"
+ by (import hollight FINITE_SING)
+
+lemma FINITE_DELETE_IMP: "finite (s::'A => bool) ==> finite (s - {x::'A})"
by (import hollight FINITE_DELETE_IMP)
-lemma FINITE_DELETE: "ALL (s::'A::type => bool) x::'A::type. FINITE (DELETE s x) = FINITE s"
+lemma FINITE_DELETE: "finite ((s::'A => bool) - {x::'A}) = finite s"
by (import hollight FINITE_DELETE)
-lemma FINITE_UNIONS: "ALL s::('q_40983::type => bool) => bool.
- FINITE s -->
- FINITE (UNIONS s) = (ALL t::'q_40983::type => bool. IN t s --> FINITE t)"
- by (import hollight FINITE_UNIONS)
-
-lemma FINITE_IMAGE_EXPAND: "ALL (f::'A::type => 'B::type) s::'A::type => bool.
- FINITE s -->
- FINITE
- (GSPEC
- (%u::'B::type.
- EX y::'B::type. SETSPEC u (EX x::'A::type. IN x s & y = f x) y))"
+lemma FINITE_FINITE_UNIONS: "finite (s::('q_48871 => bool) => bool)
+==> finite (Union s) = (ALL t::'q_48871 => bool. t : s --> finite t)"
+ by (import hollight FINITE_FINITE_UNIONS)
+
+lemma FINITE_IMAGE_EXPAND: "finite (s::'A => bool)
+==> finite
+ {u::'B. EX y::'B. (EX x::'A. x : s & y = (f::'A => 'B) x) & u = y}"
by (import hollight FINITE_IMAGE_EXPAND)
-lemma FINITE_IMAGE: "ALL (x::'A::type => 'B::type) xa::'A::type => bool.
- FINITE xa --> FINITE (IMAGE x xa)"
- by (import hollight FINITE_IMAGE)
-
-lemma FINITE_IMAGE_INJ_GENERAL: "ALL (f::'A::type => 'B::type) (x::'B::type => bool) s::'A::type => bool.
- (ALL (x::'A::type) y::'A::type. IN x s & IN y s & f x = f y --> x = y) &
- FINITE x -->
- FINITE
- (GSPEC
- (%u::'A::type. EX xa::'A::type. SETSPEC u (IN xa s & IN (f xa) x) xa))"
+lemma FINITE_IMAGE_INJ_GENERAL: "(ALL (x::'A) y::'A.
+ x : (s::'A => bool) & y : s & (f::'A => 'B) x = f y --> x = y) &
+finite (x::'B => bool)
+==> finite {u::'A. EX xa::'A. (xa : s & f xa : x) & u = xa}"
by (import hollight FINITE_IMAGE_INJ_GENERAL)
-lemma FINITE_IMAGE_INJ: "ALL (f::'A::type => 'B::type) A::'B::type => bool.
- (ALL (x::'A::type) y::'A::type. f x = f y --> x = y) & FINITE A -->
- FINITE (GSPEC (%u::'A::type. EX x::'A::type. SETSPEC u (IN (f x) A) x))"
+lemma FINITE_FINITE_PREIMAGE_GENERAL: "finite (t::'B => bool) &
+(ALL y::'B.
+ y : t -->
+ finite
+ {u::'A. EX x::'A. (x : (s::'A => bool) & (f::'A => 'B) x = y) & u = x})
+==> finite {u::'A. EX x::'A. (x : s & f x : t) & u = x}"
+ by (import hollight FINITE_FINITE_PREIMAGE_GENERAL)
+
+lemma FINITE_FINITE_PREIMAGE: "finite (t::'B => bool) &
+(ALL y::'B. y : t --> finite {u::'A. EX x::'A. (f::'A => 'B) x = y & u = x})
+==> finite {u::'A. EX x::'A. f x : t & u = x}"
+ by (import hollight FINITE_FINITE_PREIMAGE)
+
+lemma FINITE_IMAGE_INJ_EQ: "(!!(x::'A) y::'A.
+ x : (s::'A => bool) & y : s & (f::'A => 'B) x = f y ==> x = y)
+==> finite (f ` s) = finite s"
+ by (import hollight FINITE_IMAGE_INJ_EQ)
+
+lemma FINITE_IMAGE_INJ: "(ALL (x::'A) y::'A. (f::'A => 'B) x = f y --> x = y) &
+finite (A::'B => bool)
+==> finite {u::'A. EX x::'A. f x : A & u = x}"
by (import hollight FINITE_IMAGE_INJ)
-lemma INFINITE_IMAGE_INJ: "ALL f::'A::type => 'B::type.
- (ALL (x::'A::type) y::'A::type. f x = f y --> x = y) -->
- (ALL s::'A::type => bool. INFINITE s --> INFINITE (IMAGE f s))"
+lemma INFINITE_IMAGE_INJ: "[| !!(x::'A) y::'A. (f::'A => 'B) x = f y ==> x = y;
+ infinite (s::'A => bool) |]
+==> infinite (f ` s)"
by (import hollight INFINITE_IMAGE_INJ)
-lemma INFINITE_NONEMPTY: "ALL s::'q_41466::type => bool. INFINITE s --> s ~= EMPTY"
- by (import hollight INFINITE_NONEMPTY)
-
-lemma INFINITE_DIFF_FINITE: "ALL (s::'A::type => bool) t::'A::type => bool.
- INFINITE s & FINITE t --> INFINITE (DIFF s t)"
- by (import hollight INFINITE_DIFF_FINITE)
-
-lemma FINITE_SUBSET_IMAGE: "ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'B::type => bool.
- (FINITE t & SUBSET t (IMAGE f s)) =
- (EX x::'A::type => bool. FINITE x & SUBSET x s & t = IMAGE f x)"
+lemma FINITE_SUBSET_IMAGE: "(finite (t::'B => bool) & t <= (f::'A => 'B) ` (s::'A => bool)) =
+(EX x::'A => bool. finite x & x <= s & t = f ` x)"
by (import hollight FINITE_SUBSET_IMAGE)
-lemma FINITE_SUBSET_IMAGE_IMP: "ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'B::type => bool.
- FINITE t & SUBSET t (IMAGE f s) -->
- (EX s'::'A::type => bool.
- FINITE s' & SUBSET s' s & SUBSET t (IMAGE f s'))"
+lemma EXISTS_FINITE_SUBSET_IMAGE: "(EX xc::'q_49755 => bool.
+ finite xc &
+ xc <= (xa::'q_49735 => 'q_49755) ` (xb::'q_49735 => bool) &
+ (x::('q_49755 => bool) => bool) xc) =
+(EX xc::'q_49735 => bool. finite xc & xc <= xb & x (xa ` xc))"
+ by (import hollight EXISTS_FINITE_SUBSET_IMAGE)
+
+lemma FINITE_SUBSET_IMAGE_IMP: "finite (t::'B => bool) & t <= (f::'A => 'B) ` (s::'A => bool)
+==> EX s'::'A => bool. finite s' & s' <= s & t <= f ` s'"
by (import hollight FINITE_SUBSET_IMAGE_IMP)
-lemma FINITE_SUBSETS: "ALL s::'A::type => bool.
- FINITE s -->
- FINITE
- (GSPEC
- (%u::'A::type => bool.
- EX t::'A::type => bool. SETSPEC u (SUBSET t s) t))"
- by (import hollight FINITE_SUBSETS)
-
-lemma FINITE_DIFF: "ALL (s::'q_41764::type => bool) t::'q_41764::type => bool.
- FINITE s --> FINITE (DIFF s t)"
- by (import hollight FINITE_DIFF)
-
-definition FINREC :: "('q_41824 => 'q_41823 => 'q_41823)
-=> 'q_41823 => ('q_41824 => bool) => 'q_41823 => nat => bool" where
+definition
+ FINREC :: "('A => 'B => 'B) => 'B => ('A => bool) => 'B => nat => bool" where
"FINREC ==
-SOME FINREC::('q_41824::type => 'q_41823::type => 'q_41823::type)
- => 'q_41823::type
- => ('q_41824::type => bool)
- => 'q_41823::type => nat => bool.
- (ALL (f::'q_41824::type => 'q_41823::type => 'q_41823::type)
- (s::'q_41824::type => bool) (a::'q_41823::type) b::'q_41823::type.
- FINREC f b s a 0 = (s = EMPTY & a = b)) &
- (ALL (b::'q_41823::type) (s::'q_41824::type => bool) (n::nat)
- (a::'q_41823::type)
- f::'q_41824::type => 'q_41823::type => 'q_41823::type.
+SOME FINREC::('A => 'B => 'B) => 'B => ('A => bool) => 'B => nat => bool.
+ (ALL (f::'A => 'B => 'B) (s::'A => bool) (a::'B) b::'B.
+ FINREC f b s a (0::nat) = (s = {} & a = b)) &
+ (ALL (b::'B) (s::'A => bool) (n::nat) (a::'B) f::'A => 'B => 'B.
FINREC f b s a (Suc n) =
- (EX (x::'q_41824::type) c::'q_41823::type.
- IN x s & FINREC f b (DELETE s x) c n & a = f x c))"
+ (EX (x::'A) c::'B. x : s & FINREC f b (s - {x}) c n & a = f x c))"
lemma DEF_FINREC: "FINREC =
-(SOME FINREC::('q_41824::type => 'q_41823::type => 'q_41823::type)
- => 'q_41823::type
- => ('q_41824::type => bool)
- => 'q_41823::type => nat => bool.
- (ALL (f::'q_41824::type => 'q_41823::type => 'q_41823::type)
- (s::'q_41824::type => bool) (a::'q_41823::type) b::'q_41823::type.
- FINREC f b s a 0 = (s = EMPTY & a = b)) &
- (ALL (b::'q_41823::type) (s::'q_41824::type => bool) (n::nat)
- (a::'q_41823::type)
- f::'q_41824::type => 'q_41823::type => 'q_41823::type.
+(SOME FINREC::('A => 'B => 'B) => 'B => ('A => bool) => 'B => nat => bool.
+ (ALL (f::'A => 'B => 'B) (s::'A => bool) (a::'B) b::'B.
+ FINREC f b s a (0::nat) = (s = {} & a = b)) &
+ (ALL (b::'B) (s::'A => bool) (n::nat) (a::'B) f::'A => 'B => 'B.
FINREC f b s a (Suc n) =
- (EX (x::'q_41824::type) c::'q_41823::type.
- IN x s & FINREC f b (DELETE s x) c n & a = f x c)))"
+ (EX (x::'A) c::'B. x : s & FINREC f b (s - {x}) c n & a = f x c)))"
by (import hollight DEF_FINREC)
-lemma FINREC_1_LEMMA: "ALL (x::'q_41869::type => 'q_41868::type => 'q_41868::type)
- (xa::'q_41868::type) (xb::'q_41869::type => bool) xc::'q_41868::type.
- FINREC x xa xb xc (Suc 0) =
- (EX xd::'q_41869::type. xb = INSERT xd EMPTY & xc = x xd xa)"
+lemma FINREC_1_LEMMA: "FINREC (x::'q_49919 => 'q_49918 => 'q_49918) (xa::'q_49918)
+ (xb::'q_49919 => bool) (xc::'q_49918) (Suc (0::nat)) =
+(EX xd::'q_49919. xb = {xd} & xc = x xd xa)"
by (import hollight FINREC_1_LEMMA)
-lemma FINREC_SUC_LEMMA: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- (ALL (n::nat) (s::'A::type => bool) z::'B::type.
- FINREC f b s z (Suc n) -->
- (ALL x::'A::type.
- IN x s -->
- (EX w::'B::type. FINREC f b (DELETE s x) w n & z = f x w)))"
+lemma FINREC_SUC_LEMMA: "[| !!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s);
+ FINREC f (b::'B) (s::'A => bool) (z::'B) (Suc (n::nat)); (x::'A) : s |]
+==> EX w::'B. FINREC f b (s - {x}) w n & z = f x w"
by (import hollight FINREC_SUC_LEMMA)
-lemma FINREC_UNIQUE_LEMMA: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- (ALL (n1::nat) (n2::nat) (s::'A::type => bool) (a1::'B::type)
- a2::'B::type.
- FINREC f b s a1 n1 & FINREC f b s a2 n2 --> a1 = a2 & n1 = n2)"
+lemma FINREC_UNIQUE_LEMMA: "[| !!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s);
+ FINREC f (b::'B) (s::'A => bool) (a1::'B) (n1::nat) &
+ FINREC f b s (a2::'B) (n2::nat) |]
+==> a1 = a2 & n1 = n2"
by (import hollight FINREC_UNIQUE_LEMMA)
-lemma FINREC_EXISTS_LEMMA: "ALL (f::'A::type => 'B::type => 'B::type) (b::'B::type) s::'A::type => bool.
- FINITE s --> (EX a::'B::type. Ex (FINREC f b s a))"
+lemma FINREC_EXISTS_LEMMA: "finite (s::'A => bool)
+==> EX a::'B. Ex (FINREC (f::'A => 'B => 'B) (b::'B) s a)"
by (import hollight FINREC_EXISTS_LEMMA)
-lemma FINREC_FUN_LEMMA: "ALL (P::'A::type => bool) R::'A::type => 'B::type => 'C::type => bool.
- (ALL s::'A::type. P s --> (EX a::'B::type. Ex (R s a))) &
- (ALL (n1::'C::type) (n2::'C::type) (s::'A::type) (a1::'B::type)
- a2::'B::type. R s a1 n1 & R s a2 n2 --> a1 = a2 & n1 = n2) -->
- (EX x::'A::type => 'B::type.
- ALL (s::'A::type) a::'B::type. P s --> Ex (R s a) = (x s = a))"
+lemma FINREC_FUN_LEMMA: "(ALL s::'A.
+ (P::'A => bool) s -->
+ (EX a::'B. Ex ((R::'A => 'B => 'C => bool) s a))) &
+(ALL (n1::'C) (n2::'C) (s::'A) (a1::'B) a2::'B.
+ R s a1 n1 & R s a2 n2 --> a1 = a2 & n1 = n2)
+==> EX x::'A => 'B. ALL (s::'A) a::'B. P s --> Ex (R s a) = (x s = a)"
by (import hollight FINREC_FUN_LEMMA)
-lemma FINREC_FUN: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- (EX g::('A::type => bool) => 'B::type.
- g EMPTY = b &
- (ALL (s::'A::type => bool) x::'A::type.
- FINITE s & IN x s --> g s = f x (g (DELETE s x))))"
+lemma FINREC_FUN: "(!!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s))
+==> EX g::('A => bool) => 'B.
+ g {} = (b::'B) &
+ (ALL (s::'A => bool) x::'A.
+ finite s & x : s --> g s = f x (g (s - {x})))"
by (import hollight FINREC_FUN)
-lemma SET_RECURSION_LEMMA: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- (EX g::('A::type => bool) => 'B::type.
- g EMPTY = b &
- (ALL (x::'A::type) s::'A::type => bool.
- FINITE s --> g (INSERT x s) = COND (IN x s) (g s) (f x (g s))))"
+lemma SET_RECURSION_LEMMA: "(!!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s))
+==> EX g::('A => bool) => 'B.
+ g {} = (b::'B) &
+ (ALL (x::'A) s::'A => bool.
+ finite s --> g (insert x s) = (if x : s then g s else f x (g s)))"
by (import hollight SET_RECURSION_LEMMA)
-definition ITSET :: "('q_42525 => 'q_42524 => 'q_42524)
-=> ('q_42525 => bool) => 'q_42524 => 'q_42524" where
+definition
+ ITSET :: "('q_50575 => 'q_50574 => 'q_50574)
+=> ('q_50575 => bool) => 'q_50574 => 'q_50574" where
"ITSET ==
-%(u::'q_42525::type => 'q_42524::type => 'q_42524::type)
- (ua::'q_42525::type => bool) ub::'q_42524::type.
- (SOME g::('q_42525::type => bool) => 'q_42524::type.
- g EMPTY = ub &
- (ALL (x::'q_42525::type) s::'q_42525::type => bool.
- FINITE s --> g (INSERT x s) = COND (IN x s) (g s) (u x (g s))))
+%(u::'q_50575 => 'q_50574 => 'q_50574) (ua::'q_50575 => bool) ub::'q_50574.
+ (SOME g::('q_50575 => bool) => 'q_50574.
+ g {} = ub &
+ (ALL (x::'q_50575) s::'q_50575 => bool.
+ finite s -->
+ g (insert x s) = (if x : s then g s else u x (g s))))
ua"
lemma DEF_ITSET: "ITSET =
-(%(u::'q_42525::type => 'q_42524::type => 'q_42524::type)
- (ua::'q_42525::type => bool) ub::'q_42524::type.
- (SOME g::('q_42525::type => bool) => 'q_42524::type.
- g EMPTY = ub &
- (ALL (x::'q_42525::type) s::'q_42525::type => bool.
- FINITE s --> g (INSERT x s) = COND (IN x s) (g s) (u x (g s))))
+(%(u::'q_50575 => 'q_50574 => 'q_50574) (ua::'q_50575 => bool) ub::'q_50574.
+ (SOME g::('q_50575 => bool) => 'q_50574.
+ g {} = ub &
+ (ALL (x::'q_50575) s::'q_50575 => bool.
+ finite s -->
+ g (insert x s) = (if x : s then g s else u x (g s))))
ua)"
by (import hollight DEF_ITSET)
-lemma FINITE_RECURSION: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- ITSET f EMPTY b = b &
- (ALL (x::'A::type) xa::'A::type => bool.
- FINITE xa -->
- ITSET f (INSERT x xa) b =
- COND (IN x xa) (ITSET f xa b) (f x (ITSET f xa b)))"
+lemma FINITE_RECURSION: "(!!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s))
+==> ITSET f {} (b::'B) = b &
+ (ALL (x::'A) xa::'A => bool.
+ finite xa -->
+ ITSET f (insert x xa) b =
+ (if x : xa then ITSET f xa b else f x (ITSET f xa b)))"
by (import hollight FINITE_RECURSION)
-lemma FINITE_RECURSION_DELETE: "ALL (f::'A::type => 'B::type => 'B::type) b::'B::type.
- (ALL (x::'A::type) (y::'A::type) s::'B::type.
- x ~= y --> f x (f y s) = f y (f x s)) -->
- ITSET f EMPTY b = b &
- (ALL (x::'A::type) s::'A::type => bool.
- FINITE s -->
- ITSET f s b =
- COND (IN x s) (f x (ITSET f (DELETE s x) b))
- (ITSET f (DELETE s x) b))"
+lemma FINITE_RECURSION_DELETE: "(!!(x::'A) (y::'A) s::'B.
+ x ~= y ==> (f::'A => 'B => 'B) x (f y s) = f y (f x s))
+==> ITSET f {} (b::'B) = b &
+ (ALL (x::'A) s::'A => bool.
+ finite s -->
+ ITSET f s b =
+ (if x : s then f x (ITSET f (s - {x}) b) else ITSET f (s - {x}) b))"
by (import hollight FINITE_RECURSION_DELETE)
-lemma ITSET_EQ: "ALL (x::'q_42830::type => bool)
- (xa::'q_42830::type => 'q_42831::type => 'q_42831::type)
- (xb::'q_42830::type => 'q_42831::type => 'q_42831::type)
- xc::'q_42831::type.
- FINITE x &
- (ALL xc::'q_42830::type. IN xc x --> xa xc = xb xc) &
- (ALL (x::'q_42830::type) (y::'q_42830::type) s::'q_42831::type.
- x ~= y --> xa x (xa y s) = xa y (xa x s)) &
- (ALL (x::'q_42830::type) (y::'q_42830::type) s::'q_42831::type.
- x ~= y --> xb x (xb y s) = xb y (xb x s)) -->
- ITSET xa x xc = ITSET xb x xc"
+lemma ITSET_EQ: "finite (x::'q_50880 => bool) &
+(ALL xc::'q_50880.
+ xc : x -->
+ (xa::'q_50880 => 'q_50881 => 'q_50881) xc =
+ (xb::'q_50880 => 'q_50881 => 'q_50881) xc) &
+(ALL (x::'q_50880) (y::'q_50880) s::'q_50881.
+ x ~= y --> xa x (xa y s) = xa y (xa x s)) &
+(ALL (x::'q_50880) (y::'q_50880) s::'q_50881.
+ x ~= y --> xb x (xb y s) = xb y (xb x s))
+==> ITSET xa x (xc::'q_50881) = ITSET xb x xc"
by (import hollight ITSET_EQ)
-lemma SUBSET_RESTRICT: "ALL (x::'q_42864::type => bool) xa::'q_42864::type => bool.
- SUBSET
- (GSPEC
- (%u::'q_42864::type.
- EX xb::'q_42864::type. SETSPEC u (IN xb x & xa xb) xb))
- x"
+lemma SUBSET_RESTRICT: "{u::'q_50914.
+ EX xb::'q_50914.
+ (xb : (x::'q_50914 => bool) & (xa::'q_50914 => bool) xb) & u = xb}
+<= x"
by (import hollight SUBSET_RESTRICT)
-lemma FINITE_RESTRICT: "ALL (s::'A::type => bool) p::'q_42882::type.
- FINITE s -->
- FINITE
- (GSPEC
- (%u::'A::type.
- EX x::'A::type. SETSPEC u (IN x s & (P::'A::type => bool) x) x))"
+lemma FINITE_RESTRICT: "finite (s::'A => bool)
+==> finite {u::'A. EX x::'A. (x : s & (P::'A => bool) x) & u = x}"
by (import hollight FINITE_RESTRICT)
-definition CARD :: "('q_42918 => bool) => nat" where
- "CARD == %u::'q_42918::type => bool. ITSET (%x::'q_42918::type. Suc) u 0"
-
-lemma DEF_CARD: "CARD = (%u::'q_42918::type => bool. ITSET (%x::'q_42918::type. Suc) u 0)"
+definition
+ CARD :: "('q_50968 => bool) => nat" where
+ "CARD == %u::'q_50968 => bool. ITSET (%x::'q_50968. Suc) u (0::nat)"
+
+lemma DEF_CARD: "CARD = (%u::'q_50968 => bool. ITSET (%x::'q_50968. Suc) u (0::nat))"
by (import hollight DEF_CARD)
-lemma CARD_CLAUSES: "(op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((CARD::('A::type => bool) => nat) (EMPTY::'A::type => bool)) (0::nat))
- ((All::('A::type => bool) => bool)
- (%x::'A::type.
- (All::(('A::type => bool) => bool) => bool)
- (%s::'A::type => bool.
- (op -->::bool => bool => bool)
- ((FINITE::('A::type => bool) => bool) s)
- ((op =::nat => nat => bool)
- ((CARD::('A::type => bool) => nat)
- ((INSERT::'A::type
- => ('A::type => bool) => 'A::type => bool)
- x s))
- ((COND::bool => nat => nat => nat)
- ((IN::'A::type => ('A::type => bool) => bool) x s)
- ((CARD::('A::type => bool) => nat) s)
- ((Suc::nat => nat)
- ((CARD::('A::type => bool) => nat) s)))))))"
+lemma CARD_CLAUSES: "CARD {} = (0::nat) &
+(ALL (x::'A::type) s::'A::type => bool.
+ finite s -->
+ CARD (insert x s) = (if x : s then CARD s else Suc (CARD s)))"
by (import hollight CARD_CLAUSES)
-lemma CARD_UNION: "ALL (x::'A::type => bool) xa::'A::type => bool.
- FINITE x & FINITE xa & hollight.INTER x xa = EMPTY -->
- CARD (hollight.UNION x xa) = CARD x + CARD xa"
+lemma CARD_UNION: "finite (x::'A => bool) & finite (xa::'A => bool) & x Int xa = {}
+==> CARD (x Un xa) = CARD x + CARD xa"
by (import hollight CARD_UNION)
-lemma CARD_DELETE: "ALL (x::'A::type) s::'A::type => bool.
- FINITE s -->
- CARD (DELETE s x) = COND (IN x s) (CARD s - NUMERAL_BIT1 0) (CARD s)"
+lemma CARD_DELETE: "finite (s::'A => bool)
+==> CARD (s - {x::'A}) = (if x : s then CARD s - (1::nat) else CARD s)"
by (import hollight CARD_DELETE)
-lemma CARD_UNION_EQ: "ALL (s::'q_43163::type => bool) (t::'q_43163::type => bool)
- u::'q_43163::type => bool.
- FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
- CARD s + CARD t = CARD u"
+lemma CARD_UNION_EQ: "finite (u::'q_51213 => bool) &
+(s::'q_51213 => bool) Int (t::'q_51213 => bool) = {} & s Un t = u
+==> CARD s + CARD t = CARD u"
by (import hollight CARD_UNION_EQ)
-definition HAS_SIZE :: "('q_43199 => bool) => nat => bool" where
- "HAS_SIZE == %(u::'q_43199::type => bool) ua::nat. FINITE u & CARD u = ua"
-
-lemma DEF_HAS_SIZE: "HAS_SIZE = (%(u::'q_43199::type => bool) ua::nat. FINITE u & CARD u = ua)"
+lemma CARD_DIFF: "finite (s::'q_51270 => bool) & (t::'q_51270 => bool) <= s
+==> CARD (s - t) = CARD s - CARD t"
+ by (import hollight CARD_DIFF)
+
+lemma CARD_EQ_0: "finite (s::'q_51308 => bool) ==> (CARD s = (0::nat)) = (s = {})"
+ by (import hollight CARD_EQ_0)
+
+lemma FINITE_INDUCT_DELETE: "[| (P::('A => bool) => bool) {} &
+ (ALL s::'A => bool.
+ finite s & s ~= {} --> (EX x::'A. x : s & (P (s - {x}) --> P s)));
+ finite (s::'A => bool) |]
+==> P s"
+ by (import hollight FINITE_INDUCT_DELETE)
+
+definition
+ HAS_SIZE :: "('q_51427 => bool) => nat => bool" where
+ "HAS_SIZE == %(u::'q_51427 => bool) ua::nat. finite u & CARD u = ua"
+
+lemma DEF_HAS_SIZE: "HAS_SIZE = (%(u::'q_51427 => bool) ua::nat. finite u & CARD u = ua)"
by (import hollight DEF_HAS_SIZE)
-lemma HAS_SIZE_CARD: "ALL (x::'q_43218::type => bool) xa::nat. HAS_SIZE x xa --> CARD x = xa"
+lemma HAS_SIZE_CARD: "HAS_SIZE (x::'q_51446 => bool) (xa::nat) ==> CARD x = xa"
by (import hollight HAS_SIZE_CARD)
-lemma HAS_SIZE_0: "ALL (s::'A::type => bool) n::'q_43234::type. HAS_SIZE s 0 = (s = EMPTY)"
+lemma HAS_SIZE_0: "HAS_SIZE (s::'A => bool) (0::nat) = (s = {})"
by (import hollight HAS_SIZE_0)
-lemma HAS_SIZE_SUC: "ALL (s::'A::type => bool) n::nat.
- HAS_SIZE s (Suc n) =
- (s ~= EMPTY & (ALL x::'A::type. IN x s --> HAS_SIZE (DELETE s x) n))"
+lemma HAS_SIZE_SUC: "HAS_SIZE (s::'A => bool) (Suc (n::nat)) =
+(s ~= {} & (ALL x::'A. x : s --> HAS_SIZE (s - {x}) n))"
by (import hollight HAS_SIZE_SUC)
-lemma HAS_SIZE_UNION: "ALL (x::'q_43356::type => bool) (xa::'q_43356::type => bool) (xb::nat)
- xc::nat.
- HAS_SIZE x xb & HAS_SIZE xa xc & DISJOINT x xa -->
- HAS_SIZE (hollight.UNION x xa) (xb + xc)"
+lemma HAS_SIZE_UNION: "HAS_SIZE (x::'q_51584 => bool) (xb::nat) &
+HAS_SIZE (xa::'q_51584 => bool) (xc::nat) & x Int xa = {}
+==> HAS_SIZE (x Un xa) (xb + xc)"
by (import hollight HAS_SIZE_UNION)
-lemma HAS_SIZE_UNIONS: "ALL (x::'A::type => bool) (xa::'A::type => 'B::type => bool) (xb::nat)
- xc::nat.
- HAS_SIZE x xb &
- (ALL xb::'A::type. IN xb x --> HAS_SIZE (xa xb) xc) &
- (ALL (xb::'A::type) y::'A::type.
- IN xb x & IN y x & xb ~= y --> DISJOINT (xa xb) (xa y)) -->
- HAS_SIZE
- (UNIONS
- (GSPEC
- (%u::'B::type => bool.
- EX xb::'A::type. SETSPEC u (IN xb x) (xa xb))))
- (xb * xc)"
+lemma HAS_SIZE_DIFF: "HAS_SIZE (x::'q_51620 => bool) (xb::nat) &
+HAS_SIZE (xa::'q_51620 => bool) (xc::nat) & xa <= x
+==> HAS_SIZE (x - xa) (xb - xc)"
+ by (import hollight HAS_SIZE_DIFF)
+
+lemma HAS_SIZE_UNIONS: "HAS_SIZE (x::'A => bool) (xb::nat) &
+(ALL xb::'A. xb : x --> HAS_SIZE ((xa::'A => 'B => bool) xb) (xc::nat)) &
+(ALL (xb::'A) y::'A. xb : x & y : x & xb ~= y --> xa xb Int xa y = {})
+==> HAS_SIZE (Union {u::'B => bool. EX xb::'A. xb : x & u = xa xb})
+ (xb * xc)"
by (import hollight HAS_SIZE_UNIONS)
-lemma HAS_SIZE_CLAUSES: "HAS_SIZE (s::'q_43604::type => bool) 0 = (s = EMPTY) &
+lemma FINITE_HAS_SIZE: "finite (x::'q_51824 => bool) = HAS_SIZE x (CARD x)"
+ by (import hollight FINITE_HAS_SIZE)
+
+lemma HAS_SIZE_CLAUSES: "HAS_SIZE (s::'q_51886 => bool) (0::nat) = (s = {}) &
HAS_SIZE s (Suc (n::nat)) =
-(EX (a::'q_43604::type) t::'q_43604::type => bool.
- HAS_SIZE t n & ~ IN a t & s = INSERT a t)"
+(EX (a::'q_51886) t::'q_51886 => bool.
+ HAS_SIZE t n & a ~: t & s = insert a t)"
by (import hollight HAS_SIZE_CLAUSES)
-lemma CARD_SUBSET_EQ: "ALL (a::'A::type => bool) b::'A::type => bool.
- FINITE b & SUBSET a b & CARD a = CARD b --> a = b"
+lemma CARD_SUBSET_EQ: "finite (b::'A => bool) & (a::'A => bool) <= b & CARD a = CARD b ==> a = b"
by (import hollight CARD_SUBSET_EQ)
-lemma CARD_SUBSET: "ALL (a::'A::type => bool) b::'A::type => bool.
- SUBSET a b & FINITE b --> <= (CARD a) (CARD b)"
+lemma CARD_SUBSET: "(a::'A => bool) <= (b::'A => bool) & finite b ==> CARD a <= CARD b"
by (import hollight CARD_SUBSET)
-lemma CARD_SUBSET_LE: "ALL (a::'A::type => bool) b::'A::type => bool.
- FINITE b & SUBSET a b & <= (CARD b) (CARD a) --> a = b"
+lemma CARD_SUBSET_LE: "finite (b::'A => bool) & (a::'A => bool) <= b & CARD b <= CARD a ==> a = b"
by (import hollight CARD_SUBSET_LE)
-lemma CARD_EQ_0: "ALL s::'q_43920::type => bool. FINITE s --> (CARD s = 0) = (s = EMPTY)"
- by (import hollight CARD_EQ_0)
-
-lemma CARD_PSUBSET: "ALL (a::'A::type => bool) b::'A::type => bool.
- PSUBSET a b & FINITE b --> < (CARD a) (CARD b)"
+lemma SUBSET_CARD_EQ: "finite (t::'q_52197 => bool) & (s::'q_52197 => bool) <= t
+==> (CARD s = CARD t) = (s = t)"
+ by (import hollight SUBSET_CARD_EQ)
+
+lemma CARD_PSUBSET: "(a::'A => bool) < (b::'A => bool) & finite b ==> CARD a < CARD b"
by (import hollight CARD_PSUBSET)
-lemma CARD_UNION_LE: "ALL (s::'A::type => bool) t::'A::type => bool.
- FINITE s & FINITE t --> <= (CARD (hollight.UNION s t)) (CARD s + CARD t)"
+lemma CARD_UNION_LE: "finite (s::'A => bool) & finite (t::'A => bool)
+==> CARD (s Un t) <= CARD s + CARD t"
by (import hollight CARD_UNION_LE)
-lemma CARD_UNIONS_LE: "ALL (x::'A::type => bool) (xa::'A::type => 'B::type => bool) (xb::nat)
- xc::nat.
- HAS_SIZE x xb &
- (ALL xb::'A::type. IN xb x --> FINITE (xa xb) & <= (CARD (xa xb)) xc) -->
- <= (CARD
- (UNIONS
- (GSPEC
- (%u::'B::type => bool.
- EX xb::'A::type. SETSPEC u (IN xb x) (xa xb)))))
- (xb * xc)"
+lemma CARD_UNIONS_LE: "HAS_SIZE (x::'A => bool) (xb::nat) &
+(ALL xb::'A.
+ xb : x -->
+ finite ((xa::'A => 'B => bool) xb) & CARD (xa xb) <= (xc::nat))
+==> CARD (Union {u::'B => bool. EX xb::'A. xb : x & u = xa xb}) <= xb * xc"
by (import hollight CARD_UNIONS_LE)
-lemma CARD_IMAGE_INJ: "ALL (f::'A::type => 'B::type) x::'A::type => bool.
- (ALL (xa::'A::type) y::'A::type.
- IN xa x & IN y x & f xa = f y --> xa = y) &
- FINITE x -->
- CARD (IMAGE f x) = CARD x"
+lemma CARD_UNION_GEN: "finite (s::'q_52620 => bool) & finite (t::'q_52620 => bool)
+==> CARD (s Un t) = CARD s + CARD t - CARD (s Int t)"
+ by (import hollight CARD_UNION_GEN)
+
+lemma CARD_UNION_OVERLAP_EQ: "finite (s::'q_52701 => bool) & finite (t::'q_52701 => bool)
+==> (CARD (s Un t) = CARD s + CARD t) = (s Int t = {})"
+ by (import hollight CARD_UNION_OVERLAP_EQ)
+
+lemma CARD_UNION_OVERLAP: "finite (x::'q_52743 => bool) &
+finite (xa::'q_52743 => bool) & CARD (x Un xa) < CARD x + CARD xa
+==> x Int xa ~= {}"
+ by (import hollight CARD_UNION_OVERLAP)
+
+lemma CARD_IMAGE_INJ: "(ALL (xa::'A) y::'A.
+ xa : (x::'A => bool) & y : x & (f::'A => 'B) xa = f y --> xa = y) &
+finite x
+==> CARD (f ` x) = CARD x"
by (import hollight CARD_IMAGE_INJ)
-lemma HAS_SIZE_IMAGE_INJ: "ALL (x::'A::type => 'B::type) (xa::'A::type => bool) xb::nat.
- (ALL (xb::'A::type) y::'A::type.
- IN xb xa & IN y xa & x xb = x y --> xb = y) &
- HAS_SIZE xa xb -->
- HAS_SIZE (IMAGE x xa) xb"
+lemma HAS_SIZE_IMAGE_INJ: "(ALL (xb::'A) y::'A.
+ xb : (xa::'A => bool) & y : xa & (x::'A => 'B) xb = x y --> xb = y) &
+HAS_SIZE xa (xb::nat)
+==> HAS_SIZE (x ` xa) xb"
by (import hollight HAS_SIZE_IMAGE_INJ)
-lemma CARD_IMAGE_LE: "ALL (f::'A::type => 'B::type) s::'A::type => bool.
- FINITE s --> <= (CARD (IMAGE f s)) (CARD s)"
+lemma CARD_IMAGE_LE: "finite (s::'A => bool) ==> CARD ((f::'A => 'B) ` s) <= CARD s"
by (import hollight CARD_IMAGE_LE)
-lemma CHOOSE_SUBSET: "ALL s::'A::type => bool.
- FINITE s -->
- (ALL n::nat.
- <= n (CARD s) -->
- (EX t::'A::type => bool. SUBSET t s & HAS_SIZE t n))"
+lemma CARD_IMAGE_INJ_EQ: "finite (s::'A => bool) &
+(ALL x::'A. x : s --> (f::'A => 'B) x : (t::'B => bool)) &
+(ALL y::'B. y : t --> (EX! x::'A. x : s & f x = y))
+==> CARD t = CARD s"
+ by (import hollight CARD_IMAGE_INJ_EQ)
+
+lemma CARD_SUBSET_IMAGE: "finite (t::'q_52977 => bool) &
+(s::'q_52984 => bool) <= (f::'q_52977 => 'q_52984) ` t
+==> CARD s <= CARD t"
+ by (import hollight CARD_SUBSET_IMAGE)
+
+lemma HAS_SIZE_IMAGE_INJ_EQ: "(!!(x::'q_53049) y::'q_53049.
+ x : (s::'q_53049 => bool) & y : s & (f::'q_53049 => 'q_53044) x = f y
+ ==> x = y)
+==> HAS_SIZE (f ` s) (n::nat) = HAS_SIZE s n"
+ by (import hollight HAS_SIZE_IMAGE_INJ_EQ)
+
+lemma CHOOSE_SUBSET_STRONG: "(finite (s::'A => bool) ==> (n::nat) <= CARD s) ==> EX t<=s. HAS_SIZE t n"
+ by (import hollight CHOOSE_SUBSET_STRONG)
+
+lemma CHOOSE_SUBSET: "[| finite (s::'A => bool); (n::nat) <= CARD s |] ==> EX t<=s. HAS_SIZE t n"
by (import hollight CHOOSE_SUBSET)
-lemma HAS_SIZE_PRODUCT_DEPENDENT: "ALL (x::'A::type => bool) (xa::nat) (xb::'A::type => 'B::type => bool)
- xc::nat.
- HAS_SIZE x xa & (ALL xa::'A::type. IN xa x --> HAS_SIZE (xb xa) xc) -->
- HAS_SIZE
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (xa::'A::type) y::'B::type.
- SETSPEC u (IN xa x & IN y (xb xa)) (xa, y)))
- (xa * xc)"
+lemma HAS_SIZE_PRODUCT_DEPENDENT: "HAS_SIZE (x::'A => bool) (xa::nat) &
+(ALL xa::'A. xa : x --> HAS_SIZE ((xb::'A => 'B => bool) xa) (xc::nat))
+==> HAS_SIZE
+ {u::'A * 'B. EX (xa::'A) y::'B. (xa : x & y : xb xa) & u = (xa, y)}
+ (xa * xc)"
by (import hollight HAS_SIZE_PRODUCT_DEPENDENT)
-lemma FINITE_PRODUCT_DEPENDENT: "ALL (x::'A::type => bool) xa::'A::type => 'B::type => bool.
- FINITE x & (ALL xb::'A::type. IN xb x --> FINITE (xa xb)) -->
- FINITE
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (xb::'A::type) y::'B::type.
- SETSPEC u (IN xb x & IN y (xa xb)) (xb, y)))"
+lemma FINITE_PRODUCT_DEPENDENT: "finite (s::'A => bool) &
+(ALL x::'A. x : s --> finite ((t::'A => 'B => bool) x))
+==> finite
+ {u::'C.
+ EX (x::'A) y::'B. (x : s & y : t x) & u = (f::'A => 'B => 'C) x y}"
by (import hollight FINITE_PRODUCT_DEPENDENT)
-lemma FINITE_PRODUCT: "ALL (x::'A::type => bool) xa::'B::type => bool.
- FINITE x & FINITE xa -->
- FINITE
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (xb::'A::type) y::'B::type.
- SETSPEC u (IN xb x & IN y xa) (xb, y)))"
+lemma FINITE_PRODUCT: "finite (x::'A => bool) & finite (xa::'B => bool)
+==> finite {u::'A * 'B. EX (xb::'A) y::'B. (xb : x & y : xa) & u = (xb, y)}"
by (import hollight FINITE_PRODUCT)
-lemma CARD_PRODUCT: "ALL (s::'A::type => bool) t::'B::type => bool.
- FINITE s & FINITE t -->
- CARD
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (x::'A::type) y::'B::type.
- SETSPEC u (IN x s & IN y t) (x, y))) =
- CARD s * CARD t"
+lemma CARD_PRODUCT: "finite (s::'A => bool) & finite (t::'B => bool)
+==> CARD {u::'A * 'B. EX (x::'A) y::'B. (x : s & y : t) & u = (x, y)} =
+ CARD s * CARD t"
by (import hollight CARD_PRODUCT)
-lemma HAS_SIZE_PRODUCT: "ALL (x::'A::type => bool) (xa::nat) (xb::'B::type => bool) xc::nat.
- HAS_SIZE x xa & HAS_SIZE xb xc -->
- HAS_SIZE
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (xa::'A::type) y::'B::type.
- SETSPEC u (IN xa x & IN y xb) (xa, y)))
- (xa * xc)"
+lemma HAS_SIZE_PRODUCT: "HAS_SIZE (x::'A => bool) (xa::nat) & HAS_SIZE (xb::'B => bool) (xc::nat)
+==> HAS_SIZE
+ {u::'A * 'B. EX (xa::'A) y::'B. (xa : x & y : xb) & u = (xa, y)}
+ (xa * xc)"
by (import hollight HAS_SIZE_PRODUCT)
-lemma HAS_SIZE_FUNSPACE: "ALL (d::'B::type) (n::nat) (t::'B::type => bool) (m::nat)
- s::'A::type => bool.
- HAS_SIZE s m & HAS_SIZE t n -->
- HAS_SIZE
- (GSPEC
- (%u::'A::type => 'B::type.
- EX f::'A::type => 'B::type.
- SETSPEC u
- ((ALL x::'A::type. IN x s --> IN (f x) t) &
- (ALL x::'A::type. ~ IN x s --> f x = d))
- f))
- (EXP n m)"
+definition
+ CROSS :: "('q_53759 => bool) => ('q_53758 => bool) => 'q_53759 * 'q_53758 => bool" where
+ "CROSS ==
+%(u::'q_53759 => bool) ua::'q_53758 => bool.
+ {ub::'q_53759 * 'q_53758.
+ EX (x::'q_53759) y::'q_53758. (x : u & y : ua) & ub = (x, y)}"
+
+lemma DEF_CROSS: "CROSS =
+(%(u::'q_53759 => bool) ua::'q_53758 => bool.
+ {ub::'q_53759 * 'q_53758.
+ EX (x::'q_53759) y::'q_53758. (x : u & y : ua) & ub = (x, y)})"
+ by (import hollight DEF_CROSS)
+
+lemma IN_CROSS: "((x::'q_53795, xa::'q_53798)
+ : CROSS (xb::'q_53795 => bool) (xc::'q_53798 => bool)) =
+(x : xb & xa : xc)"
+ by (import hollight IN_CROSS)
+
+lemma HAS_SIZE_CROSS: "HAS_SIZE (x::'q_53823 => bool) (xb::nat) &
+HAS_SIZE (xa::'q_53826 => bool) (xc::nat)
+==> HAS_SIZE (CROSS x xa) (xb * xc)"
+ by (import hollight HAS_SIZE_CROSS)
+
+lemma FINITE_CROSS: "finite (x::'q_53851 => bool) & finite (xa::'q_53853 => bool)
+==> finite (CROSS x xa)"
+ by (import hollight FINITE_CROSS)
+
+lemma CARD_CROSS: "finite (x::'q_53874 => bool) & finite (xa::'q_53876 => bool)
+==> CARD (CROSS x xa) = CARD x * CARD xa"
+ by (import hollight CARD_CROSS)
+
+lemma CROSS_EQ_EMPTY: "(CROSS (x::'q_53917 => bool) (xa::'q_53921 => bool) = {}) =
+(x = {} | xa = {})"
+ by (import hollight CROSS_EQ_EMPTY)
+
+lemma HAS_SIZE_FUNSPACE: "HAS_SIZE (s::'A => bool) (m::nat) & HAS_SIZE (t::'B => bool) (n::nat)
+==> HAS_SIZE
+ {u::'A => 'B.
+ EX f::'A => 'B.
+ ((ALL x::'A. x : s --> f x : t) &
+ (ALL x::'A. x ~: s --> f x = (d::'B))) &
+ u = f}
+ (n ^ m)"
by (import hollight HAS_SIZE_FUNSPACE)
-lemma CARD_FUNSPACE: "ALL (s::'q_45275::type => bool) t::'q_45272::type => bool.
- FINITE s & FINITE t -->
- CARD
- (GSPEC
- (%u::'q_45275::type => 'q_45272::type.
- EX f::'q_45275::type => 'q_45272::type.
- SETSPEC u
- ((ALL x::'q_45275::type. IN x s --> IN (f x) t) &
- (ALL x::'q_45275::type.
- ~ IN x s --> f x = (d::'q_45272::type)))
- f)) =
- EXP (CARD t) (CARD s)"
+lemma CARD_FUNSPACE: "finite (s::'q_54227 => bool) & finite (t::'q_54224 => bool)
+==> CARD
+ {u::'q_54227 => 'q_54224.
+ EX f::'q_54227 => 'q_54224.
+ ((ALL x::'q_54227. x : s --> f x : t) &
+ (ALL x::'q_54227. x ~: s --> f x = (d::'q_54224))) &
+ u = f} =
+ CARD t ^ CARD s"
by (import hollight CARD_FUNSPACE)
-lemma FINITE_FUNSPACE: "ALL (s::'q_45341::type => bool) t::'q_45338::type => bool.
- FINITE s & FINITE t -->
- FINITE
- (GSPEC
- (%u::'q_45341::type => 'q_45338::type.
- EX f::'q_45341::type => 'q_45338::type.
- SETSPEC u
- ((ALL x::'q_45341::type. IN x s --> IN (f x) t) &
- (ALL x::'q_45341::type.
- ~ IN x s --> f x = (d::'q_45338::type)))
- f))"
+lemma FINITE_FUNSPACE: "finite (s::'q_54293 => bool) & finite (t::'q_54290 => bool)
+==> finite
+ {u::'q_54293 => 'q_54290.
+ EX f::'q_54293 => 'q_54290.
+ ((ALL x::'q_54293. x : s --> f x : t) &
+ (ALL x::'q_54293. x ~: s --> f x = (d::'q_54290))) &
+ u = f}"
by (import hollight FINITE_FUNSPACE)
-lemma HAS_SIZE_POWERSET: "ALL (s::'A::type => bool) n::nat.
- HAS_SIZE s n -->
- HAS_SIZE
- (GSPEC
- (%u::'A::type => bool.
- EX t::'A::type => bool. SETSPEC u (SUBSET t s) t))
- (EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) n)"
+lemma HAS_SIZE_POWERSET: "HAS_SIZE (s::'A => bool) (n::nat)
+==> HAS_SIZE {u::'A => bool. EX t<=s. u = t} ((2::nat) ^ n)"
by (import hollight HAS_SIZE_POWERSET)
-lemma CARD_POWERSET: "ALL s::'A::type => bool.
- FINITE s -->
- CARD
- (GSPEC
- (%u::'A::type => bool.
- EX t::'A::type => bool. SETSPEC u (SUBSET t s) t)) =
- EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) (CARD s)"
+lemma CARD_POWERSET: "finite (s::'A => bool)
+==> CARD {u::'A => bool. EX t<=s. u = t} = (2::nat) ^ CARD s"
by (import hollight CARD_POWERSET)
-lemma FINITE_POWERSET: "ALL s::'A::type => bool.
- FINITE s -->
- FINITE
- (GSPEC
- (%u::'A::type => bool.
- EX t::'A::type => bool. SETSPEC u (SUBSET t s) t))"
+lemma FINITE_POWERSET: "finite (s::'A => bool) ==> finite {u::'A => bool. EX t<=s. u = t}"
by (import hollight FINITE_POWERSET)
-lemma CARD_GE_REFL: "ALL s::'A::type => bool. CARD_GE s s"
- by (import hollight CARD_GE_REFL)
-
-lemma CARD_GE_TRANS: "ALL (s::'A::type => bool) (t::'B::type => bool) u::'C::type => bool.
- CARD_GE s t & CARD_GE t u --> CARD_GE s u"
- by (import hollight CARD_GE_TRANS)
-
-lemma FINITE_HAS_SIZE_LEMMA: "ALL s::'A::type => bool.
- FINITE s -->
- (EX n::nat. CARD_GE (GSPEC (%u::nat. EX x::nat. SETSPEC u (< x n) x)) s)"
- by (import hollight FINITE_HAS_SIZE_LEMMA)
-
-lemma HAS_SIZE_NUMSEG_LT: "ALL n::nat. HAS_SIZE (GSPEC (%u::nat. EX m::nat. SETSPEC u (< m n) m)) n"
+lemma FINITE_UNIONS: "finite (Union (s::('A => bool) => bool)) =
+(finite s & (ALL t::'A => bool. t : s --> finite t))"
+ by (import hollight FINITE_UNIONS)
+
+lemma POWERSET_CLAUSES: "{x::'q_54515 => bool. EX xa<={}. x = xa} = {{}} &
+(ALL (x::'A) xa::'A => bool.
+ {xb::'A => bool. EX xc<=insert x xa. xb = xc} =
+ {u::'A => bool. EX s<=xa. u = s} Un
+ insert x ` {u::'A => bool. EX s<=xa. u = s})"
+ by (import hollight POWERSET_CLAUSES)
+
+lemma HAS_SIZE_NUMSEG_LT: "HAS_SIZE {u. EX m<n. u = m} n"
by (import hollight HAS_SIZE_NUMSEG_LT)
-lemma CARD_NUMSEG_LT: "ALL x::nat. CARD (GSPEC (%u::nat. EX m::nat. SETSPEC u (< m x) m)) = x"
+lemma CARD_NUMSEG_LT: "CARD {u. EX m<x. u = m} = x"
by (import hollight CARD_NUMSEG_LT)
-lemma FINITE_NUMSEG_LT: "ALL x::nat. FINITE (GSPEC (%u::nat. EX m::nat. SETSPEC u (< m x) m))"
+lemma FINITE_NUMSEG_LT: "finite {u::nat. EX m<x::nat. u = m}"
by (import hollight FINITE_NUMSEG_LT)
-lemma HAS_SIZE_NUMSEG_LE: "ALL x::nat.
- HAS_SIZE (GSPEC (%xa::nat. EX xb::nat. SETSPEC xa (<= xb x) xb))
- (x + NUMERAL_BIT1 0)"
+lemma HAS_SIZE_NUMSEG_LE: "HAS_SIZE {xa. EX xb<=x. xa = xb} (x + 1)"
by (import hollight HAS_SIZE_NUMSEG_LE)
-lemma FINITE_NUMSEG_LE: "ALL x::nat. FINITE (GSPEC (%u::nat. EX m::nat. SETSPEC u (<= m x) m))"
+lemma FINITE_NUMSEG_LE: "finite {u::nat. EX m<=x::nat. u = m}"
by (import hollight FINITE_NUMSEG_LE)
-lemma CARD_NUMSEG_LE: "ALL x::nat.
- CARD (GSPEC (%u::nat. EX m::nat. SETSPEC u (<= m x) m)) =
- x + NUMERAL_BIT1 0"
+lemma CARD_NUMSEG_LE: "CARD {u. EX m<=x. u = m} = x + 1"
by (import hollight CARD_NUMSEG_LE)
-lemma num_FINITE: "ALL s::nat => bool. FINITE s = (EX a::nat. ALL x::nat. IN x s --> <= x a)"
+lemma num_FINITE: "finite (s::nat => bool) = (EX a::nat. ALL x::nat. x : s --> x <= a)"
by (import hollight num_FINITE)
-lemma num_FINITE_AVOID: "ALL s::nat => bool. FINITE s --> (EX a::nat. ~ IN a s)"
+lemma num_FINITE_AVOID: "finite (s::nat => bool) ==> EX a::nat. a ~: s"
by (import hollight num_FINITE_AVOID)
-lemma num_INFINITE: "(INFINITE::(nat => bool) => bool) (hollight.UNIV::nat => bool)"
- by (import hollight num_INFINITE)
-
-lemma HAS_SIZE_INDEX: "ALL (x::'A::type => bool) n::nat.
- HAS_SIZE x n -->
- (EX f::nat => 'A::type.
- (ALL m::nat. < m n --> IN (f m) x) &
- (ALL xa::'A::type. IN xa x --> (EX! m::nat. < m n & f m = xa)))"
+lemma FINITE_REAL_INTERVAL: "(ALL a. infinite {u. EX x. real_lt a x & u = x}) &
+(ALL a. infinite {u. EX x. real_le a x & u = x}) &
+(ALL b. infinite {u. EX x. real_lt x b & u = x}) &
+(ALL b. infinite {u. EX x. real_le x b & u = x}) &
+(ALL x xa.
+ finite {u. EX xb. (real_lt x xb & real_lt xb xa) & u = xb} =
+ real_le xa x) &
+(ALL a b.
+ finite {u. EX x. (real_le a x & real_lt x b) & u = x} = real_le b a) &
+(ALL a b.
+ finite {u. EX x. (real_lt a x & real_le x b) & u = x} = real_le b a) &
+(ALL a b.
+ finite {u. EX x. (real_le a x & real_le x b) & u = x} = real_le b a)"
+ by (import hollight FINITE_REAL_INTERVAL)
+
+lemma real_INFINITE: "(infinite::(hollight.real => bool) => bool) (UNIV::hollight.real => bool)"
+ by (import hollight real_INFINITE)
+
+lemma HAS_SIZE_INDEX: "HAS_SIZE (x::'A => bool) (n::nat)
+==> EX f::nat => 'A.
+ (ALL m<n. f m : x) &
+ (ALL xa::'A. xa : x --> (EX! m::nat. m < n & f m = xa))"
by (import hollight HAS_SIZE_INDEX)
-definition set_of_list :: "'q_45968 hollight.list => 'q_45968 => bool" where
- "set_of_list ==
-SOME set_of_list::'q_45968::type hollight.list => 'q_45968::type => bool.
- set_of_list NIL = EMPTY &
- (ALL (h::'q_45968::type) t::'q_45968::type hollight.list.
- set_of_list (CONS h t) = INSERT h (set_of_list t))"
-
-lemma DEF_set_of_list: "set_of_list =
-(SOME set_of_list::'q_45968::type hollight.list => 'q_45968::type => bool.
- set_of_list NIL = EMPTY &
- (ALL (h::'q_45968::type) t::'q_45968::type hollight.list.
- set_of_list (CONS h t) = INSERT h (set_of_list t)))"
- by (import hollight DEF_set_of_list)
-
-definition list_of_set :: "('q_45986 => bool) => 'q_45986 hollight.list" where
- "list_of_set ==
-%u::'q_45986::type => bool.
- SOME l::'q_45986::type hollight.list.
- set_of_list l = u & LENGTH l = CARD u"
-
-lemma DEF_list_of_set: "list_of_set =
-(%u::'q_45986::type => bool.
- SOME l::'q_45986::type hollight.list.
- set_of_list l = u & LENGTH l = CARD u)"
- by (import hollight DEF_list_of_set)
-
-lemma LIST_OF_SET_PROPERTIES: "ALL x::'A::type => bool.
- FINITE x -->
- set_of_list (list_of_set x) = x & LENGTH (list_of_set x) = CARD x"
- by (import hollight LIST_OF_SET_PROPERTIES)
-
-lemma SET_OF_LIST_OF_SET: "ALL s::'q_46035::type => bool. FINITE s --> set_of_list (list_of_set s) = s"
- by (import hollight SET_OF_LIST_OF_SET)
-
-lemma LENGTH_LIST_OF_SET: "ALL s::'q_46051::type => bool. FINITE s --> LENGTH (list_of_set s) = CARD s"
- by (import hollight LENGTH_LIST_OF_SET)
-
-lemma MEM_LIST_OF_SET: "ALL s::'A::type => bool.
- FINITE s --> (ALL x::'A::type. MEM x (list_of_set s) = IN x s)"
- by (import hollight MEM_LIST_OF_SET)
-
-lemma FINITE_SET_OF_LIST: "ALL l::'q_46096::type hollight.list. FINITE (set_of_list l)"
- by (import hollight FINITE_SET_OF_LIST)
-
-lemma IN_SET_OF_LIST: "ALL (x::'q_46114::type) l::'q_46114::type hollight.list.
- IN x (set_of_list l) = MEM x l"
- by (import hollight IN_SET_OF_LIST)
-
-lemma SET_OF_LIST_APPEND: "ALL (x::'q_46139::type hollight.list) xa::'q_46139::type hollight.list.
- set_of_list (APPEND x xa) =
- hollight.UNION (set_of_list x) (set_of_list xa)"
- by (import hollight SET_OF_LIST_APPEND)
-
-definition pairwise :: "('q_46198 => 'q_46198 => bool) => ('q_46198 => bool) => bool" where
+definition
+ pairwise :: "('q_55938 => 'q_55938 => bool) => ('q_55938 => bool) => bool" where
"pairwise ==
-%(u::'q_46198::type => 'q_46198::type => bool) ua::'q_46198::type => bool.
- ALL (x::'q_46198::type) y::'q_46198::type.
- IN x ua & IN y ua & x ~= y --> u x y"
+%(u::'q_55938 => 'q_55938 => bool) ua::'q_55938 => bool.
+ ALL (x::'q_55938) y::'q_55938. x : ua & y : ua & x ~= y --> u x y"
lemma DEF_pairwise: "pairwise =
-(%(u::'q_46198::type => 'q_46198::type => bool) ua::'q_46198::type => bool.
- ALL (x::'q_46198::type) y::'q_46198::type.
- IN x ua & IN y ua & x ~= y --> u x y)"
+(%(u::'q_55938 => 'q_55938 => bool) ua::'q_55938 => bool.
+ ALL (x::'q_55938) y::'q_55938. x : ua & y : ua & x ~= y --> u x y)"
by (import hollight DEF_pairwise)
-definition PAIRWISE :: "('q_46220 => 'q_46220 => bool) => 'q_46220 hollight.list => bool" where
+definition
+ PAIRWISE :: "('A => 'A => bool) => 'A list => bool" where
"PAIRWISE ==
-SOME PAIRWISE::('q_46220::type => 'q_46220::type => bool)
- => 'q_46220::type hollight.list => bool.
- (ALL r::'q_46220::type => 'q_46220::type => bool.
- PAIRWISE r NIL = True) &
- (ALL (h::'q_46220::type) (r::'q_46220::type => 'q_46220::type => bool)
- t::'q_46220::type hollight.list.
- PAIRWISE r (CONS h t) = (ALL_list (r h) t & PAIRWISE r t))"
+SOME PAIRWISE::('A => 'A => bool) => 'A list => bool.
+ (ALL r::'A => 'A => bool. PAIRWISE r [] = True) &
+ (ALL (h::'A) (r::'A => 'A => bool) t::'A list.
+ PAIRWISE r (h # t) = (list_all (r h) t & PAIRWISE r t))"
lemma DEF_PAIRWISE: "PAIRWISE =
-(SOME PAIRWISE::('q_46220::type => 'q_46220::type => bool)
- => 'q_46220::type hollight.list => bool.
- (ALL r::'q_46220::type => 'q_46220::type => bool.
- PAIRWISE r NIL = True) &
- (ALL (h::'q_46220::type) (r::'q_46220::type => 'q_46220::type => bool)
- t::'q_46220::type hollight.list.
- PAIRWISE r (CONS h t) = (ALL_list (r h) t & PAIRWISE r t)))"
+(SOME PAIRWISE::('A => 'A => bool) => 'A list => bool.
+ (ALL r::'A => 'A => bool. PAIRWISE r [] = True) &
+ (ALL (h::'A) (r::'A => 'A => bool) t::'A list.
+ PAIRWISE r (h # t) = (list_all (r h) t & PAIRWISE r t)))"
by (import hollight DEF_PAIRWISE)
-typedef (open) ('A) finite_image = "(Collect::('A::type => bool) => 'A::type set)
- (%x::'A::type.
- (op |::bool => bool => bool)
- ((op =::'A::type => 'A::type => bool) x
- ((Eps::('A::type => bool) => 'A::type) (%z::'A::type. True::bool)))
- ((FINITE::('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool)))" morphisms "dest_finite_image" "mk_finite_image"
- apply (rule light_ex_imp_nonempty[where t="(Eps::('A::type => bool) => 'A::type)
- (%x::'A::type.
- (op |::bool => bool => bool)
- ((op =::'A::type => 'A::type => bool) x
- ((Eps::('A::type => bool) => 'A::type) (%z::'A::type. True::bool)))
- ((FINITE::('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool)))"])
+lemma PAIRWISE_EMPTY: "pairwise (x::'q_55973 => 'q_55973 => bool) {} = True"
+ by (import hollight PAIRWISE_EMPTY)
+
+lemma PAIRWISE_SING: "pairwise (x::'q_55991 => 'q_55991 => bool) {xa::'q_55991} = True"
+ by (import hollight PAIRWISE_SING)
+
+lemma PAIRWISE_MONO: "pairwise (x::'q_56011 => 'q_56011 => bool) (xa::'q_56011 => bool) &
+(xb::'q_56011 => bool) <= xa
+==> pairwise x xb"
+ by (import hollight PAIRWISE_MONO)
+
+lemma SURJECTIVE_IFF_INJECTIVE_GEN: "finite (s::'A => bool) &
+finite (t::'B => bool) & CARD s = CARD t & (f::'A => 'B) ` s <= t
+==> (ALL y::'B. y : t --> (EX x::'A. x : s & f x = y)) =
+ (ALL (x::'A) y::'A. x : s & y : s & f x = f y --> x = y)"
+ by (import hollight SURJECTIVE_IFF_INJECTIVE_GEN)
+
+lemma SURJECTIVE_IFF_INJECTIVE: "finite (x::'A => bool) & (xa::'A => 'A) ` x <= x
+==> (ALL y::'A. y : x --> (EX xb::'A. xb : x & xa xb = y)) =
+ (ALL (xb::'A) y::'A. xb : x & y : x & xa xb = xa y --> xb = y)"
+ by (import hollight SURJECTIVE_IFF_INJECTIVE)
+
+lemma IMAGE_IMP_INJECTIVE_GEN: "[| finite (s::'A => bool) &
+ CARD s = CARD (t::'B => bool) & (f::'A => 'B) ` s = t;
+ (x::'A) : s & (y::'A) : s & f x = f y |]
+==> x = y"
+ by (import hollight IMAGE_IMP_INJECTIVE_GEN)
+
+lemma IMAGE_IMP_INJECTIVE: "[| finite (s::'q_56387 => bool) & (f::'q_56387 => 'q_56387) ` s = s;
+ (x::'q_56387) : s & (y::'q_56387) : s & f x = f y |]
+==> x = y"
+ by (import hollight IMAGE_IMP_INJECTIVE)
+
+lemma CARD_LE_INJ: "finite (x::'A => bool) & finite (xa::'B => bool) & CARD x <= CARD xa
+==> EX f::'A => 'B.
+ f ` x <= xa &
+ (ALL (xa::'A) y::'A. xa : x & y : x & f xa = f y --> xa = y)"
+ by (import hollight CARD_LE_INJ)
+
+lemma FORALL_IN_CLAUSES: "(ALL x::'q_56493 => bool. (ALL xa::'q_56493. xa : {} --> x xa) = True) &
+(ALL (x::'q_56533 => bool) (xa::'q_56533) xb::'q_56533 => bool.
+ (ALL xc::'q_56533. xc : insert xa xb --> x xc) =
+ (x xa & (ALL xa::'q_56533. xa : xb --> x xa)))"
+ by (import hollight FORALL_IN_CLAUSES)
+
+lemma EXISTS_IN_CLAUSES: "(ALL x::'q_56553 => bool. (EX xa::'q_56553. xa : {} & x xa) = False) &
+(ALL (x::'q_56593 => bool) (xa::'q_56593) xb::'q_56593 => bool.
+ (EX xc::'q_56593. xc : insert xa xb & x xc) =
+ (x xa | (EX xa::'q_56593. xa : xb & x xa)))"
+ by (import hollight EXISTS_IN_CLAUSES)
+
+lemma SURJECTIVE_ON_RIGHT_INVERSE: "(ALL xb::'q_56650.
+ xb : (xa::'q_56650 => bool) -->
+ (EX xa::'q_56649.
+ xa : (s::'q_56649 => bool) & (x::'q_56649 => 'q_56650) xa = xb)) =
+(EX g::'q_56650 => 'q_56649.
+ ALL y::'q_56650. y : xa --> g y : s & x (g y) = y)"
+ by (import hollight SURJECTIVE_ON_RIGHT_INVERSE)
+
+lemma INJECTIVE_ON_LEFT_INVERSE: "(ALL (xb::'q_56743) y::'q_56743.
+ xb : (xa::'q_56743 => bool) &
+ y : xa & (x::'q_56743 => 'q_56746) xb = x y -->
+ xb = y) =
+(EX xb::'q_56746 => 'q_56743. ALL xc::'q_56743. xc : xa --> xb (x xc) = xc)"
+ by (import hollight INJECTIVE_ON_LEFT_INVERSE)
+
+lemma BIJECTIVE_ON_LEFT_RIGHT_INVERSE: "(!!x::'q_56878.
+ x : (s::'q_56878 => bool)
+ ==> (f::'q_56878 => 'q_56877) x : (t::'q_56877 => bool))
+==> ((ALL (x::'q_56878) y::'q_56878. x : s & y : s & f x = f y --> x = y) &
+ (ALL x::'q_56877. x : t --> (EX xa::'q_56878. xa : s & f xa = x))) =
+ (EX g::'q_56877 => 'q_56878.
+ (ALL y::'q_56877. y : t --> g y : s) &
+ (ALL y::'q_56877. y : t --> f (g y) = y) &
+ (ALL x::'q_56878. x : s --> g (f x) = x))"
+ by (import hollight BIJECTIVE_ON_LEFT_RIGHT_INVERSE)
+
+lemma SURJECTIVE_RIGHT_INVERSE: "(ALL y::'q_56902. EX x::'q_56905. (f::'q_56905 => 'q_56902) x = y) =
+(EX g::'q_56902 => 'q_56905. ALL y::'q_56902. f (g y) = y)"
+ by (import hollight SURJECTIVE_RIGHT_INVERSE)
+
+lemma INJECTIVE_LEFT_INVERSE: "(ALL (x::'q_56939) xa::'q_56939.
+ (f::'q_56939 => 'q_56942) x = f xa --> x = xa) =
+(EX g::'q_56942 => 'q_56939. ALL x::'q_56939. g (f x) = x)"
+ by (import hollight INJECTIVE_LEFT_INVERSE)
+
+lemma BIJECTIVE_LEFT_RIGHT_INVERSE: "((ALL (x::'A) y::'A. (f::'A => 'B) x = f y --> x = y) &
+ (ALL y::'B. EX x::'A. f x = y)) =
+(EX g::'B => 'A. (ALL y::'B. f (g y) = y) & (ALL x::'A. g (f x) = x))"
+ by (import hollight BIJECTIVE_LEFT_RIGHT_INVERSE)
+
+lemma FUNCTION_FACTORS_RIGHT: "(ALL xb::'q_57046.
+ EX y::'q_57034.
+ (xa::'q_57034 => 'q_57047) y = (x::'q_57046 => 'q_57047) xb) =
+(EX xb::'q_57046 => 'q_57034. x = xa o xb)"
+ by (import hollight FUNCTION_FACTORS_RIGHT)
+
+lemma FUNCTION_FACTORS_LEFT: "(ALL (xb::'q_57119) y::'q_57119.
+ (xa::'q_57119 => 'q_57099) xb = xa y -->
+ (x::'q_57119 => 'q_57120) xb = x y) =
+(EX xb::'q_57099 => 'q_57120. x = xb o xa)"
+ by (import hollight FUNCTION_FACTORS_LEFT)
+
+lemma SURJECTIVE_FORALL_THM: "(ALL y::'B. EX x::'A. (f::'A => 'B) x = y) =
+(ALL P::'B => bool. (ALL x::'A. P (f x)) = All P)"
+ by (import hollight SURJECTIVE_FORALL_THM)
+
+lemma SURJECTIVE_EXISTS_THM: "(ALL y::'B. EX x::'A. (f::'A => 'B) x = y) =
+(ALL P::'B => bool. (EX x::'A. P (f x)) = Ex P)"
+ by (import hollight SURJECTIVE_EXISTS_THM)
+
+lemma SURJECTIVE_IMAGE_THM: "(ALL y::'B. EX x::'A. (f::'A => 'B) x = y) =
+(ALL x::'B => bool.
+ f ` {u::'A. EX xa::'A. x (f xa) & u = xa} =
+ {u::'B. EX xa::'B. x xa & u = xa})"
+ by (import hollight SURJECTIVE_IMAGE_THM)
+
+lemma IMAGE_INJECTIVE_IMAGE_OF_SUBSET: "EX x<=s::'A => bool.
+ (f::'A => 'B) ` s = f ` x &
+ (ALL (xa::'A) y::'A. xa : x & y : x & f xa = f y --> xa = y)"
+ by (import hollight IMAGE_INJECTIVE_IMAGE_OF_SUBSET)
+
+lemma INJECTIVE_ON_IMAGE: "(ALL (s::'A => bool) t::'A => bool.
+ s <= (u::'A => bool) & t <= u & (f::'A => 'B) ` s = f ` t --> s = t) =
+(ALL (x::'A) y::'A. x : u & y : u & f x = f y --> x = y)"
+ by (import hollight INJECTIVE_ON_IMAGE)
+
+lemma INJECTIVE_IMAGE: "(ALL (s::'A => bool) t::'A => bool. (f::'A => 'B) ` s = f ` t --> s = t) =
+(ALL (x::'A) y::'A. f x = f y --> x = y)"
+ by (import hollight INJECTIVE_IMAGE)
+
+lemma SURJECTIVE_ON_IMAGE: "(ALL t<=v::'B => bool. EX s<=u::'A => bool. (f::'A => 'B) ` s = t) =
+(ALL y::'B. y : v --> (EX x::'A. x : u & f x = y))"
+ by (import hollight SURJECTIVE_ON_IMAGE)
+
+lemma SURJECTIVE_IMAGE: "(ALL t::'B => bool. EX s::'A => bool. (f::'A => 'B) ` s = t) =
+(ALL y::'B. EX x::'A. f x = y)"
+ by (import hollight SURJECTIVE_IMAGE)
+
+lemma CARD_EQ_BIJECTION: "finite (s::'A => bool) & finite (t::'B => bool) & CARD s = CARD t
+==> EX f::'A => 'B.
+ (ALL x::'A. x : s --> f x : t) &
+ (ALL y::'B. y : t --> (EX x::'A. x : s & f x = y)) &
+ (ALL (x::'A) y::'A. x : s & y : s & f x = f y --> x = y)"
+ by (import hollight CARD_EQ_BIJECTION)
+
+lemma CARD_EQ_BIJECTIONS: "finite (s::'A => bool) & finite (t::'B => bool) & CARD s = CARD t
+==> EX (f::'A => 'B) g::'B => 'A.
+ (ALL x::'A. x : s --> f x : t & g (f x) = x) &
+ (ALL y::'B. y : t --> g y : s & f (g y) = y)"
+ by (import hollight CARD_EQ_BIJECTIONS)
+
+lemma BIJECTIONS_HAS_SIZE: "(ALL x::'A.
+ x : (s::'A => bool) -->
+ (f::'A => 'B) x : (t::'B => bool) & (g::'B => 'A) (f x) = x) &
+(ALL y::'B. y : t --> g y : s & f (g y) = y) & HAS_SIZE s (n::nat)
+==> HAS_SIZE t n"
+ by (import hollight BIJECTIONS_HAS_SIZE)
+
+lemma BIJECTIONS_HAS_SIZE_EQ: "(ALL x::'A.
+ x : (s::'A => bool) -->
+ (f::'A => 'B) x : (t::'B => bool) & (g::'B => 'A) (f x) = x) &
+(ALL y::'B. y : t --> g y : s & f (g y) = y)
+==> HAS_SIZE s (n::nat) = HAS_SIZE t n"
+ by (import hollight BIJECTIONS_HAS_SIZE_EQ)
+
+lemma BIJECTIONS_CARD_EQ: "(finite (s::'A => bool) | finite (t::'B => bool)) &
+(ALL x::'A. x : s --> (f::'A => 'B) x : t & (g::'B => 'A) (f x) = x) &
+(ALL y::'B. y : t --> g y : s & f (g y) = y)
+==> CARD s = CARD t"
+ by (import hollight BIJECTIONS_CARD_EQ)
+
+lemma WF_FINITE: "(ALL x::'A. ~ (u_556::'A => 'A => bool) x x) &
+(ALL (x::'A) (y::'A) z::'A. u_556 x y & u_556 y z --> u_556 x z) &
+(ALL x::'A. finite {u::'A. EX y::'A. u_556 y x & u = y})
+==> wfP u_556"
+ by (import hollight WF_FINITE)
+
+consts
+ "<=_c" :: "('q_58200 => bool) => ('q_58195 => bool) => bool" ("<='_c")
+
+defs
+ "<=_c_def": "<=_c ==
+%(u::'q_58200 => bool) ua::'q_58195 => bool.
+ EX f::'q_58200 => 'q_58195.
+ (ALL x::'q_58200. x : u --> f x : ua) &
+ (ALL (x::'q_58200) y::'q_58200. x : u & y : u & f x = f y --> x = y)"
+
+lemma DEF__lessthan__equal__c: "<=_c =
+(%(u::'q_58200 => bool) ua::'q_58195 => bool.
+ EX f::'q_58200 => 'q_58195.
+ (ALL x::'q_58200. x : u --> f x : ua) &
+ (ALL (x::'q_58200) y::'q_58200. x : u & y : u & f x = f y --> x = y))"
+ by (import hollight DEF__lessthan__equal__c)
+
+consts
+ "<_c" :: "('q_58212 => bool) => ('q_58213 => bool) => bool" ("<'_c")
+
+defs
+ "<_c_def": "<_c == %(u::'q_58212 => bool) ua::'q_58213 => bool. <=_c u ua & ~ <=_c ua u"
+
+lemma DEF__lessthan__c: "<_c = (%(u::'q_58212 => bool) ua::'q_58213 => bool. <=_c u ua & ~ <=_c ua u)"
+ by (import hollight DEF__lessthan__c)
+
+consts
+ "=_c" :: "('q_58264 => bool) => ('q_58261 => bool) => bool" ("='_c")
+
+defs
+ "=_c_def": "=_c ==
+%(u::'q_58264 => bool) ua::'q_58261 => bool.
+ EX f::'q_58264 => 'q_58261.
+ (ALL x::'q_58264. x : u --> f x : ua) &
+ (ALL y::'q_58261. y : ua --> (EX! x::'q_58264. x : u & f x = y))"
+
+lemma DEF__equal__c: "=_c =
+(%(u::'q_58264 => bool) ua::'q_58261 => bool.
+ EX f::'q_58264 => 'q_58261.
+ (ALL x::'q_58264. x : u --> f x : ua) &
+ (ALL y::'q_58261. y : ua --> (EX! x::'q_58264. x : u & f x = y)))"
+ by (import hollight DEF__equal__c)
+
+consts
+ ">=_c" :: "('q_58273 => bool) => ('q_58272 => bool) => bool" (">='_c")
+
+defs
+ ">=_c_def": ">=_c == %(u::'q_58273 => bool) ua::'q_58272 => bool. <=_c ua u"
+
+lemma DEF__greaterthan__equal__c: ">=_c = (%(u::'q_58273 => bool) ua::'q_58272 => bool. <=_c ua u)"
+ by (import hollight DEF__greaterthan__equal__c)
+
+consts
+ ">_c" :: "('q_58282 => bool) => ('q_58281 => bool) => bool" (">'_c")
+
+defs
+ ">_c_def": ">_c == %(u::'q_58282 => bool) ua::'q_58281 => bool. <_c ua u"
+
+lemma DEF__greaterthan__c: ">_c = (%(u::'q_58282 => bool) ua::'q_58281 => bool. <_c ua u)"
+ by (import hollight DEF__greaterthan__c)
+
+lemma LE_C: "<=_c (x::'q_58320 => bool) (xa::'q_58323 => bool) =
+(EX xb::'q_58323 => 'q_58320.
+ ALL xc::'q_58320. xc : x --> (EX x::'q_58323. x : xa & xb x = xc))"
+ by (import hollight LE_C)
+
+lemma GE_C: ">=_c (x::'q_58364 => bool) (xa::'q_58361 => bool) =
+(EX f::'q_58364 => 'q_58361.
+ ALL y::'q_58361. y : xa --> (EX xa::'q_58364. xa : x & y = f xa))"
+ by (import hollight GE_C)
+
+definition
+ COUNTABLE :: "('q_58372 => bool) => bool" where
+ "(op ==::(('q_58372::type => bool) => bool)
+ => (('q_58372::type => bool) => bool) => prop)
+ (COUNTABLE::('q_58372::type => bool) => bool)
+ ((>=_c::(nat => bool) => ('q_58372::type => bool) => bool)
+ (UNIV::nat => bool))"
+
+lemma DEF_COUNTABLE: "(op =::(('q_58372::type => bool) => bool)
+ => (('q_58372::type => bool) => bool) => bool)
+ (COUNTABLE::('q_58372::type => bool) => bool)
+ ((>=_c::(nat => bool) => ('q_58372::type => bool) => bool)
+ (UNIV::nat => bool))"
+ by (import hollight DEF_COUNTABLE)
+
+lemma NUMSEG_COMBINE_R: "(x::nat) <= (xa::nat) + (1::nat) & xa <= (xb::nat)
+==> {x..xa} Un {xa + (1::nat)..xb} = {x..xb}"
+ by (import hollight NUMSEG_COMBINE_R)
+
+lemma NUMSEG_COMBINE_L: "(x::nat) <= (xa::nat) & xa <= (xb::nat) + (1::nat)
+==> {x..xa - (1::nat)} Un {xa..xb} = {x..xb}"
+ by (import hollight NUMSEG_COMBINE_L)
+
+lemma NUMSEG_LREC: "(x::nat) <= (xa::nat) ==> insert x {x + (1::nat)..xa} = {x..xa}"
+ by (import hollight NUMSEG_LREC)
+
+lemma NUMSEG_RREC: "(x::nat) <= (xa::nat) ==> insert xa {x..xa - (1::nat)} = {x..xa}"
+ by (import hollight NUMSEG_RREC)
+
+lemma IN_NUMSEG_0: "((x::nat) : {0::nat..xa::nat}) = (x <= xa)"
+ by (import hollight IN_NUMSEG_0)
+
+lemma NUMSEG_EMPTY: "({x::nat..xa::nat} = {}) = (xa < x)"
+ by (import hollight NUMSEG_EMPTY)
+
+lemma CARD_NUMSEG_LEMMA: "CARD {m..m + d} = d + 1"
+ by (import hollight CARD_NUMSEG_LEMMA)
+
+lemma CARD_NUMSEG: "CARD {m..n} = n + 1 - m"
+ by (import hollight CARD_NUMSEG)
+
+lemma HAS_SIZE_NUMSEG: "HAS_SIZE {x..xa} (xa + 1 - x)"
+ by (import hollight HAS_SIZE_NUMSEG)
+
+lemma CARD_NUMSEG_1: "CARD {1..x} = x"
+ by (import hollight CARD_NUMSEG_1)
+
+lemma HAS_SIZE_NUMSEG_1: "HAS_SIZE {1..x} x"
+ by (import hollight HAS_SIZE_NUMSEG_1)
+
+lemma NUMSEG_CLAUSES: "(ALL m::nat. {m..0::nat} = (if m = (0::nat) then {0::nat} else {})) &
+(ALL (m::nat) n::nat.
+ {m..Suc n} = (if m <= Suc n then insert (Suc n) {m..n} else {m..n}))"
+ by (import hollight NUMSEG_CLAUSES)
+
+lemma FINITE_INDEX_NUMSEG: "finite (s::'A => bool) =
+(EX f::nat => 'A.
+ (ALL (i::nat) j::nat.
+ i : {1::nat..CARD s} & j : {1::nat..CARD s} & f i = f j --> i = j) &
+ s = f ` {1::nat..CARD s})"
+ by (import hollight FINITE_INDEX_NUMSEG)
+
+lemma FINITE_INDEX_NUMBERS: "finite (s::'A => bool) =
+(EX (k::nat => bool) f::nat => 'A.
+ (ALL (i::nat) j::nat. i : k & j : k & f i = f j --> i = j) &
+ finite k & s = f ` k)"
+ by (import hollight FINITE_INDEX_NUMBERS)
+
+lemma DISJOINT_NUMSEG: "({x::nat..xa::nat} Int {xb::nat..xc::nat} = {}) =
+(xa < xb | xc < x | xa < x | xc < xb)"
+ by (import hollight DISJOINT_NUMSEG)
+
+lemma NUMSEG_ADD_SPLIT: "(x::nat) <= (xa::nat) + (1::nat)
+==> {x..xa + (xb::nat)} = {x..xa} Un {xa + (1::nat)..xa + xb}"
+ by (import hollight NUMSEG_ADD_SPLIT)
+
+lemma SUBSET_NUMSEG: "({m::nat..n::nat} <= {p::nat..q::nat}) = (n < m | p <= m & n <= q)"
+ by (import hollight SUBSET_NUMSEG)
+
+lemma NUMSEG_LE: "{u::nat. EX xa<=x::nat. u = xa} = {0::nat..x}"
+ by (import hollight NUMSEG_LE)
+
+lemma NUMSEG_LT: "{u::nat. EX x<n::nat. u = x} =
+(if n = (0::nat) then {} else {0::nat..n - (1::nat)})"
+ by (import hollight NUMSEG_LT)
+
+lemma TOPOLOGICAL_SORT: "[| (ALL (x::'A) y::'A.
+ (u_556::'A => 'A => bool) x y & u_556 y x --> x = y) &
+ (ALL (x::'A) (y::'A) z::'A. u_556 x y & u_556 y z --> u_556 x z);
+ HAS_SIZE (s::'A => bool) (n::nat) |]
+==> EX f::nat => 'A.
+ s = f ` {1::nat..n} &
+ (ALL (j::nat) k::nat.
+ j : {1::nat..n} & k : {1::nat..n} & j < k -->
+ ~ u_556 (f k) (f j))"
+ by (import hollight TOPOLOGICAL_SORT)
+
+lemma FINITE_INTSEG: "(ALL l r. finite {u. EX x. (int_le l x & int_le x r) & u = x}) &
+(ALL l r. finite {u. EX x. (int_le l x & int_lt x r) & u = x}) &
+(ALL l r. finite {u. EX x. (int_lt l x & int_le x r) & u = x}) &
+(ALL l r. finite {u. EX x. (int_lt l x & int_lt x r) & u = x})"
+ by (import hollight FINITE_INTSEG)
+
+definition
+ neutral :: "('q_59899 => 'q_59899 => 'q_59899) => 'q_59899" where
+ "neutral ==
+%u::'q_59899 => 'q_59899 => 'q_59899.
+ SOME x::'q_59899. ALL y::'q_59899. u x y = y & u y x = y"
+
+lemma DEF_neutral: "neutral =
+(%u::'q_59899 => 'q_59899 => 'q_59899.
+ SOME x::'q_59899. ALL y::'q_59899. u x y = y & u y x = y)"
+ by (import hollight DEF_neutral)
+
+definition
+ monoidal :: "('A => 'A => 'A) => bool" where
+ "monoidal ==
+%u::'A => 'A => 'A.
+ (ALL (x::'A) y::'A. u x y = u y x) &
+ (ALL (x::'A) (y::'A) z::'A. u x (u y z) = u (u x y) z) &
+ (ALL x::'A. u (neutral u) x = x)"
+
+lemma DEF_monoidal: "monoidal =
+(%u::'A => 'A => 'A.
+ (ALL (x::'A) y::'A. u x y = u y x) &
+ (ALL (x::'A) (y::'A) z::'A. u x (u y z) = u (u x y) z) &
+ (ALL x::'A. u (neutral u) x = x))"
+ by (import hollight DEF_monoidal)
+
+lemma MONOIDAL_AC: "monoidal (x::'q_60055 => 'q_60055 => 'q_60055)
+==> (ALL a::'q_60055. x (neutral x) a = a) &
+ (ALL a::'q_60055. x a (neutral x) = a) &
+ (ALL (a::'q_60055) b::'q_60055. x a b = x b a) &
+ (ALL (a::'q_60055) (b::'q_60055) c::'q_60055.
+ x (x a b) c = x a (x b c)) &
+ (ALL (a::'q_60055) (b::'q_60055) c::'q_60055. x a (x b c) = x b (x a c))"
+ by (import hollight MONOIDAL_AC)
+
+definition
+ support :: "('B => 'B => 'B) => ('A => 'B) => ('A => bool) => 'A => bool" where
+ "support ==
+%(u::'B => 'B => 'B) (ua::'A => 'B) ub::'A => bool.
+ {uc::'A. EX x::'A. (x : ub & ua x ~= neutral u) & uc = x}"
+
+lemma DEF_support: "support =
+(%(u::'B => 'B => 'B) (ua::'A => 'B) ub::'A => bool.
+ {uc::'A. EX x::'A. (x : ub & ua x ~= neutral u) & uc = x})"
+ by (import hollight DEF_support)
+
+definition
+ iterate :: "('q_60113 => 'q_60113 => 'q_60113)
+=> ('A => bool) => ('A => 'q_60113) => 'q_60113" where
+ "iterate ==
+%(u::'q_60113 => 'q_60113 => 'q_60113) (ua::'A => bool) ub::'A => 'q_60113.
+ if finite (support u ub ua)
+ then ITSET (%x::'A. u (ub x)) (support u ub ua) (neutral u)
+ else neutral u"
+
+lemma DEF_iterate: "iterate =
+(%(u::'q_60113 => 'q_60113 => 'q_60113) (ua::'A => bool) ub::'A => 'q_60113.
+ if finite (support u ub ua)
+ then ITSET (%x::'A. u (ub x)) (support u ub ua) (neutral u)
+ else neutral u)"
+ by (import hollight DEF_iterate)
+
+lemma IN_SUPPORT: "((xb::'q_60163)
+ : support (x::'q_60160 => 'q_60160 => 'q_60160) (xa::'q_60163 => 'q_60160)
+ (xc::'q_60163 => bool)) =
+(xb : xc & xa xb ~= neutral x)"
+ by (import hollight IN_SUPPORT)
+
+lemma SUPPORT_SUPPORT: "support (x::'q_60185 => 'q_60185 => 'q_60185) (xa::'q_60196 => 'q_60185)
+ (support x xa (xb::'q_60196 => bool)) =
+support x xa xb"
+ by (import hollight SUPPORT_SUPPORT)
+
+lemma SUPPORT_EMPTY: "(ALL xc::'q_60235.
+ xc : (xb::'q_60235 => bool) -->
+ (xa::'q_60235 => 'q_60221) xc =
+ neutral (x::'q_60221 => 'q_60221 => 'q_60221)) =
+(support x xa xb = {})"
+ by (import hollight SUPPORT_EMPTY)
+
+lemma SUPPORT_SUBSET: "support (x::'q_60255 => 'q_60255 => 'q_60255) (xa::'q_60256 => 'q_60255)
+ (xb::'q_60256 => bool)
+<= xb"
+ by (import hollight SUPPORT_SUBSET)
+
+lemma FINITE_SUPPORT: "finite (s::'q_60273 => bool)
+==> finite
+ (support (u::'q_60279 => 'q_60279 => 'q_60279)
+ (f::'q_60273 => 'q_60279) s)"
+ by (import hollight FINITE_SUPPORT)
+
+lemma SUPPORT_CLAUSES: "(ALL x::'q_60297 => 'q_60530.
+ support (u_4372::'q_60530 => 'q_60530 => 'q_60530) x {} = {}) &
+(ALL (x::'q_60345 => 'q_60530) (xa::'q_60345) xb::'q_60345 => bool.
+ support u_4372 x (insert xa xb) =
+ (if x xa = neutral u_4372 then support u_4372 x xb
+ else insert xa (support u_4372 x xb))) &
+(ALL (x::'q_60378 => 'q_60530) (xa::'q_60378) xb::'q_60378 => bool.
+ support u_4372 x (xb - {xa}) = support u_4372 x xb - {xa}) &
+(ALL (x::'q_60416 => 'q_60530) (xa::'q_60416 => bool) xb::'q_60416 => bool.
+ support u_4372 x (xa Un xb) =
+ support u_4372 x xa Un support u_4372 x xb) &
+(ALL (x::'q_60454 => 'q_60530) (xa::'q_60454 => bool) xb::'q_60454 => bool.
+ support u_4372 x (xa Int xb) =
+ support u_4372 x xa Int support u_4372 x xb) &
+(ALL (x::'q_60492 => 'q_60530) (xa::'q_60492 => bool) xb::'q_60492 => bool.
+ support u_4372 x (xa - xb) =
+ support u_4372 x xa - support u_4372 x xb) &
+(ALL (x::'q_60529 => 'q_60520) (xa::'q_60520 => 'q_60530)
+ xb::'q_60529 => bool.
+ support u_4372 xa (x ` xb) = x ` support u_4372 (xa o x) xb)"
+ by (import hollight SUPPORT_CLAUSES)
+
+lemma SUPPORT_DELTA: "support (x::'q_60556 => 'q_60556 => 'q_60556)
+ (%xa::'q_60584.
+ if xa = (xc::'q_60584) then (xb::'q_60584 => 'q_60556) xa
+ else neutral x)
+ (xa::'q_60584 => bool) =
+(if xc : xa then support x xb {xc} else {})"
+ by (import hollight SUPPORT_DELTA)
+
+lemma FINITE_SUPPORT_DELTA: "finite
+ (support (x::'q_60605 => 'q_60605 => 'q_60605)
+ (%xc::'q_60614.
+ if xc = (xb::'q_60614) then (xa::'q_60614 => 'q_60605) xc
+ else neutral x)
+ (s::'q_60614 => bool))"
+ by (import hollight FINITE_SUPPORT_DELTA)
+
+lemma ITERATE_SUPPORT: "iterate (x::'q_60630 => 'q_60630 => 'q_60630)
+ (support x (xa::'q_60642 => 'q_60630) (xb::'q_60642 => bool)) xa =
+iterate x xb xa"
+ by (import hollight ITERATE_SUPPORT)
+
+lemma ITERATE_EXPAND_CASES: "iterate (x::'q_60661 => 'q_60661 => 'q_60661) (xb::'q_60667 => bool)
+ (xa::'q_60667 => 'q_60661) =
+(if finite (support x xa xb) then iterate x (support x xa xb) xa
+ else neutral x)"
+ by (import hollight ITERATE_EXPAND_CASES)
+
+lemma ITERATE_CLAUSES_GEN: "monoidal (u_4372::'B => 'B => 'B)
+==> (ALL f::'A => 'B. iterate u_4372 {} f = neutral u_4372) &
+ (ALL (f::'A => 'B) (x::'A) s::'A => bool.
+ monoidal u_4372 & finite (support u_4372 f s) -->
+ iterate u_4372 (insert x s) f =
+ (if x : s then iterate u_4372 s f
+ else u_4372 (f x) (iterate u_4372 s f)))"
+ by (import hollight ITERATE_CLAUSES_GEN)
+
+lemma ITERATE_CLAUSES: "monoidal (x::'q_60857 => 'q_60857 => 'q_60857)
+==> (ALL f::'q_60815 => 'q_60857. iterate x {} f = neutral x) &
+ (ALL (f::'q_60859 => 'q_60857) (xa::'q_60859) s::'q_60859 => bool.
+ finite s -->
+ iterate x (insert xa s) f =
+ (if xa : s then iterate x s f else x (f xa) (iterate x s f)))"
+ by (import hollight ITERATE_CLAUSES)
+
+lemma ITERATE_UNION: "[| monoidal (u_4372::'q_60945 => 'q_60945 => 'q_60945);
+ finite (s::'q_60930 => bool) &
+ finite (x::'q_60930 => bool) & s Int x = {} |]
+==> iterate u_4372 (s Un x) (f::'q_60930 => 'q_60945) =
+ u_4372 (iterate u_4372 s f) (iterate u_4372 x f)"
+ by (import hollight ITERATE_UNION)
+
+lemma ITERATE_UNION_GEN: "[| monoidal (x::'B => 'B => 'B);
+ finite (support x (xa::'A => 'B) (xb::'A => bool)) &
+ finite (support x xa (xc::'A => bool)) &
+ support x xa xb Int support x xa xc = {} |]
+==> iterate x (xb Un xc) xa = x (iterate x xb xa) (iterate x xc xa)"
+ by (import hollight ITERATE_UNION_GEN)
+
+lemma ITERATE_DIFF: "[| monoidal (u::'q_61087 => 'q_61087 => 'q_61087);
+ finite (s::'q_61083 => bool) & (t::'q_61083 => bool) <= s |]
+==> u (iterate u (s - t) (f::'q_61083 => 'q_61087)) (iterate u t f) =
+ iterate u s f"
+ by (import hollight ITERATE_DIFF)
+
+lemma ITERATE_DIFF_GEN: "[| monoidal (x::'B => 'B => 'B);
+ finite (support x (xa::'A => 'B) (xb::'A => bool)) &
+ support x xa (xc::'A => bool) <= support x xa xb |]
+==> x (iterate x (xb - xc) xa) (iterate x xc xa) = iterate x xb xa"
+ by (import hollight ITERATE_DIFF_GEN)
+
+lemma ITERATE_INCL_EXCL: "[| monoidal (u_4372::'q_61316 => 'q_61316 => 'q_61316);
+ finite (s::'q_61298 => bool) & finite (t::'q_61298 => bool) |]
+==> u_4372 (iterate u_4372 s (f::'q_61298 => 'q_61316))
+ (iterate u_4372 t f) =
+ u_4372 (iterate u_4372 (s Un t) f) (iterate u_4372 (s Int t) f)"
+ by (import hollight ITERATE_INCL_EXCL)
+
+lemma ITERATE_CLOSED: "[| monoidal (u_4372::'B => 'B => 'B);
+ (P::'B => bool) (neutral u_4372) &
+ (ALL (x::'B) y::'B. P x & P y --> P (u_4372 x y));
+ !!x::'A.
+ x : (s::'A => bool) & (f::'A => 'B) x ~= neutral u_4372 ==> P (f x) |]
+==> P (iterate u_4372 s f)"
+ by (import hollight ITERATE_CLOSED)
+
+lemma ITERATE_RELATED: "[| monoidal (u_4372::'B => 'B => 'B);
+ (R::'B => 'B => bool) (neutral u_4372) (neutral u_4372) &
+ (ALL (x1::'B) (y1::'B) (x2::'B) y2::'B.
+ R x1 x2 & R y1 y2 --> R (u_4372 x1 y1) (u_4372 x2 y2));
+ finite (x::'A => bool) &
+ (ALL xa::'A. xa : x --> R ((f::'A => 'B) xa) ((g::'A => 'B) xa)) |]
+==> R (iterate u_4372 x f) (iterate u_4372 x g)"
+ by (import hollight ITERATE_RELATED)
+
+lemma ITERATE_EQ_NEUTRAL: "[| monoidal (u_4372::'B => 'B => 'B);
+ !!x::'A. x : (s::'A => bool) ==> (f::'A => 'B) x = neutral u_4372 |]
+==> iterate u_4372 s f = neutral u_4372"
+ by (import hollight ITERATE_EQ_NEUTRAL)
+
+lemma ITERATE_SING: "monoidal (x::'B => 'B => 'B) ==> iterate x {xa::'A} (f::'A => 'B) = f xa"
+ by (import hollight ITERATE_SING)
+
+lemma ITERATE_DELETE: "[| monoidal (u::'B => 'B => 'B); finite (s::'A => bool) & (a::'A) : s |]
+==> u ((f::'A => 'B) a) (iterate u (s - {a}) f) = iterate u s f"
+ by (import hollight ITERATE_DELETE)
+
+lemma ITERATE_DELTA: "monoidal (u_4372::'q_61672 => 'q_61672 => 'q_61672)
+==> iterate u_4372 (xb::'q_61691 => bool)
+ (%xb::'q_61691.
+ if xb = (xa::'q_61691) then (x::'q_61691 => 'q_61672) xb
+ else neutral u_4372) =
+ (if xa : xb then x xa else neutral u_4372)"
+ by (import hollight ITERATE_DELTA)
+
+lemma ITERATE_IMAGE: "[| monoidal (u_4372::'C => 'C => 'C);
+ !!(x::'A) y::'A.
+ x : (s::'A => bool) & y : s & (f::'A => 'B) x = f y ==> x = y |]
+==> iterate u_4372 (f ` s) (g::'B => 'C) = iterate u_4372 s (g o f)"
+ by (import hollight ITERATE_IMAGE)
+
+lemma ITERATE_BIJECTION: "[| monoidal (u_4372::'B => 'B => 'B);
+ (ALL x::'A. x : (s::'A => bool) --> (p::'A => 'A) x : s) &
+ (ALL y::'A. y : s --> (EX! x::'A. x : s & p x = y)) |]
+==> iterate u_4372 s (f::'A => 'B) = iterate u_4372 s (f o p)"
+ by (import hollight ITERATE_BIJECTION)
+
+lemma ITERATE_ITERATE_PRODUCT: "[| monoidal (u_4372::'C => 'C => 'C);
+ finite (x::'A => bool) &
+ (ALL i::'A. i : x --> finite ((xa::'A => 'B => bool) i)) |]
+==> iterate u_4372 x
+ (%i::'A. iterate u_4372 (xa i) ((xb::'A => 'B => 'C) i)) =
+ iterate u_4372
+ {u::'A * 'B. EX (i::'A) j::'B. (i : x & j : xa i) & u = (i, j)}
+ (SOME f::'A * 'B => 'C. ALL (i::'A) j::'B. f (i, j) = xb i j)"
+ by (import hollight ITERATE_ITERATE_PRODUCT)
+
+lemma ITERATE_EQ: "[| monoidal (u_4372::'B => 'B => 'B);
+ !!x::'A. x : (s::'A => bool) ==> (f::'A => 'B) x = (g::'A => 'B) x |]
+==> iterate u_4372 s f = iterate u_4372 s g"
+ by (import hollight ITERATE_EQ)
+
+lemma ITERATE_EQ_GENERAL: "[| monoidal (u_4372::'C => 'C => 'C);
+ (ALL y::'B.
+ y : (t::'B => bool) -->
+ (EX! x::'A. x : (s::'A => bool) & (h::'A => 'B) x = y)) &
+ (ALL x::'A. x : s --> h x : t & (g::'B => 'C) (h x) = (f::'A => 'C) x) |]
+==> iterate u_4372 s f = iterate u_4372 t g"
+ by (import hollight ITERATE_EQ_GENERAL)
+
+lemma ITERATE_EQ_GENERAL_INVERSES: "[| monoidal (u_4372::'C => 'C => 'C);
+ (ALL y::'B.
+ y : (t::'B => bool) -->
+ (k::'B => 'A) y : (s::'A => bool) & (h::'A => 'B) (k y) = y) &
+ (ALL x::'A.
+ x : s -->
+ h x : t & k (h x) = x & (g::'B => 'C) (h x) = (f::'A => 'C) x) |]
+==> iterate u_4372 s f = iterate u_4372 t g"
+ by (import hollight ITERATE_EQ_GENERAL_INVERSES)
+
+lemma ITERATE_INJECTION: "[| monoidal (u_4372::'B => 'B => 'B);
+ finite (s::'A => bool) &
+ (ALL x::'A. x : s --> (p::'A => 'A) x : s) &
+ (ALL (x::'A) y::'A. x : s & y : s & p x = p y --> x = y) |]
+==> iterate u_4372 s ((f::'A => 'B) o p) = iterate u_4372 s f"
+ by (import hollight ITERATE_INJECTION)
+
+lemma ITERATE_UNION_NONZERO: "[| monoidal (u_4372::'B => 'B => 'B);
+ finite (s::'A => bool) &
+ finite (t::'A => bool) &
+ (ALL x::'A. x : s Int t --> (f::'A => 'B) x = neutral u_4372) |]
+==> iterate u_4372 (s Un t) f =
+ u_4372 (iterate u_4372 s f) (iterate u_4372 t f)"
+ by (import hollight ITERATE_UNION_NONZERO)
+
+lemma ITERATE_OP: "[| monoidal (u_4372::'q_62649 => 'q_62649 => 'q_62649);
+ finite (s::'q_62648 => bool) |]
+==> iterate u_4372 s
+ (%x::'q_62648.
+ u_4372 ((f::'q_62648 => 'q_62649) x)
+ ((g::'q_62648 => 'q_62649) x)) =
+ u_4372 (iterate u_4372 s f) (iterate u_4372 s g)"
+ by (import hollight ITERATE_OP)
+
+lemma ITERATE_SUPERSET: "[| monoidal (u_4372::'B => 'B => 'B);
+ (u::'A => bool) <= (v::'A => bool) &
+ (ALL x::'A. x : v & x ~: u --> (f::'A => 'B) x = neutral u_4372) |]
+==> iterate u_4372 v f = iterate u_4372 u f"
+ by (import hollight ITERATE_SUPERSET)
+
+lemma ITERATE_IMAGE_NONZERO: "[| monoidal (u_4372::'C => 'C => 'C);
+ finite (x::'A => bool) &
+ (ALL (xa::'A) y::'A.
+ xa : x & y : x & xa ~= y & (f::'A => 'B) xa = f y -->
+ (g::'B => 'C) (f xa) = neutral u_4372) |]
+==> iterate u_4372 (f ` x) g = iterate u_4372 x (g o f)"
+ by (import hollight ITERATE_IMAGE_NONZERO)
+
+lemma ITERATE_CASES: "[| monoidal (u_4372::'B => 'B => 'B); finite (s::'A => bool) |]
+==> iterate u_4372 s
+ (%x::'A.
+ if (P::'A => bool) x then (f::'A => 'B) x else (g::'A => 'B) x) =
+ u_4372 (iterate u_4372 {u::'A. EX x::'A. (x : s & P x) & u = x} f)
+ (iterate u_4372 {u::'A. EX x::'A. (x : s & ~ P x) & u = x} g)"
+ by (import hollight ITERATE_CASES)
+
+lemma ITERATE_OP_GEN: "[| monoidal (u_4372::'B => 'B => 'B);
+ finite (support u_4372 (f::'A => 'B) (s::'A => bool)) &
+ finite (support u_4372 (g::'A => 'B) s) |]
+==> iterate u_4372 s (%x::'A. u_4372 (f x) (g x)) =
+ u_4372 (iterate u_4372 s f) (iterate u_4372 s g)"
+ by (import hollight ITERATE_OP_GEN)
+
+lemma ITERATE_CLAUSES_NUMSEG: "monoidal (x::'q_63246 => 'q_63246 => 'q_63246)
+==> (ALL xa::nat.
+ iterate x {xa..0::nat} (f::nat => 'q_63246) =
+ (if xa = (0::nat) then f (0::nat) else neutral x)) &
+ (ALL (xa::nat) xb::nat.
+ iterate x {xa..Suc xb} f =
+ (if xa <= Suc xb then x (iterate x {xa..xb} f) (f (Suc xb))
+ else iterate x {xa..xb} f))"
+ by (import hollight ITERATE_CLAUSES_NUMSEG)
+
+lemma ITERATE_PAIR: "monoidal (u_4372::'q_63421 => 'q_63421 => 'q_63421)
+==> iterate u_4372 {(2::nat) * (m::nat)..(2::nat) * (n::nat) + (1::nat)}
+ (f::nat => 'q_63421) =
+ iterate u_4372 {m..n}
+ (%i::nat. u_4372 (f ((2::nat) * i)) (f ((2::nat) * i + (1::nat))))"
+ by (import hollight ITERATE_PAIR)
+
+definition
+ nsum :: "('q_63439 => bool) => ('q_63439 => nat) => nat" where
+ "(op ==::(('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ => (('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ => prop)
+ (nsum::('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ ((iterate::(nat => nat => nat)
+ => ('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ (op +::nat => nat => nat))"
+
+lemma DEF_nsum: "(op =::(('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ => (('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ => bool)
+ (nsum::('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ ((iterate::(nat => nat => nat)
+ => ('q_63439::type => bool) => ('q_63439::type => nat) => nat)
+ (op +::nat => nat => nat))"
+ by (import hollight DEF_nsum)
+
+lemma NEUTRAL_ADD: "neutral op + = (0::nat)"
+ by (import hollight NEUTRAL_ADD)
+
+lemma NEUTRAL_MUL: "neutral op * = (1::nat)"
+ by (import hollight NEUTRAL_MUL)
+
+lemma MONOIDAL_ADD: "(monoidal::(nat => nat => nat) => bool) (op +::nat => nat => nat)"
+ by (import hollight MONOIDAL_ADD)
+
+lemma MONOIDAL_MUL: "(monoidal::(nat => nat => nat) => bool) (op *::nat => nat => nat)"
+ by (import hollight MONOIDAL_MUL)
+
+lemma NSUM_CLAUSES: "(ALL x::'q_63477 => nat. nsum {} x = (0::nat)) &
+(ALL (x::'q_63516) (xa::'q_63516 => nat) xb::'q_63516 => bool.
+ finite xb -->
+ nsum (insert x xb) xa =
+ (if x : xb then nsum xb xa else xa x + nsum xb xa))"
+ by (import hollight NSUM_CLAUSES)
+
+lemma NSUM_UNION: "finite (xa::'q_63542 => bool) &
+finite (xb::'q_63542 => bool) & xa Int xb = {}
+==> nsum (xa Un xb) (x::'q_63542 => nat) = nsum xa x + nsum xb x"
+ by (import hollight NSUM_UNION)
+
+lemma NSUM_DIFF: "finite (s::'q_63597 => bool) & (t::'q_63597 => bool) <= s
+==> nsum (s - t) (f::'q_63597 => nat) = nsum s f - nsum t f"
+ by (import hollight NSUM_DIFF)
+
+lemma NSUM_INCL_EXCL: "finite (x::'A => bool) & finite (xa::'A => bool)
+==> nsum x (xb::'A => nat) + nsum xa xb =
+ nsum (x Un xa) xb + nsum (x Int xa) xb"
+ by (import hollight NSUM_INCL_EXCL)
+
+lemma NSUM_SUPPORT: "nsum (support op + (x::'q_63686 => nat) (xa::'q_63686 => bool)) x =
+nsum xa x"
+ by (import hollight NSUM_SUPPORT)
+
+lemma NSUM_ADD: "finite (xb::'q_63720 => bool)
+==> nsum xb
+ (%xb::'q_63720. (x::'q_63720 => nat) xb + (xa::'q_63720 => nat) xb) =
+ nsum xb x + nsum xb xa"
+ by (import hollight NSUM_ADD)
+
+lemma NSUM_ADD_GEN: "finite
+ {xa::'q_63807.
+ EX xc::'q_63807.
+ (xc : (xb::'q_63807 => bool) & (x::'q_63807 => nat) xc ~= (0::nat)) &
+ xa = xc} &
+finite
+ {x::'q_63807.
+ EX xc::'q_63807.
+ (xc : xb & (xa::'q_63807 => nat) xc ~= (0::nat)) & x = xc}
+==> nsum xb (%xb::'q_63807. x xb + xa xb) = nsum xb x + nsum xb xa"
+ by (import hollight NSUM_ADD_GEN)
+
+lemma NSUM_EQ_0: "(!!xb::'A. xb : (xa::'A => bool) ==> (x::'A => nat) xb = (0::nat))
+==> nsum xa x = (0::nat)"
+ by (import hollight NSUM_EQ_0)
+
+lemma NSUM_0: "nsum (x::'A => bool) (%n::'A. 0::nat) = (0::nat)"
+ by (import hollight NSUM_0)
+
+lemma NSUM_LMUL: "nsum (s::'A => bool) (%x::'A. (c::nat) * (f::'A => nat) x) = c * nsum s f"
+ by (import hollight NSUM_LMUL)
+
+lemma NSUM_RMUL: "nsum (xb::'A => bool) (%xb::'A. (x::'A => nat) xb * (xa::nat)) =
+nsum xb x * xa"
+ by (import hollight NSUM_RMUL)
+
+lemma NSUM_LE: "finite (xb::'q_63997 => bool) &
+(ALL xc::'q_63997.
+ xc : xb --> (x::'q_63997 => nat) xc <= (xa::'q_63997 => nat) xc)
+==> nsum xb x <= nsum xb xa"
+ by (import hollight NSUM_LE)
+
+lemma NSUM_LT: "finite (s::'A => bool) &
+(ALL x::'A. x : s --> (f::'A => nat) x <= (g::'A => nat) x) &
+(EX x::'A. x : s & f x < g x)
+==> nsum s f < nsum s g"
+ by (import hollight NSUM_LT)
+
+lemma NSUM_LT_ALL: "finite (s::'q_64119 => bool) &
+s ~= {} &
+(ALL x::'q_64119. x : s --> (f::'q_64119 => nat) x < (g::'q_64119 => nat) x)
+==> nsum s f < nsum s g"
+ by (import hollight NSUM_LT_ALL)
+
+lemma NSUM_EQ: "(!!xc::'q_64157.
+ xc : (xb::'q_64157 => bool)
+ ==> (x::'q_64157 => nat) xc = (xa::'q_64157 => nat) xc)
+==> nsum xb x = nsum xb xa"
+ by (import hollight NSUM_EQ)
+
+lemma NSUM_CONST: "finite (s::'q_64187 => bool) ==> nsum s (%n::'q_64187. c::nat) = CARD s * c"
+ by (import hollight NSUM_CONST)
+
+lemma NSUM_POS_BOUND: "[| finite (x::'A => bool) & nsum x (f::'A => nat) <= (b::nat);
+ (xa::'A) : x |]
+==> f xa <= b"
+ by (import hollight NSUM_POS_BOUND)
+
+lemma NSUM_EQ_0_IFF: "finite (s::'q_64296 => bool)
+==> (nsum s (f::'q_64296 => nat) = (0::nat)) =
+ (ALL x::'q_64296. x : s --> f x = (0::nat))"
+ by (import hollight NSUM_EQ_0_IFF)
+
+lemma NSUM_DELETE: "finite (xa::'q_64325 => bool) & (xb::'q_64325) : xa
+==> (x::'q_64325 => nat) xb + nsum (xa - {xb}) x = nsum xa x"
+ by (import hollight NSUM_DELETE)
+
+lemma NSUM_SING: "nsum {xa::'q_64354} (x::'q_64354 => nat) = x xa"
+ by (import hollight NSUM_SING)
+
+lemma NSUM_DELTA: "nsum (x::'A => bool) (%x::'A. if x = (xa::'A) then b::nat else (0::nat)) =
+(if xa : x then b else (0::nat))"
+ by (import hollight NSUM_DELTA)
+
+lemma NSUM_SWAP: "finite (x::'A => bool) & finite (xa::'B => bool)
+==> nsum x (%i::'A. nsum xa ((f::'A => 'B => nat) i)) =
+ nsum xa (%j::'B. nsum x (%i::'A. f i j))"
+ by (import hollight NSUM_SWAP)
+
+lemma NSUM_IMAGE: "(!!(xa::'q_64490) y::'q_64490.
+ xa : (xb::'q_64490 => bool) &
+ y : xb & (x::'q_64490 => 'q_64466) xa = x y
+ ==> xa = y)
+==> nsum (x ` xb) (xa::'q_64466 => nat) = nsum xb (xa o x)"
+ by (import hollight NSUM_IMAGE)
+
+lemma NSUM_SUPERSET: "(xa::'A => bool) <= (xb::'A => bool) &
+(ALL xc::'A. xc : xb & xc ~: xa --> (x::'A => nat) xc = (0::nat))
+==> nsum xb x = nsum xa x"
+ by (import hollight NSUM_SUPERSET)
+
+lemma NSUM_UNION_RZERO: "finite (u::'A => bool) &
+(ALL x::'A. x : (v::'A => bool) & x ~: u --> (f::'A => nat) x = (0::nat))
+==> nsum (u Un v) f = nsum u f"
+ by (import hollight NSUM_UNION_RZERO)
+
+lemma NSUM_UNION_LZERO: "finite (v::'A => bool) &
+(ALL x::'A. x : (u::'A => bool) & x ~: v --> (f::'A => nat) x = (0::nat))
+==> nsum (u Un v) f = nsum v f"
+ by (import hollight NSUM_UNION_LZERO)
+
+lemma NSUM_RESTRICT: "finite (s::'q_64681 => bool)
+==> nsum s
+ (%x::'q_64681. if x : s then (f::'q_64681 => nat) x else (0::nat)) =
+ nsum s f"
+ by (import hollight NSUM_RESTRICT)
+
+lemma NSUM_BOUND: "finite (x::'A => bool) &
+(ALL xc::'A. xc : x --> (xa::'A => nat) xc <= (xb::nat))
+==> nsum x xa <= CARD x * xb"
+ by (import hollight NSUM_BOUND)
+
+lemma NSUM_BOUND_GEN: "finite (x::'A => bool) &
+x ~= {} & (ALL xa::'A. xa : x --> (f::'A => nat) xa <= (b::nat) div CARD x)
+==> nsum x f <= b"
+ by (import hollight NSUM_BOUND_GEN)
+
+lemma NSUM_BOUND_LT: "finite (s::'A => bool) &
+(ALL x::'A. x : s --> (f::'A => nat) x <= (b::nat)) &
+(EX x::'A. x : s & f x < b)
+==> nsum s f < CARD s * b"
+ by (import hollight NSUM_BOUND_LT)
+
+lemma NSUM_BOUND_LT_ALL: "finite (s::'q_64899 => bool) &
+s ~= {} & (ALL x::'q_64899. x : s --> (f::'q_64899 => nat) x < (b::nat))
+==> nsum s f < CARD s * b"
+ by (import hollight NSUM_BOUND_LT_ALL)
+
+lemma NSUM_BOUND_LT_GEN: "finite (s::'A => bool) &
+s ~= {} & (ALL x::'A. x : s --> (f::'A => nat) x < (b::nat) div CARD s)
+==> nsum s f < b"
+ by (import hollight NSUM_BOUND_LT_GEN)
+
+lemma NSUM_UNION_EQ: "finite (u::'q_65000 => bool) &
+(s::'q_65000 => bool) Int (t::'q_65000 => bool) = {} & s Un t = u
+==> nsum s (f::'q_65000 => nat) + nsum t f = nsum u f"
+ by (import hollight NSUM_UNION_EQ)
+
+lemma NSUM_EQ_SUPERSET: "finite (t::'A => bool) &
+t <= (s::'A => bool) &
+(ALL x::'A. x : t --> (f::'A => nat) x = (g::'A => nat) x) &
+(ALL x::'A. x : s & x ~: t --> f x = (0::nat))
+==> nsum s f = nsum t g"
+ by (import hollight NSUM_EQ_SUPERSET)
+
+lemma NSUM_RESTRICT_SET: "nsum
+ {u::'A. EX xb::'A. (xb : (xa::'A => bool) & (x::'A => bool) xb) & u = xb}
+ (xb::'A => nat) =
+nsum xa (%xa::'A. if x xa then xb xa else (0::nat))"
+ by (import hollight NSUM_RESTRICT_SET)
+
+lemma NSUM_NSUM_RESTRICT: "finite (s::'q_65257 => bool) & finite (t::'q_65256 => bool)
+==> nsum s
+ (%x::'q_65257.
+ nsum
+ {u::'q_65256.
+ EX y::'q_65256.
+ (y : t & (R::'q_65257 => 'q_65256 => bool) x y) & u = y}
+ ((f::'q_65257 => 'q_65256 => nat) x)) =
+ nsum t
+ (%y::'q_65256.
+ nsum {u::'q_65257. EX x::'q_65257. (x : s & R x y) & u = x}
+ (%x::'q_65257. f x y))"
+ by (import hollight NSUM_NSUM_RESTRICT)
+
+lemma CARD_EQ_NSUM: "finite (x::'q_65276 => bool) ==> CARD x = nsum x (%x::'q_65276. 1::nat)"
+ by (import hollight CARD_EQ_NSUM)
+
+lemma NSUM_MULTICOUNT_GEN: "finite (s::'A => bool) &
+finite (t::'B => bool) &
+(ALL j::'B.
+ j : t -->
+ CARD {u::'A. EX i::'A. (i : s & (R::'A => 'B => bool) i j) & u = i} =
+ (k::'B => nat) j)
+==> nsum s (%i::'A. CARD {u::'B. EX j::'B. (j : t & R i j) & u = j}) =
+ nsum t k"
+ by (import hollight NSUM_MULTICOUNT_GEN)
+
+lemma NSUM_MULTICOUNT: "finite (s::'A => bool) &
+finite (t::'B => bool) &
+(ALL j::'B.
+ j : t -->
+ CARD {u::'A. EX i::'A. (i : s & (R::'A => 'B => bool) i j) & u = i} =
+ (k::nat))
+==> nsum s (%i::'A. CARD {u::'B. EX j::'B. (j : t & R i j) & u = j}) =
+ k * CARD t"
+ by (import hollight NSUM_MULTICOUNT)
+
+lemma NSUM_IMAGE_GEN: "finite (s::'A => bool)
+==> nsum s (g::'A => nat) =
+ nsum ((f::'A => 'B) ` s)
+ (%y::'B. nsum {u::'A. EX x::'A. (x : s & f x = y) & u = x} g)"
+ by (import hollight NSUM_IMAGE_GEN)
+
+lemma NSUM_GROUP: "finite (s::'A => bool) & (f::'A => 'B) ` s <= (t::'B => bool)
+==> nsum t
+ (%y::'B.
+ nsum {u::'A. EX x::'A. (x : s & f x = y) & u = x} (g::'A => nat)) =
+ nsum s g"
+ by (import hollight NSUM_GROUP)
+
+lemma NSUM_SUBSET: "finite (u::'A => bool) &
+finite (v::'A => bool) &
+(ALL x::'A. x : u - v --> (f::'A => nat) x = (0::nat))
+==> nsum u f <= nsum v f"
+ by (import hollight NSUM_SUBSET)
+
+lemma NSUM_SUBSET_SIMPLE: "finite (v::'q_65804 => bool) & (u::'q_65804 => bool) <= v
+==> nsum u (f::'q_65804 => nat) <= nsum v f"
+ by (import hollight NSUM_SUBSET_SIMPLE)
+
+lemma NSUM_IMAGE_NONZERO: "finite (xb::'A => bool) &
+(ALL (xc::'A) xd::'A.
+ xc : xb & xd : xb & xc ~= xd & (xa::'A => 'B) xc = xa xd -->
+ (x::'B => nat) (xa xc) = (0::nat))
+==> nsum (xa ` xb) x = nsum xb (x o xa)"
+ by (import hollight NSUM_IMAGE_NONZERO)
+
+lemma NSUM_BIJECTION: "(ALL x::'A. x : (xb::'A => bool) --> (xa::'A => 'A) x : xb) &
+(ALL y::'A. y : xb --> (EX! x::'A. x : xb & xa x = y))
+==> nsum xb (x::'A => nat) = nsum xb (x o xa)"
+ by (import hollight NSUM_BIJECTION)
+
+lemma NSUM_NSUM_PRODUCT: "finite (x::'A => bool) &
+(ALL i::'A. i : x --> finite ((xa::'A => 'B => bool) i))
+==> nsum x (%x::'A. nsum (xa x) ((xb::'A => 'B => nat) x)) =
+ nsum {u::'A * 'B. EX (i::'A) j::'B. (i : x & j : xa i) & u = (i, j)}
+ (SOME f::'A * 'B => nat. ALL (i::'A) j::'B. f (i, j) = xb i j)"
+ by (import hollight NSUM_NSUM_PRODUCT)
+
+lemma NSUM_EQ_GENERAL: "(ALL y::'B.
+ y : (xa::'B => bool) -->
+ (EX! xa::'A. xa : (x::'A => bool) & (xd::'A => 'B) xa = y)) &
+(ALL xe::'A.
+ xe : x --> xd xe : xa & (xc::'B => nat) (xd xe) = (xb::'A => nat) xe)
+==> nsum x xb = nsum xa xc"
+ by (import hollight NSUM_EQ_GENERAL)
+
+lemma NSUM_EQ_GENERAL_INVERSES: "(ALL y::'B.
+ y : (xa::'B => bool) -->
+ (xe::'B => 'A) y : (x::'A => bool) & (xd::'A => 'B) (xe y) = y) &
+(ALL xf::'A.
+ xf : x -->
+ xd xf : xa &
+ xe (xd xf) = xf & (xc::'B => nat) (xd xf) = (xb::'A => nat) xf)
+==> nsum x xb = nsum xa xc"
+ by (import hollight NSUM_EQ_GENERAL_INVERSES)
+
+lemma NSUM_INJECTION: "finite (xb::'q_66274 => bool) &
+(ALL x::'q_66274. x : xb --> (xa::'q_66274 => 'q_66274) x : xb) &
+(ALL (x::'q_66274) y::'q_66274. x : xb & y : xb & xa x = xa y --> x = y)
+==> nsum xb ((x::'q_66274 => nat) o xa) = nsum xb x"
+ by (import hollight NSUM_INJECTION)
+
+lemma NSUM_UNION_NONZERO: "finite (xa::'q_66317 => bool) &
+finite (xb::'q_66317 => bool) &
+(ALL xc::'q_66317. xc : xa Int xb --> (x::'q_66317 => nat) xc = (0::nat))
+==> nsum (xa Un xb) x = nsum xa x + nsum xb x"
+ by (import hollight NSUM_UNION_NONZERO)
+
+lemma NSUM_UNIONS_NONZERO: "finite (x::('A => bool) => bool) &
+(ALL t::'A => bool. t : x --> finite t) &
+(ALL (t1::'A => bool) (t2::'A => bool) xa::'A.
+ t1 : x & t2 : x & t1 ~= t2 & xa : t1 & xa : t2 -->
+ (f::'A => nat) xa = (0::nat))
+==> nsum (Union x) f = nsum x (%t::'A => bool. nsum t f)"
+ by (import hollight NSUM_UNIONS_NONZERO)
+
+lemma NSUM_CASES: "finite (x::'A => bool)
+==> nsum x
+ (%x::'A.
+ if (xa::'A => bool) x then (xb::'A => nat) x
+ else (xc::'A => nat) x) =
+ nsum {u::'A. EX xb::'A. (xb : x & xa xb) & u = xb} xb +
+ nsum {u::'A. EX xb::'A. (xb : x & ~ xa xb) & u = xb} xc"
+ by (import hollight NSUM_CASES)
+
+lemma NSUM_CLOSED: "(P::nat => bool) (0::nat) &
+(ALL (x::nat) y::nat. P x & P y --> P (x + y)) &
+(ALL a::'A. a : (s::'A => bool) --> P ((f::'A => nat) a))
+==> P (nsum s f)"
+ by (import hollight NSUM_CLOSED)
+
+lemma NSUM_ADD_NUMSEG: "nsum {xb::nat..xc::nat} (%i::nat. (x::nat => nat) i + (xa::nat => nat) i) =
+nsum {xb..xc} x + nsum {xb..xc} xa"
+ by (import hollight NSUM_ADD_NUMSEG)
+
+lemma NSUM_LE_NUMSEG: "(!!i::nat.
+ (xb::nat) <= i & i <= (xc::nat)
+ ==> (x::nat => nat) i <= (xa::nat => nat) i)
+==> nsum {xb..xc} x <= nsum {xb..xc} xa"
+ by (import hollight NSUM_LE_NUMSEG)
+
+lemma NSUM_EQ_NUMSEG: "(!!i::nat.
+ (m::nat) <= i & i <= (n::nat) ==> (f::nat => nat) i = (g::nat => nat) i)
+==> nsum {m..n} f = nsum {m..n} g"
+ by (import hollight NSUM_EQ_NUMSEG)
+
+lemma NSUM_CONST_NUMSEG: "nsum {xa..xb} (%n. x) = (xb + 1 - xa) * x"
+ by (import hollight NSUM_CONST_NUMSEG)
+
+lemma NSUM_EQ_0_NUMSEG: "(!!i::nat. (m::nat) <= i & i <= (n::nat) ==> (x::nat => nat) i = (0::nat))
+==> nsum {m..n} x = (0::nat)"
+ by (import hollight NSUM_EQ_0_NUMSEG)
+
+lemma NSUM_EQ_0_IFF_NUMSEG: "(nsum {xa::nat..xb::nat} (x::nat => nat) = (0::nat)) =
+(ALL i::nat. xa <= i & i <= xb --> x i = (0::nat))"
+ by (import hollight NSUM_EQ_0_IFF_NUMSEG)
+
+lemma NSUM_TRIV_NUMSEG: "(n::nat) < (m::nat) ==> nsum {m..n} (f::nat => nat) = (0::nat)"
+ by (import hollight NSUM_TRIV_NUMSEG)
+
+lemma NSUM_SING_NUMSEG: "nsum {xa::nat..xa} (x::nat => nat) = x xa"
+ by (import hollight NSUM_SING_NUMSEG)
+
+lemma NSUM_CLAUSES_NUMSEG: "(ALL m. nsum {m..0} f = (if m = 0 then f 0 else 0)) &
+(ALL m n.
+ nsum {m..Suc n} f =
+ (if m <= Suc n then nsum {m..n} f + f (Suc n) else nsum {m..n} f))"
+ by (import hollight NSUM_CLAUSES_NUMSEG)
+
+lemma NSUM_SWAP_NUMSEG: "nsum {a::nat..b::nat}
+ (%i::nat. nsum {c::nat..d::nat} ((f::nat => nat => nat) i)) =
+nsum {c..d} (%j::nat. nsum {a..b} (%i::nat. f i j))"
+ by (import hollight NSUM_SWAP_NUMSEG)
+
+lemma NSUM_ADD_SPLIT: "(xa::nat) <= (xb::nat) + (1::nat)
+==> nsum {xa..xb + (xc::nat)} (x::nat => nat) =
+ nsum {xa..xb} x + nsum {xb + (1::nat)..xb + xc} x"
+ by (import hollight NSUM_ADD_SPLIT)
+
+lemma NSUM_OFFSET: "nsum {(xb::nat) + (x::nat)..(xc::nat) + x} (xa::nat => nat) =
+nsum {xb..xc} (%i::nat. xa (i + x))"
+ by (import hollight NSUM_OFFSET)
+
+lemma NSUM_OFFSET_0: "(xa::nat) <= (xb::nat)
+==> nsum {xa..xb} (x::nat => nat) =
+ nsum {0::nat..xb - xa} (%i::nat. x (i + xa))"
+ by (import hollight NSUM_OFFSET_0)
+
+lemma NSUM_CLAUSES_LEFT: "(xa::nat) <= (xb::nat)
+==> nsum {xa..xb} (x::nat => nat) = x xa + nsum {xa + (1::nat)..xb} x"
+ by (import hollight NSUM_CLAUSES_LEFT)
+
+lemma NSUM_CLAUSES_RIGHT: "(0::nat) < (n::nat) & (m::nat) <= n
+==> nsum {m..n} (f::nat => nat) = nsum {m..n - (1::nat)} f + f n"
+ by (import hollight NSUM_CLAUSES_RIGHT)
+
+lemma NSUM_PAIR: "nsum {(2::nat) * (m::nat)..(2::nat) * (n::nat) + (1::nat)} (f::nat => nat) =
+nsum {m..n} (%i::nat. f ((2::nat) * i) + f ((2::nat) * i + (1::nat)))"
+ by (import hollight NSUM_PAIR)
+
+lemma CARD_UNIONS: "finite (x::('A => bool) => bool) &
+(ALL t::'A => bool. t : x --> finite t) &
+(ALL (t::'A => bool) u::'A => bool. t : x & u : x & t ~= u --> t Int u = {})
+==> CARD (Union x) = nsum x CARD"
+ by (import hollight CARD_UNIONS)
+
+consts
+ sum :: "('q_67488 => bool) => ('q_67488 => hollight.real) => hollight.real"
+
+defs
+ sum_def: "(op ==::(('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ => (('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ => prop)
+ (hollight.sum::('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ ((iterate::(hollight.real => hollight.real => hollight.real)
+ => ('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ (real_add::hollight.real => hollight.real => hollight.real))"
+
+lemma DEF_sum: "(op =::(('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ => (('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ => bool)
+ (hollight.sum::('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ ((iterate::(hollight.real => hollight.real => hollight.real)
+ => ('q_67488::type => bool)
+ => ('q_67488::type => hollight.real) => hollight.real)
+ (real_add::hollight.real => hollight.real => hollight.real))"
+ by (import hollight DEF_sum)
+
+lemma NEUTRAL_REAL_ADD: "neutral real_add = real_of_num 0"
+ by (import hollight NEUTRAL_REAL_ADD)
+
+lemma NEUTRAL_REAL_MUL: "neutral real_mul = real_of_num 1"
+ by (import hollight NEUTRAL_REAL_MUL)
+
+lemma MONOIDAL_REAL_ADD: "monoidal real_add"
+ by (import hollight MONOIDAL_REAL_ADD)
+
+lemma MONOIDAL_REAL_MUL: "monoidal real_mul"
+ by (import hollight MONOIDAL_REAL_MUL)
+
+lemma SUM_CLAUSES: "(ALL x::'q_67530 => hollight.real.
+ hollight.sum {} x = real_of_num (0::nat)) &
+(ALL (x::'q_67571) (xa::'q_67571 => hollight.real) xb::'q_67571 => bool.
+ finite xb -->
+ hollight.sum (insert x xb) xa =
+ (if x : xb then hollight.sum xb xa
+ else real_add (xa x) (hollight.sum xb xa)))"
+ by (import hollight SUM_CLAUSES)
+
+lemma SUM_UNION: "finite (xa::'q_67597 => bool) &
+finite (xb::'q_67597 => bool) & xa Int xb = {}
+==> hollight.sum (xa Un xb) (x::'q_67597 => hollight.real) =
+ real_add (hollight.sum xa x) (hollight.sum xb x)"
+ by (import hollight SUM_UNION)
+
+lemma SUM_DIFF: "finite (xa::'q_67637 => bool) & (xb::'q_67637 => bool) <= xa
+==> hollight.sum (xa - xb) (x::'q_67637 => hollight.real) =
+ real_sub (hollight.sum xa x) (hollight.sum xb x)"
+ by (import hollight SUM_DIFF)
+
+lemma SUM_INCL_EXCL: "finite (x::'A => bool) & finite (xa::'A => bool)
+==> real_add (hollight.sum x (xb::'A => hollight.real))
+ (hollight.sum xa xb) =
+ real_add (hollight.sum (x Un xa) xb) (hollight.sum (x Int xa) xb)"
+ by (import hollight SUM_INCL_EXCL)
+
+lemma SUM_SUPPORT: "hollight.sum
+ (support real_add (x::'q_67726 => hollight.real) (xa::'q_67726 => bool))
+ x =
+hollight.sum xa x"
+ by (import hollight SUM_SUPPORT)
+
+lemma SUM_ADD: "finite (xb::'q_67760 => bool)
+==> hollight.sum xb
+ (%xb::'q_67760.
+ real_add ((x::'q_67760 => hollight.real) xb)
+ ((xa::'q_67760 => hollight.real) xb)) =
+ real_add (hollight.sum xb x) (hollight.sum xb xa)"
+ by (import hollight SUM_ADD)
+
+lemma SUM_ADD_GEN: "finite
+ {xa::'q_67851.
+ EX xc::'q_67851.
+ (xc : (xb::'q_67851 => bool) &
+ (x::'q_67851 => hollight.real) xc ~= real_of_num (0::nat)) &
+ xa = xc} &
+finite
+ {x::'q_67851.
+ EX xc::'q_67851.
+ (xc : xb &
+ (xa::'q_67851 => hollight.real) xc ~= real_of_num (0::nat)) &
+ x = xc}
+==> hollight.sum xb (%xb::'q_67851. real_add (x xb) (xa xb)) =
+ real_add (hollight.sum xb x) (hollight.sum xb xa)"
+ by (import hollight SUM_ADD_GEN)
+
+lemma SUM_EQ_0: "(!!xb::'A.
+ xb : (xa::'A => bool)
+ ==> (x::'A => hollight.real) xb = real_of_num (0::nat))
+==> hollight.sum xa x = real_of_num (0::nat)"
+ by (import hollight SUM_EQ_0)
+
+lemma SUM_0: "hollight.sum (x::'A => bool) (%n::'A. real_of_num (0::nat)) =
+real_of_num (0::nat)"
+ by (import hollight SUM_0)
+
+lemma SUM_LMUL: "hollight.sum (s::'A => bool)
+ (%x::'A. real_mul (c::hollight.real) ((f::'A => hollight.real) x)) =
+real_mul c (hollight.sum s f)"
+ by (import hollight SUM_LMUL)
+
+lemma SUM_RMUL: "hollight.sum (xb::'A => bool)
+ (%xb::'A. real_mul ((x::'A => hollight.real) xb) (xa::hollight.real)) =
+real_mul (hollight.sum xb x) xa"
+ by (import hollight SUM_RMUL)
+
+lemma SUM_NEG: "hollight.sum (xa::'q_68051 => bool)
+ (%xa::'q_68051. real_neg ((x::'q_68051 => hollight.real) xa)) =
+real_neg (hollight.sum xa x)"
+ by (import hollight SUM_NEG)
+
+lemma SUM_SUB: "finite (xb::'q_68086 => bool)
+==> hollight.sum xb
+ (%xb::'q_68086.
+ real_sub ((x::'q_68086 => hollight.real) xb)
+ ((xa::'q_68086 => hollight.real) xb)) =
+ real_sub (hollight.sum xb x) (hollight.sum xb xa)"
+ by (import hollight SUM_SUB)
+
+lemma SUM_LE: "finite (xb::'q_68128 => bool) &
+(ALL xc::'q_68128.
+ xc : xb -->
+ real_le ((x::'q_68128 => hollight.real) xc)
+ ((xa::'q_68128 => hollight.real) xc))
+==> real_le (hollight.sum xb x) (hollight.sum xb xa)"
+ by (import hollight SUM_LE)
+
+lemma SUM_LT: "finite (s::'A => bool) &
+(ALL x::'A.
+ x : s -->
+ real_le ((f::'A => hollight.real) x) ((g::'A => hollight.real) x)) &
+(EX x::'A. x : s & real_lt (f x) (g x))
+==> real_lt (hollight.sum s f) (hollight.sum s g)"
+ by (import hollight SUM_LT)
+
+lemma SUM_LT_ALL: "finite (s::'q_68250 => bool) &
+s ~= {} &
+(ALL x::'q_68250.
+ x : s -->
+ real_lt ((f::'q_68250 => hollight.real) x)
+ ((g::'q_68250 => hollight.real) x))
+==> real_lt (hollight.sum s f) (hollight.sum s g)"
+ by (import hollight SUM_LT_ALL)
+
+lemma SUM_EQ: "(!!xc::'q_68288.
+ xc : (xb::'q_68288 => bool)
+ ==> (x::'q_68288 => hollight.real) xc =
+ (xa::'q_68288 => hollight.real) xc)
+==> hollight.sum xb x = hollight.sum xb xa"
+ by (import hollight SUM_EQ)
+
+lemma SUM_ABS: "finite (s::'q_68347 => bool)
+==> real_le (real_abs (hollight.sum s (f::'q_68347 => hollight.real)))
+ (hollight.sum s (%x::'q_68347. real_abs (f x)))"
+ by (import hollight SUM_ABS)
+
+lemma SUM_ABS_LE: "finite (s::'A => bool) &
+(ALL x::'A.
+ x : s -->
+ real_le (real_abs ((f::'A => hollight.real) x))
+ ((g::'A => hollight.real) x))
+==> real_le (real_abs (hollight.sum s f)) (hollight.sum s g)"
+ by (import hollight SUM_ABS_LE)
+
+lemma SUM_CONST: "finite (s::'q_68423 => bool)
+==> hollight.sum s (%n::'q_68423. c::hollight.real) =
+ real_mul (real_of_num (CARD s)) c"
+ by (import hollight SUM_CONST)
+
+lemma SUM_POS_LE: "finite (xa::'q_68465 => bool) &
+(ALL xb::'q_68465.
+ xb : xa -->
+ real_le (real_of_num (0::nat)) ((x::'q_68465 => hollight.real) xb))
+==> real_le (real_of_num (0::nat)) (hollight.sum xa x)"
+ by (import hollight SUM_POS_LE)
+
+lemma SUM_POS_BOUND: "[| finite (x::'A => bool) &
+ (ALL xa::'A.
+ xa : x -->
+ real_le (real_of_num (0::nat)) ((f::'A => hollight.real) xa)) &
+ real_le (hollight.sum x f) (b::hollight.real);
+ (xa::'A) : x |]
+==> real_le (f xa) b"
+ by (import hollight SUM_POS_BOUND)
+
+lemma SUM_POS_EQ_0: "[| finite (xa::'q_68612 => bool) &
+ (ALL xb::'q_68612.
+ xb : xa -->
+ real_le (real_of_num (0::nat)) ((x::'q_68612 => hollight.real) xb)) &
+ hollight.sum xa x = real_of_num (0::nat);
+ (xb::'q_68612) : xa |]
+==> x xb = real_of_num (0::nat)"
+ by (import hollight SUM_POS_EQ_0)
+
+lemma SUM_ZERO_EXISTS: "finite (s::'A => bool) &
+hollight.sum s (u::'A => hollight.real) = real_of_num (0::nat)
+==> (ALL i::'A. i : s --> u i = real_of_num (0::nat)) |
+ (EX (j::'A) k::'A.
+ j : s &
+ real_lt (u j) (real_of_num (0::nat)) &
+ k : s & real_gt (u k) (real_of_num (0::nat)))"
+ by (import hollight SUM_ZERO_EXISTS)
+
+lemma SUM_DELETE: "finite (xa::'q_68854 => bool) & (xb::'q_68854) : xa
+==> hollight.sum (xa - {xb}) (x::'q_68854 => hollight.real) =
+ real_sub (hollight.sum xa x) (x xb)"
+ by (import hollight SUM_DELETE)
+
+lemma SUM_DELETE_CASES: "finite (s::'q_68907 => bool)
+==> hollight.sum (s - {a::'q_68907}) (f::'q_68907 => hollight.real) =
+ (if a : s then real_sub (hollight.sum s f) (f a) else hollight.sum s f)"
+ by (import hollight SUM_DELETE_CASES)
+
+lemma SUM_SING: "hollight.sum {xa::'q_68930} (x::'q_68930 => hollight.real) = x xa"
+ by (import hollight SUM_SING)
+
+lemma SUM_DELTA: "hollight.sum (x::'A => bool)
+ (%x::'A. if x = (xa::'A) then b::hollight.real else real_of_num (0::nat)) =
+(if xa : x then b else real_of_num (0::nat))"
+ by (import hollight SUM_DELTA)
+
+lemma SUM_SWAP: "finite (x::'A => bool) & finite (xa::'B => bool)
+==> hollight.sum x
+ (%i::'A. hollight.sum xa ((f::'A => 'B => hollight.real) i)) =
+ hollight.sum xa (%j::'B. hollight.sum x (%i::'A. f i j))"
+ by (import hollight SUM_SWAP)
+
+lemma SUM_IMAGE: "(!!(xa::'q_69070) y::'q_69070.
+ xa : (xb::'q_69070 => bool) &
+ y : xb & (x::'q_69070 => 'q_69046) xa = x y
+ ==> xa = y)
+==> hollight.sum (x ` xb) (xa::'q_69046 => hollight.real) =
+ hollight.sum xb (xa o x)"
+ by (import hollight SUM_IMAGE)
+
+lemma SUM_SUPERSET: "(xa::'A => bool) <= (xb::'A => bool) &
+(ALL xc::'A.
+ xc : xb & xc ~: xa -->
+ (x::'A => hollight.real) xc = real_of_num (0::nat))
+==> hollight.sum xb x = hollight.sum xa x"
+ by (import hollight SUM_SUPERSET)
+
+lemma SUM_UNION_RZERO: "finite (u::'A => bool) &
+(ALL x::'A.
+ x : (v::'A => bool) & x ~: u -->
+ (f::'A => hollight.real) x = real_of_num (0::nat))
+==> hollight.sum (u Un v) f = hollight.sum u f"
+ by (import hollight SUM_UNION_RZERO)
+
+lemma SUM_UNION_LZERO: "finite (v::'A => bool) &
+(ALL x::'A.
+ x : (u::'A => bool) & x ~: v -->
+ (f::'A => hollight.real) x = real_of_num (0::nat))
+==> hollight.sum (u Un v) f = hollight.sum v f"
+ by (import hollight SUM_UNION_LZERO)
+
+lemma SUM_RESTRICT: "finite (s::'q_69267 => bool)
+==> hollight.sum s
+ (%x::'q_69267.
+ if x : s then (f::'q_69267 => hollight.real) x
+ else real_of_num (0::nat)) =
+ hollight.sum s f"
+ by (import hollight SUM_RESTRICT)
+
+lemma SUM_BOUND: "finite (x::'A => bool) &
+(ALL xc::'A.
+ xc : x --> real_le ((xa::'A => hollight.real) xc) (xb::hollight.real))
+==> real_le (hollight.sum x xa) (real_mul (real_of_num (CARD x)) xb)"
+ by (import hollight SUM_BOUND)
+
+lemma SUM_BOUND_GEN: "finite (s::'A => bool) &
+s ~= {} &
+(ALL x::'A.
+ x : s -->
+ real_le ((f::'A => hollight.real) x)
+ (real_div (b::hollight.real) (real_of_num (CARD s))))
+==> real_le (hollight.sum s f) b"
+ by (import hollight SUM_BOUND_GEN)
+
+lemma SUM_ABS_BOUND: "finite (s::'A => bool) &
+(ALL x::'A.
+ x : s -->
+ real_le (real_abs ((f::'A => hollight.real) x)) (b::hollight.real))
+==> real_le (real_abs (hollight.sum s f))
+ (real_mul (real_of_num (CARD s)) b)"
+ by (import hollight SUM_ABS_BOUND)
+
+lemma SUM_BOUND_LT: "finite (s::'A => bool) &
+(ALL x::'A.
+ x : s --> real_le ((f::'A => hollight.real) x) (b::hollight.real)) &
+(EX x::'A. x : s & real_lt (f x) b)
+==> real_lt (hollight.sum s f) (real_mul (real_of_num (CARD s)) b)"
+ by (import hollight SUM_BOUND_LT)
+
+lemma SUM_BOUND_LT_ALL: "finite (s::'q_69531 => bool) &
+s ~= {} &
+(ALL x::'q_69531.
+ x : s --> real_lt ((f::'q_69531 => hollight.real) x) (b::hollight.real))
+==> real_lt (hollight.sum s f) (real_mul (real_of_num (CARD s)) b)"
+ by (import hollight SUM_BOUND_LT_ALL)
+
+lemma SUM_BOUND_LT_GEN: "finite (s::'A => bool) &
+s ~= {} &
+(ALL x::'A.
+ x : s -->
+ real_lt ((f::'A => hollight.real) x)
+ (real_div (b::hollight.real) (real_of_num (CARD s))))
+==> real_lt (hollight.sum s f) b"
+ by (import hollight SUM_BOUND_LT_GEN)
+
+lemma SUM_UNION_EQ: "finite (u::'q_69614 => bool) &
+(s::'q_69614 => bool) Int (t::'q_69614 => bool) = {} & s Un t = u
+==> real_add (hollight.sum s (f::'q_69614 => hollight.real))
+ (hollight.sum t f) =
+ hollight.sum u f"
+ by (import hollight SUM_UNION_EQ)
+
+lemma SUM_EQ_SUPERSET: "finite (t::'A => bool) &
+t <= (s::'A => bool) &
+(ALL x::'A.
+ x : t --> (f::'A => hollight.real) x = (g::'A => hollight.real) x) &
+(ALL x::'A. x : s & x ~: t --> f x = real_of_num (0::nat))
+==> hollight.sum s f = hollight.sum t g"
+ by (import hollight SUM_EQ_SUPERSET)
+
+lemma SUM_RESTRICT_SET: "hollight.sum
+ {u::'q_69783.
+ EX xb::'q_69783.
+ (xb : (xa::'q_69783 => bool) & (x::'q_69783 => bool) xb) & u = xb}
+ (xb::'q_69783 => hollight.real) =
+hollight.sum xa
+ (%xa::'q_69783. if x xa then xb xa else real_of_num (0::nat))"
+ by (import hollight SUM_RESTRICT_SET)
+
+lemma SUM_SUM_RESTRICT: "finite (s::'q_69875 => bool) & finite (t::'q_69874 => bool)
+==> hollight.sum s
+ (%x::'q_69875.
+ hollight.sum
+ {u::'q_69874.
+ EX y::'q_69874.
+ (y : t & (R::'q_69875 => 'q_69874 => bool) x y) & u = y}
+ ((f::'q_69875 => 'q_69874 => hollight.real) x)) =
+ hollight.sum t
+ (%y::'q_69874.
+ hollight.sum {u::'q_69875. EX x::'q_69875. (x : s & R x y) & u = x}
+ (%x::'q_69875. f x y))"
+ by (import hollight SUM_SUM_RESTRICT)
+
+lemma CARD_EQ_SUM: "finite (x::'q_69896 => bool)
+==> real_of_num (CARD x) =
+ hollight.sum x (%x::'q_69896. real_of_num (1::nat))"
+ by (import hollight CARD_EQ_SUM)
+
+lemma SUM_MULTICOUNT_GEN: "finite (s::'A => bool) &
+finite (t::'B => bool) &
+(ALL j::'B.
+ j : t -->
+ CARD {u::'A. EX i::'A. (i : s & (R::'A => 'B => bool) i j) & u = i} =
+ (k::'B => nat) j)
+==> hollight.sum s
+ (%i::'A.
+ real_of_num (CARD {u::'B. EX j::'B. (j : t & R i j) & u = j})) =
+ hollight.sum t (%i::'B. real_of_num (k i))"
+ by (import hollight SUM_MULTICOUNT_GEN)
+
+lemma SUM_MULTICOUNT: "finite (s::'A => bool) &
+finite (t::'B => bool) &
+(ALL j::'B.
+ j : t -->
+ CARD {u::'A. EX i::'A. (i : s & (R::'A => 'B => bool) i j) & u = i} =
+ (k::nat))
+==> hollight.sum s
+ (%i::'A.
+ real_of_num (CARD {u::'B. EX j::'B. (j : t & R i j) & u = j})) =
+ real_of_num (k * CARD t)"
+ by (import hollight SUM_MULTICOUNT)
+
+lemma SUM_IMAGE_GEN: "finite (s::'A => bool)
+==> hollight.sum s (g::'A => hollight.real) =
+ hollight.sum ((f::'A => 'B) ` s)
+ (%y::'B. hollight.sum {u::'A. EX x::'A. (x : s & f x = y) & u = x} g)"
+ by (import hollight SUM_IMAGE_GEN)
+
+lemma SUM_GROUP: "finite (s::'A => bool) & (f::'A => 'B) ` s <= (t::'B => bool)
+==> hollight.sum t
+ (%y::'B.
+ hollight.sum {u::'A. EX x::'A. (x : s & f x = y) & u = x}
+ (g::'A => hollight.real)) =
+ hollight.sum s g"
+ by (import hollight SUM_GROUP)
+
+lemma REAL_OF_NUM_SUM: "finite (s::'q_70361 => bool)
+==> real_of_num (nsum s (f::'q_70361 => nat)) =
+ hollight.sum s (%x::'q_70361. real_of_num (f x))"
+ by (import hollight REAL_OF_NUM_SUM)
+
+lemma SUM_SUBSET: "finite (u::'A => bool) &
+finite (v::'A => bool) &
+(ALL x::'A.
+ x : u - v -->
+ real_le ((f::'A => hollight.real) x) (real_of_num (0::nat))) &
+(ALL x::'A. x : v - u --> real_le (real_of_num (0::nat)) (f x))
+==> real_le (hollight.sum u f) (hollight.sum v f)"
+ by (import hollight SUM_SUBSET)
+
+lemma SUM_SUBSET_SIMPLE: "finite (v::'A => bool) &
+(u::'A => bool) <= v &
+(ALL x::'A.
+ x : v - u -->
+ real_le (real_of_num (0::nat)) ((f::'A => hollight.real) x))
+==> real_le (hollight.sum u f) (hollight.sum v f)"
+ by (import hollight SUM_SUBSET_SIMPLE)
+
+lemma SUM_IMAGE_NONZERO: "finite (xb::'A => bool) &
+(ALL (xc::'A) xd::'A.
+ xc : xb & xd : xb & xc ~= xd & (xa::'A => 'B) xc = xa xd -->
+ (x::'B => hollight.real) (xa xc) = real_of_num (0::nat))
+==> hollight.sum (xa ` xb) x = hollight.sum xb (x o xa)"
+ by (import hollight SUM_IMAGE_NONZERO)
+
+lemma SUM_BIJECTION: "(ALL x::'A. x : (xb::'A => bool) --> (xa::'A => 'A) x : xb) &
+(ALL y::'A. y : xb --> (EX! x::'A. x : xb & xa x = y))
+==> hollight.sum xb (x::'A => hollight.real) = hollight.sum xb (x o xa)"
+ by (import hollight SUM_BIJECTION)
+
+lemma SUM_SUM_PRODUCT: "finite (x::'A => bool) &
+(ALL i::'A. i : x --> finite ((xa::'A => 'B => bool) i))
+==> hollight.sum x
+ (%x::'A. hollight.sum (xa x) ((xb::'A => 'B => hollight.real) x)) =
+ hollight.sum
+ {u::'A * 'B. EX (i::'A) j::'B. (i : x & j : xa i) & u = (i, j)}
+ (SOME f::'A * 'B => hollight.real.
+ ALL (i::'A) j::'B. f (i, j) = xb i j)"
+ by (import hollight SUM_SUM_PRODUCT)
+
+lemma SUM_EQ_GENERAL: "(ALL y::'B.
+ y : (xa::'B => bool) -->
+ (EX! xa::'A. xa : (x::'A => bool) & (xd::'A => 'B) xa = y)) &
+(ALL xe::'A.
+ xe : x -->
+ xd xe : xa &
+ (xc::'B => hollight.real) (xd xe) = (xb::'A => hollight.real) xe)
+==> hollight.sum x xb = hollight.sum xa xc"
+ by (import hollight SUM_EQ_GENERAL)
+
+lemma SUM_EQ_GENERAL_INVERSES: "(ALL y::'B.
+ y : (xa::'B => bool) -->
+ (xe::'B => 'A) y : (x::'A => bool) & (xd::'A => 'B) (xe y) = y) &
+(ALL xf::'A.
+ xf : x -->
+ xd xf : xa &
+ xe (xd xf) = xf &
+ (xc::'B => hollight.real) (xd xf) = (xb::'A => hollight.real) xf)
+==> hollight.sum x xb = hollight.sum xa xc"
+ by (import hollight SUM_EQ_GENERAL_INVERSES)
+
+lemma SUM_INJECTION: "finite (xb::'q_71007 => bool) &
+(ALL x::'q_71007. x : xb --> (xa::'q_71007 => 'q_71007) x : xb) &
+(ALL (x::'q_71007) y::'q_71007. x : xb & y : xb & xa x = xa y --> x = y)
+==> hollight.sum xb ((x::'q_71007 => hollight.real) o xa) =
+ hollight.sum xb x"
+ by (import hollight SUM_INJECTION)
+
+lemma SUM_UNION_NONZERO: "finite (xa::'q_71050 => bool) &
+finite (xb::'q_71050 => bool) &
+(ALL xc::'q_71050.
+ xc : xa Int xb -->
+ (x::'q_71050 => hollight.real) xc = real_of_num (0::nat))
+==> hollight.sum (xa Un xb) x =
+ real_add (hollight.sum xa x) (hollight.sum xb x)"
+ by (import hollight SUM_UNION_NONZERO)
+
+lemma SUM_UNIONS_NONZERO: "finite (x::('A => bool) => bool) &
+(ALL t::'A => bool. t : x --> finite t) &
+(ALL (t1::'A => bool) (t2::'A => bool) xa::'A.
+ t1 : x & t2 : x & t1 ~= t2 & xa : t1 & xa : t2 -->
+ (f::'A => hollight.real) xa = real_of_num (0::nat))
+==> hollight.sum (Union x) f =
+ hollight.sum x (%t::'A => bool. hollight.sum t f)"
+ by (import hollight SUM_UNIONS_NONZERO)
+
+lemma SUM_CASES: "finite (x::'A => bool)
+==> hollight.sum x
+ (%x::'A.
+ if (xa::'A => bool) x then (xb::'A => hollight.real) x
+ else (xc::'A => hollight.real) x) =
+ real_add (hollight.sum {u::'A. EX xb::'A. (xb : x & xa xb) & u = xb} xb)
+ (hollight.sum {u::'A. EX xb::'A. (xb : x & ~ xa xb) & u = xb} xc)"
+ by (import hollight SUM_CASES)
+
+lemma SUM_CASES_1: "finite (s::'q_71319 => bool) & (a::'q_71319) : s
+==> hollight.sum s
+ (%x::'q_71319.
+ if x = a then y::hollight.real
+ else (f::'q_71319 => hollight.real) x) =
+ real_add (hollight.sum s f) (real_sub y (f a))"
+ by (import hollight SUM_CASES_1)
+
+lemma SUM_LE_INCLUDED: "finite (s::'A => bool) &
+finite (t::'B => bool) &
+(ALL y::'B.
+ y : t --> real_le (real_of_num (0::nat)) ((g::'B => hollight.real) y)) &
+(ALL x::'A.
+ x : s -->
+ (EX y::'B.
+ y : t &
+ (i::'B => 'A) y = x & real_le ((f::'A => hollight.real) x) (g y)))
+==> real_le (hollight.sum s f) (hollight.sum t g)"
+ by (import hollight SUM_LE_INCLUDED)
+
+lemma SUM_IMAGE_LE: "finite (s::'A => bool) &
+(ALL x::'A.
+ x : s -->
+ real_le (real_of_num (0::nat))
+ ((g::'B => hollight.real) ((f::'A => 'B) x)))
+==> real_le (hollight.sum (f ` s) g) (hollight.sum s (g o f))"
+ by (import hollight SUM_IMAGE_LE)
+
+lemma SUM_CLOSED: "(P::hollight.real => bool) (real_of_num (0::nat)) &
+(ALL (x::hollight.real) y::hollight.real. P x & P y --> P (real_add x y)) &
+(ALL a::'A. a : (s::'A => bool) --> P ((f::'A => hollight.real) a))
+==> P (hollight.sum s f)"
+ by (import hollight SUM_CLOSED)
+
+lemma SUM_ADD_NUMSEG: "hollight.sum {xb::nat..xc::nat}
+ (%i::nat.
+ real_add ((x::nat => hollight.real) i)
+ ((xa::nat => hollight.real) i)) =
+real_add (hollight.sum {xb..xc} x) (hollight.sum {xb..xc} xa)"
+ by (import hollight SUM_ADD_NUMSEG)
+
+lemma SUM_SUB_NUMSEG: "hollight.sum {xb::nat..xc::nat}
+ (%i::nat.
+ real_sub ((x::nat => hollight.real) i)
+ ((xa::nat => hollight.real) i)) =
+real_sub (hollight.sum {xb..xc} x) (hollight.sum {xb..xc} xa)"
+ by (import hollight SUM_SUB_NUMSEG)
+
+lemma SUM_LE_NUMSEG: "(!!i::nat.
+ (xb::nat) <= i & i <= (xc::nat)
+ ==> real_le ((x::nat => hollight.real) i)
+ ((xa::nat => hollight.real) i))
+==> real_le (hollight.sum {xb..xc} x) (hollight.sum {xb..xc} xa)"
+ by (import hollight SUM_LE_NUMSEG)
+
+lemma SUM_EQ_NUMSEG: "(!!i::nat.
+ (m::nat) <= i & i <= (n::nat)
+ ==> (f::nat => hollight.real) i = (g::nat => hollight.real) i)
+==> hollight.sum {m..n} f = hollight.sum {m..n} g"
+ by (import hollight SUM_EQ_NUMSEG)
+
+lemma SUM_ABS_NUMSEG: "real_le
+ (real_abs (hollight.sum {xa::nat..xb::nat} (x::nat => hollight.real)))
+ (hollight.sum {xa..xb} (%i::nat. real_abs (x i)))"
+ by (import hollight SUM_ABS_NUMSEG)
+
+lemma SUM_CONST_NUMSEG: "hollight.sum {xa..xb} (%n. x) = real_mul (real_of_num (xb + 1 - xa)) x"
+ by (import hollight SUM_CONST_NUMSEG)
+
+lemma SUM_EQ_0_NUMSEG: "(!!i::nat.
+ (m::nat) <= i & i <= (n::nat)
+ ==> (x::nat => hollight.real) i = real_of_num (0::nat))
+==> hollight.sum {m..n} x = real_of_num (0::nat)"
+ by (import hollight SUM_EQ_0_NUMSEG)
+
+lemma SUM_TRIV_NUMSEG: "(n::nat) < (m::nat)
+==> hollight.sum {m..n} (f::nat => hollight.real) = real_of_num (0::nat)"
+ by (import hollight SUM_TRIV_NUMSEG)
+
+lemma SUM_POS_LE_NUMSEG: "(!!p::nat.
+ (x::nat) <= p & p <= (xa::nat)
+ ==> real_le (real_of_num (0::nat)) ((xb::nat => hollight.real) p))
+==> real_le (real_of_num (0::nat)) (hollight.sum {x..xa} xb)"
+ by (import hollight SUM_POS_LE_NUMSEG)
+
+lemma SUM_POS_EQ_0_NUMSEG: "[| (ALL p::nat.
+ (m::nat) <= p & p <= (n::nat) -->
+ real_le (real_of_num (0::nat)) ((f::nat => hollight.real) p)) &
+ hollight.sum {m..n} f = real_of_num (0::nat);
+ m <= (p::nat) & p <= n |]
+==> f p = real_of_num (0::nat)"
+ by (import hollight SUM_POS_EQ_0_NUMSEG)
+
+lemma SUM_SING_NUMSEG: "hollight.sum {xa::nat..xa} (x::nat => hollight.real) = x xa"
+ by (import hollight SUM_SING_NUMSEG)
+
+lemma SUM_CLAUSES_NUMSEG: "(ALL m. hollight.sum {m..0} f = (if m = 0 then f 0 else real_of_num 0)) &
+(ALL m n.
+ hollight.sum {m..Suc n} f =
+ (if m <= Suc n then real_add (hollight.sum {m..n} f) (f (Suc n))
+ else hollight.sum {m..n} f))"
+ by (import hollight SUM_CLAUSES_NUMSEG)
+
+lemma SUM_SWAP_NUMSEG: "hollight.sum {a::nat..b::nat}
+ (%i::nat.
+ hollight.sum {c::nat..d::nat} ((f::nat => nat => hollight.real) i)) =
+hollight.sum {c..d} (%j::nat. hollight.sum {a..b} (%i::nat. f i j))"
+ by (import hollight SUM_SWAP_NUMSEG)
+
+lemma SUM_ADD_SPLIT: "(xa::nat) <= (xb::nat) + (1::nat)
+==> hollight.sum {xa..xb + (xc::nat)} (x::nat => hollight.real) =
+ real_add (hollight.sum {xa..xb} x)
+ (hollight.sum {xb + (1::nat)..xb + xc} x)"
+ by (import hollight SUM_ADD_SPLIT)
+
+lemma SUM_OFFSET: "hollight.sum {(xb::nat) + (x::nat)..(xc::nat) + x}
+ (xa::nat => hollight.real) =
+hollight.sum {xb..xc} (%i::nat. xa (i + x))"
+ by (import hollight SUM_OFFSET)
+
+lemma SUM_OFFSET_0: "(xa::nat) <= (xb::nat)
+==> hollight.sum {xa..xb} (x::nat => hollight.real) =
+ hollight.sum {0::nat..xb - xa} (%i::nat. x (i + xa))"
+ by (import hollight SUM_OFFSET_0)
+
+lemma SUM_CLAUSES_LEFT: "(xa::nat) <= (xb::nat)
+==> hollight.sum {xa..xb} (x::nat => hollight.real) =
+ real_add (x xa) (hollight.sum {xa + (1::nat)..xb} x)"
+ by (import hollight SUM_CLAUSES_LEFT)
+
+lemma SUM_CLAUSES_RIGHT: "(0::nat) < (n::nat) & (m::nat) <= n
+==> hollight.sum {m..n} (f::nat => hollight.real) =
+ real_add (hollight.sum {m..n - (1::nat)} f) (f n)"
+ by (import hollight SUM_CLAUSES_RIGHT)
+
+lemma SUM_PAIR: "hollight.sum {(2::nat) * (m::nat)..(2::nat) * (n::nat) + (1::nat)}
+ (f::nat => hollight.real) =
+hollight.sum {m..n}
+ (%i::nat. real_add (f ((2::nat) * i)) (f ((2::nat) * i + (1::nat))))"
+ by (import hollight SUM_PAIR)
+
+lemma REAL_OF_NUM_SUM_NUMSEG: "real_of_num (nsum {xa::nat..xb::nat} (x::nat => nat)) =
+hollight.sum {xa..xb} (%i::nat. real_of_num (x i))"
+ by (import hollight REAL_OF_NUM_SUM_NUMSEG)
+
+lemma SUM_PARTIAL_SUC: "hollight.sum {m::nat..n::nat}
+ (%k::nat.
+ real_mul ((f::nat => hollight.real) k)
+ (real_sub ((g::nat => hollight.real) (k + (1::nat))) (g k))) =
+(if m <= n
+ then real_sub
+ (real_sub (real_mul (f (n + (1::nat))) (g (n + (1::nat))))
+ (real_mul (f m) (g m)))
+ (hollight.sum {m..n}
+ (%k::nat.
+ real_mul (g (k + (1::nat)))
+ (real_sub (f (k + (1::nat))) (f k))))
+ else real_of_num (0::nat))"
+ by (import hollight SUM_PARTIAL_SUC)
+
+lemma SUM_PARTIAL_PRE: "hollight.sum {m::nat..n::nat}
+ (%k::nat.
+ real_mul ((f::nat => hollight.real) k)
+ (real_sub ((g::nat => hollight.real) k) (g (k - (1::nat))))) =
+(if m <= n
+ then real_sub
+ (real_sub (real_mul (f (n + (1::nat))) (g n))
+ (real_mul (f m) (g (m - (1::nat)))))
+ (hollight.sum {m..n}
+ (%k::nat. real_mul (g k) (real_sub (f (k + (1::nat))) (f k))))
+ else real_of_num (0::nat))"
+ by (import hollight SUM_PARTIAL_PRE)
+
+lemma SUM_DIFFS: "hollight.sum {x::nat..xa::nat}
+ (%x::nat. real_sub ((f::nat => hollight.real) x) (f (x + (1::nat)))) =
+(if x <= xa then real_sub (f x) (f (xa + (1::nat)))
+ else real_of_num (0::nat))"
+ by (import hollight SUM_DIFFS)
+
+lemma SUM_DIFFS_ALT: "hollight.sum {m::nat..n::nat}
+ (%x::nat. real_sub ((f::nat => hollight.real) (x + (1::nat))) (f x)) =
+(if m <= n then real_sub (f (n + (1::nat))) (f m) else real_of_num (0::nat))"
+ by (import hollight SUM_DIFFS_ALT)
+
+lemma SUM_COMBINE_R: "(m::nat) <= (n::nat) + (1::nat) & n <= (p::nat)
+==> real_add (hollight.sum {m..n} (f::nat => hollight.real))
+ (hollight.sum {n + (1::nat)..p} f) =
+ hollight.sum {m..p} f"
+ by (import hollight SUM_COMBINE_R)
+
+lemma SUM_COMBINE_L: "(0::nat) < (n::nat) & (m::nat) <= n & n <= (p::nat) + (1::nat)
+==> real_add (hollight.sum {m..n - (1::nat)} (f::nat => hollight.real))
+ (hollight.sum {n..p} f) =
+ hollight.sum {m..p} f"
+ by (import hollight SUM_COMBINE_L)
+
+lemma REAL_SUB_POW: "1 <= xb
+==> real_sub (real_pow x xb) (real_pow xa xb) =
+ real_mul (real_sub x xa)
+ (hollight.sum {0..xb - 1}
+ (%i. real_mul (real_pow x i) (real_pow xa (xb - 1 - i))))"
+ by (import hollight REAL_SUB_POW)
+
+lemma REAL_SUB_POW_R1: "1 <= n
+==> real_sub (real_pow x n) (real_of_num 1) =
+ real_mul (real_sub x (real_of_num 1))
+ (hollight.sum {0..n - 1} (real_pow x))"
+ by (import hollight REAL_SUB_POW_R1)
+
+lemma REAL_SUB_POW_L1: "1 <= xa
+==> real_sub (real_of_num 1) (real_pow x xa) =
+ real_mul (real_sub (real_of_num 1) x)
+ (hollight.sum {0..xa - 1} (real_pow x))"
+ by (import hollight REAL_SUB_POW_L1)
+
+definition
+ dimindex :: "('A => bool) => nat" where
+ "(op ==::(('A::type => bool) => nat) => (('A::type => bool) => nat) => prop)
+ (dimindex::('A::type => bool) => nat)
+ (%u::'A::type => bool.
+ (If::bool => nat => nat => nat)
+ ((finite::('A::type => bool) => bool) (UNIV::'A::type => bool))
+ ((CARD::('A::type => bool) => nat) (UNIV::'A::type => bool)) (1::nat))"
+
+lemma DEF_dimindex: "(op =::(('A::type => bool) => nat) => (('A::type => bool) => nat) => bool)
+ (dimindex::('A::type => bool) => nat)
+ (%u::'A::type => bool.
+ (If::bool => nat => nat => nat)
+ ((finite::('A::type => bool) => bool) (UNIV::'A::type => bool))
+ ((CARD::('A::type => bool) => nat) (UNIV::'A::type => bool)) (1::nat))"
+ by (import hollight DEF_dimindex)
+
+lemma DIMINDEX_NONZERO: "dimindex (s::'A => bool) ~= (0::nat)"
+ by (import hollight DIMINDEX_NONZERO)
+
+lemma DIMINDEX_GE_1: "(1::nat) <= dimindex (x::'A => bool)"
+ by (import hollight DIMINDEX_GE_1)
+
+lemma DIMINDEX_UNIV: "(op =::nat => nat => bool)
+ ((dimindex::('A::type => bool) => nat) (x::'A::type => bool))
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool))"
+ by (import hollight DIMINDEX_UNIV)
+
+lemma DIMINDEX_UNIQUE: "(op ==>::prop => prop => prop)
+ ((Trueprop::bool => prop)
+ ((HAS_SIZE::('A::type => bool) => nat => bool) (UNIV::'A::type => bool)
+ (n::nat)))
+ ((Trueprop::bool => prop)
+ ((op =::nat => nat => bool)
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool)) n))"
+ by (import hollight DIMINDEX_UNIQUE)
+
+typedef (open) ('A) finite_image = "{x::nat. x : dotdot (NUMERAL (NUMERAL_BIT1 (0::nat))) (dimindex UNIV)}" morphisms "dest_finite_image" "finite_index"
+ apply (rule light_ex_imp_nonempty[where t="SOME x::nat. x : dotdot (NUMERAL (NUMERAL_BIT1 (0::nat))) (dimindex UNIV)"])
by (import hollight TYDEF_finite_image)
syntax
dest_finite_image :: _
syntax
- mk_finite_image :: _
+ finite_index :: _
lemmas "TYDEF_finite_image_@intern" = typedef_hol2hollight
[where a="a :: 'A finite_image" and r=r ,
@@ -6914,160 +6644,61 @@
lemma FINITE_IMAGE_IMAGE: "(op =::('A::type finite_image => bool)
=> ('A::type finite_image => bool) => bool)
- (hollight.UNIV::'A::type finite_image => bool)
- ((IMAGE::('A::type => 'A::type finite_image)
- => ('A::type => bool) => 'A::type finite_image => bool)
- (mk_finite_image::'A::type => 'A::type finite_image)
- ((COND::bool
- => ('A::type => bool) => ('A::type => bool) => 'A::type => bool)
- ((FINITE::('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool))
- (hollight.UNIV::'A::type => bool)
- ((INSERT::'A::type => ('A::type => bool) => 'A::type => bool)
- ((Eps::('A::type => bool) => 'A::type) (%z::'A::type. True::bool))
- (EMPTY::'A::type => bool))))"
+ (UNIV::'A::type finite_image => bool)
+ ((op `::(nat => 'A::type finite_image)
+ => (nat => bool) => 'A::type finite_image => bool)
+ (finite_index::nat => 'A::type finite_image)
+ ((atLeastAtMost::nat => nat => nat => bool) (1::nat)
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool))))"
by (import hollight FINITE_IMAGE_IMAGE)
-definition dimindex :: "('A => bool) => nat" where
- "(op ==::(('A::type => bool) => nat) => (('A::type => bool) => nat) => prop)
- (dimindex::('A::type => bool) => nat)
- (%u::'A::type => bool.
- (COND::bool => nat => nat => nat)
- ((FINITE::('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool))
- ((CARD::('A::type => bool) => nat) (hollight.UNIV::'A::type => bool))
- ((NUMERAL_BIT1::nat => nat) (0::nat)))"
-
-lemma DEF_dimindex: "(op =::(('A::type => bool) => nat) => (('A::type => bool) => nat) => bool)
- (dimindex::('A::type => bool) => nat)
- (%u::'A::type => bool.
- (COND::bool => nat => nat => nat)
- ((FINITE::('A::type => bool) => bool)
- (hollight.UNIV::'A::type => bool))
- ((CARD::('A::type => bool) => nat) (hollight.UNIV::'A::type => bool))
- ((NUMERAL_BIT1::nat => nat) (0::nat)))"
- by (import hollight DEF_dimindex)
-
-lemma HAS_SIZE_FINITE_IMAGE: "(All::(('A::type => bool) => bool) => bool)
- (%s::'A::type => bool.
- (HAS_SIZE::('A::type finite_image => bool) => nat => bool)
- (hollight.UNIV::'A::type finite_image => bool)
- ((dimindex::('A::type => bool) => nat) s))"
+lemma HAS_SIZE_FINITE_IMAGE: "(HAS_SIZE::('A::type finite_image => bool) => nat => bool)
+ (UNIV::'A::type finite_image => bool)
+ ((dimindex::('A::type => bool) => nat) (s::'A::type => bool))"
by (import hollight HAS_SIZE_FINITE_IMAGE)
-lemma CARD_FINITE_IMAGE: "(All::(('A::type => bool) => bool) => bool)
- (%s::'A::type => bool.
- (op =::nat => nat => bool)
- ((CARD::('A::type finite_image => bool) => nat)
- (hollight.UNIV::'A::type finite_image => bool))
- ((dimindex::('A::type => bool) => nat) s))"
+lemma CARD_FINITE_IMAGE: "(op =::nat => nat => bool)
+ ((CARD::('A::type finite_image => bool) => nat)
+ (UNIV::'A::type finite_image => bool))
+ ((dimindex::('A::type => bool) => nat) (s::'A::type => bool))"
by (import hollight CARD_FINITE_IMAGE)
-lemma FINITE_FINITE_IMAGE: "(FINITE::('A::type finite_image => bool) => bool)
- (hollight.UNIV::'A::type finite_image => bool)"
+lemma FINITE_FINITE_IMAGE: "(finite::('A::type finite_image => bool) => bool)
+ (UNIV::'A::type finite_image => bool)"
by (import hollight FINITE_FINITE_IMAGE)
-lemma DIMINDEX_NONZERO: "ALL s::'A::type => bool. dimindex s ~= 0"
- by (import hollight DIMINDEX_NONZERO)
-
-lemma DIMINDEX_GE_1: "ALL x::'A::type => bool. <= (NUMERAL_BIT1 0) (dimindex x)"
- by (import hollight DIMINDEX_GE_1)
-
-lemma DIMINDEX_FINITE_IMAGE: "ALL (s::'A::type finite_image => bool) t::'A::type => bool.
- dimindex s = dimindex t"
+lemma DIMINDEX_FINITE_IMAGE: "dimindex (s::'A finite_image => bool) = dimindex (t::'A => bool)"
by (import hollight DIMINDEX_FINITE_IMAGE)
-definition finite_index :: "nat => 'A" where
- "(op ==::(nat => 'A::type) => (nat => 'A::type) => prop)
- (finite_index::nat => 'A::type)
- ((Eps::((nat => 'A::type) => bool) => nat => 'A::type)
- (%f::nat => 'A::type.
- (All::('A::type => bool) => bool)
- (%x::'A::type.
- (Ex1::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) n)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) n
- ((dimindex::('A::type => bool) => nat)
- (hollight.UNIV::'A::type => bool)))
- ((op =::'A::type => 'A::type => bool) (f n) x))))))"
-
-lemma DEF_finite_index: "(op =::(nat => 'A::type) => (nat => 'A::type) => bool)
- (finite_index::nat => 'A::type)
- ((Eps::((nat => 'A::type) => bool) => nat => 'A::type)
- (%f::nat => 'A::type.
- (All::('A::type => bool) => bool)
- (%x::'A::type.
- (Ex1::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) n)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) n
- ((dimindex::('A::type => bool) => nat)
- (hollight.UNIV::'A::type => bool)))
- ((op =::'A::type => 'A::type => bool) (f n) x))))))"
- by (import hollight DEF_finite_index)
-
-lemma FINITE_INDEX_WORKS_FINITE: "(op -->::bool => bool => bool)
- ((FINITE::('N::type => bool) => bool) (hollight.UNIV::'N::type => bool))
- ((All::('N::type => bool) => bool)
- (%x::'N::type.
- (Ex1::(nat => bool) => bool)
- (%xa::nat.
- (op &::bool => bool => bool)
- ((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat))
- xa)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) xa
- ((dimindex::('N::type => bool) => nat)
- (hollight.UNIV::'N::type => bool)))
- ((op =::'N::type => 'N::type => bool)
- ((finite_index::nat => 'N::type) xa) x)))))"
- by (import hollight FINITE_INDEX_WORKS_FINITE)
-
-lemma FINITE_INDEX_WORKS: "(All::('A::type finite_image => bool) => bool)
- (%i::'A::type finite_image.
- (Ex1::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat))
- n)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) n
- ((dimindex::('A::type => bool) => nat)
- (hollight.UNIV::'A::type => bool)))
- ((op =::'A::type finite_image => 'A::type finite_image => bool)
- ((finite_index::nat => 'A::type finite_image) n) i))))"
+lemma FINITE_INDEX_WORKS: "(Ex1::(nat => bool) => bool)
+ (%xa::nat.
+ (op &::bool => bool => bool) ((op <=::nat => nat => bool) (1::nat) xa)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) xa
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool)))
+ ((op =::'A::type finite_image => 'A::type finite_image => bool)
+ ((finite_index::nat => 'A::type finite_image) xa)
+ (x::'A::type finite_image))))"
by (import hollight FINITE_INDEX_WORKS)
-lemma FINITE_INDEX_INJ: "(All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat))
- x)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) x
- ((dimindex::('A::type => bool) => nat)
- (hollight.UNIV::'A::type => bool)))
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) xa)
- ((<=::nat => nat => bool) xa
- ((dimindex::('A::type => bool) => nat)
- (hollight.UNIV::'A::type => bool))))))
- ((op =::bool => bool => bool)
- ((op =::'A::type => 'A::type => bool)
- ((finite_index::nat => 'A::type) x)
- ((finite_index::nat => 'A::type) xa))
- ((op =::nat => nat => bool) x xa))))"
+lemma FINITE_INDEX_INJ: "(op ==>::prop => prop => prop)
+ ((Trueprop::bool => prop)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) (i::nat))
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) i
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool)))
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) (j::nat))
+ ((op <=::nat => nat => bool) j
+ ((dimindex::('A::type => bool) => nat)
+ (UNIV::'A::type => bool)))))))
+ ((Trueprop::bool => prop)
+ ((op =::bool => bool => bool)
+ ((op =::'A::type finite_image => 'A::type finite_image => bool)
+ ((finite_index::nat => 'A::type finite_image) i)
+ ((finite_index::nat => 'A::type finite_image) j))
+ ((op =::nat => nat => bool) i j)))"
by (import hollight FINITE_INDEX_INJ)
lemma FORALL_FINITE_INDEX: "(op =::bool => bool => bool)
@@ -7077,19 +6708,15 @@
(%i::nat.
(op -->::bool => bool => bool)
((op &::bool => bool => bool)
- ((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat)) i)
- ((<=::nat => nat => bool) i
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
((dimindex::('N::type => bool) => nat)
- (hollight.UNIV::'N::type => bool))))
+ (UNIV::'N::type => bool))))
(P ((finite_index::nat => 'N::type finite_image) i))))"
by (import hollight FORALL_FINITE_INDEX)
-typedef (open) ('A, 'B) cart = "(Collect::(('B::type finite_image => 'A::type) => bool)
- => ('B::type finite_image => 'A::type) set)
- (%f::'B::type finite_image => 'A::type. True::bool)" morphisms "dest_cart" "mk_cart"
- apply (rule light_ex_imp_nonempty[where t="(Eps::(('B::type finite_image => 'A::type) => bool)
- => 'B::type finite_image => 'A::type)
- (%f::'B::type finite_image => 'A::type. True::bool)"])
+typedef (open) ('A, 'B) cart = "{f. True}" morphisms "dest_cart" "mk_cart"
+ apply (rule light_ex_imp_nonempty[where t="SOME f. True"])
by (import hollight TYDEF_cart)
syntax
@@ -7103,42 +6730,32 @@
OF type_definition_cart]
consts
- "$" :: "('q_46627, 'q_46634) cart => nat => 'q_46627" ("$")
+ "$" :: "('q_73536, 'q_73546) cart => nat => 'q_73536" ("$")
defs
- "$_def": "$ ==
-%(u::('q_46627::type, 'q_46634::type) cart) ua::nat.
- dest_cart u (finite_index ua)"
-
-lemma "DEF_$": "$ =
-(%(u::('q_46627::type, 'q_46634::type) cart) ua::nat.
- dest_cart u (finite_index ua))"
+ "$_def": "$ == %(u::('q_73536, 'q_73546) cart) ua::nat. dest_cart u (finite_index ua)"
+
+lemma "DEF_$": "$ = (%(u::('q_73536, 'q_73546) cart) ua::nat. dest_cart u (finite_index ua))"
by (import hollight "DEF_$")
-lemma CART_EQ: "(All::(('A::type, 'B::type) cart => bool) => bool)
- (%x::('A::type, 'B::type) cart.
- (All::(('A::type, 'B::type) cart => bool) => bool)
- (%y::('A::type, 'B::type) cart.
- (op =::bool => bool => bool)
- ((op =::('A::type, 'B::type) cart
- => ('A::type, 'B::type) cart => bool)
- x y)
- ((All::(nat => bool) => bool)
- (%xa::nat.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) xa)
- ((<=::nat => nat => bool) xa
- ((dimindex::('B::type => bool) => nat)
- (hollight.UNIV::'B::type => bool))))
- ((op =::'A::type => 'A::type => bool)
- (($::('A::type, 'B::type) cart => nat => 'A::type) x xa)
- (($::('A::type, 'B::type) cart => nat => 'A::type) y
- xa))))))"
+lemma CART_EQ: "(op =::bool => bool => bool)
+ ((op =::('A::type, 'B::type) cart => ('A::type, 'B::type) cart => bool)
+ (x::('A::type, 'B::type) cart) (y::('A::type, 'B::type) cart))
+ ((All::(nat => bool) => bool)
+ (%xa::nat.
+ (op -->::bool => bool => bool)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) xa)
+ ((op <=::nat => nat => bool) xa
+ ((dimindex::('B::type => bool) => nat)
+ (UNIV::'B::type => bool))))
+ ((op =::'A::type => 'A::type => bool)
+ (($::('A::type, 'B::type) cart => nat => 'A::type) x xa)
+ (($::('A::type, 'B::type) cart => nat => 'A::type) y xa))))"
by (import hollight CART_EQ)
-definition lambda :: "(nat => 'A) => ('A, 'B) cart" where
+definition
+ lambda :: "(nat => 'A) => ('A, 'B) cart" where
"(op ==::((nat => 'A::type) => ('A::type, 'B::type) cart)
=> ((nat => 'A::type) => ('A::type, 'B::type) cart) => prop)
(lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
@@ -7149,11 +6766,10 @@
(%i::nat.
(op -->::bool => bool => bool)
((op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) i)
- ((<=::nat => nat => bool) i
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
((dimindex::('B::type => bool) => nat)
- (hollight.UNIV::'B::type => bool))))
+ (UNIV::'B::type => bool))))
((op =::'A::type => 'A::type => bool)
(($::('A::type, 'B::type) cart => nat => 'A::type) f i)
(u i)))))"
@@ -7168,64 +6784,73 @@
(%i::nat.
(op -->::bool => bool => bool)
((op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) i)
- ((<=::nat => nat => bool) i
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
((dimindex::('B::type => bool) => nat)
- (hollight.UNIV::'B::type => bool))))
+ (UNIV::'B::type => bool))))
((op =::'A::type => 'A::type => bool)
(($::('A::type, 'B::type) cart => nat => 'A::type) f i)
(u i)))))"
by (import hollight DEF_lambda)
-lemma LAMBDA_BETA: "(All::(nat => bool) => bool)
- (%x::nat.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat)) x)
- ((<=::nat => nat => bool) x
- ((dimindex::('B::type => bool) => nat)
- (hollight.UNIV::'B::type => bool))))
- ((op =::'A::type => 'A::type => bool)
- (($::('A::type, 'B::type) cart => nat => 'A::type)
- ((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
- (g::nat => 'A::type))
- x)
- (g x)))"
+lemma LAMBDA_BETA: "(op ==>::prop => prop => prop)
+ ((Trueprop::bool => prop)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) (x::nat))
+ ((op <=::nat => nat => bool) x
+ ((dimindex::('B::type => bool) => nat) (UNIV::'B::type => bool)))))
+ ((Trueprop::bool => prop)
+ ((op =::'A::type => 'A::type => bool)
+ (($::('A::type, 'B::type) cart => nat => 'A::type)
+ ((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
+ (g::nat => 'A::type))
+ x)
+ (g x)))"
by (import hollight LAMBDA_BETA)
-lemma LAMBDA_UNIQUE: "(All::(('A::type, 'B::type) cart => bool) => bool)
- (%x::('A::type, 'B::type) cart.
- (All::((nat => 'A::type) => bool) => bool)
- (%xa::nat => 'A::type.
- (op =::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%i::nat.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((<=::nat => nat => bool)
- ((NUMERAL_BIT1::nat => nat) (0::nat)) i)
- ((<=::nat => nat => bool) i
- ((dimindex::('B::type => bool) => nat)
- (hollight.UNIV::'B::type => bool))))
- ((op =::'A::type => 'A::type => bool)
- (($::('A::type, 'B::type) cart => nat => 'A::type) x i)
- (xa i))))
- ((op =::('A::type, 'B::type) cart
- => ('A::type, 'B::type) cart => bool)
- ((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart) xa)
- x)))"
+lemma LAMBDA_UNIQUE: "(op =::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%i::nat.
+ (op -->::bool => bool => bool)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
+ ((dimindex::('B::type => bool) => nat)
+ (UNIV::'B::type => bool))))
+ ((op =::'A::type => 'A::type => bool)
+ (($::('A::type, 'B::type) cart => nat => 'A::type)
+ (x::('A::type, 'B::type) cart) i)
+ ((xa::nat => 'A::type) i))))
+ ((op =::('A::type, 'B::type) cart => ('A::type, 'B::type) cart => bool)
+ ((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart) xa) x)"
by (import hollight LAMBDA_UNIQUE)
-lemma LAMBDA_ETA: "ALL x::('q_46825::type, 'q_46829::type) cart. lambda ($ x) = x"
+lemma LAMBDA_ETA: "lambda ($ (x::('q_73734, 'q_73738) cart)) = x"
by (import hollight LAMBDA_ETA)
-typedef (open) ('A, 'B) finite_sum = "(Collect::('A::type finite_image + 'B::type finite_image => bool)
- => ('A::type finite_image + 'B::type finite_image) set)
- (%x::'A::type finite_image + 'B::type finite_image. True::bool)" morphisms "dest_finite_sum" "mk_finite_sum"
- apply (rule light_ex_imp_nonempty[where t="(Eps::('A::type finite_image + 'B::type finite_image => bool)
- => 'A::type finite_image + 'B::type finite_image)
- (%x::'A::type finite_image + 'B::type finite_image. True::bool)"])
+lemma FINITE_INDEX_INRANGE: "(Ex::(nat => bool) => bool)
+ (%xa::nat.
+ (op &::bool => bool => bool) ((op <=::nat => nat => bool) (1::nat) xa)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) xa
+ ((dimindex::('N::type => bool) => nat) (UNIV::'N::type => bool)))
+ ((All::(('A::type, 'N::type) cart => bool) => bool)
+ (%xb::('A::type, 'N::type) cart.
+ (op =::'A::type => 'A::type => bool)
+ (($::('A::type, 'N::type) cart => nat => 'A::type) xb
+ (x::nat))
+ (($::('A::type, 'N::type) cart => nat => 'A::type) xb xa)))))"
+ by (import hollight FINITE_INDEX_INRANGE)
+
+lemma CART_EQ_FULL: "((x::('A, 'N) cart) = (y::('A, 'N) cart)) = (ALL i::nat. $ x i = $ y i)"
+ by (import hollight CART_EQ_FULL)
+
+typedef (open) ('A, 'B) finite_sum = "{x::nat.
+ x : dotdot (NUMERAL (NUMERAL_BIT1 (0::nat)))
+ (dimindex UNIV + dimindex UNIV)}" morphisms "dest_finite_sum" "mk_finite_sum"
+ apply (rule light_ex_imp_nonempty[where t="SOME x::nat.
+ x : dotdot (NUMERAL (NUMERAL_BIT1 (0::nat)))
+ (dimindex UNIV + dimindex UNIV)"])
by (import hollight TYDEF_finite_sum)
syntax
@@ -7238,7 +6863,8 @@
[where a="a :: ('A, 'B) finite_sum" and r=r ,
OF type_definition_finite_sum]
-definition pastecart :: "('A, 'M) cart => ('A, 'N) cart => ('A, ('M, 'N) finite_sum) cart" where
+definition
+ pastecart :: "('A, 'M) cart => ('A, 'N) cart => ('A, ('M, 'N) finite_sum) cart" where
"(op ==::(('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
@@ -7253,15 +6879,15 @@
(lambda::(nat => 'A::type)
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
(%i::nat.
- (COND::bool => 'A::type => 'A::type => 'A::type)
- ((<=::nat => nat => bool) i
+ (If::bool => 'A::type => 'A::type => 'A::type)
+ ((op <=::nat => nat => bool) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool)))
+ (UNIV::'M::type => bool)))
(($::('A::type, 'M::type) cart => nat => 'A::type) u i)
(($::('A::type, 'N::type) cart => nat => 'A::type) ua
((op -::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool))))))"
+ (UNIV::'M::type => bool))))))"
lemma DEF_pastecart: "(op =::(('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
@@ -7277,26 +6903,26 @@
(lambda::(nat => 'A::type)
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
(%i::nat.
- (COND::bool => 'A::type => 'A::type => 'A::type)
- ((<=::nat => nat => bool) i
+ (If::bool => 'A::type => 'A::type => 'A::type)
+ ((op <=::nat => nat => bool) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool)))
+ (UNIV::'M::type => bool)))
(($::('A::type, 'M::type) cart => nat => 'A::type) u i)
(($::('A::type, 'N::type) cart => nat => 'A::type) ua
((op -::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool))))))"
+ (UNIV::'M::type => bool))))))"
by (import hollight DEF_pastecart)
-definition fstcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'M) cart" where
- "fstcart ==
-%u::('A::type, ('M::type, 'N::type) finite_sum) cart. lambda ($ u)"
-
-lemma DEF_fstcart: "fstcart =
-(%u::('A::type, ('M::type, 'N::type) finite_sum) cart. lambda ($ u))"
+definition
+ fstcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'M) cart" where
+ "fstcart == %u::('A, ('M, 'N) finite_sum) cart. lambda ($ u)"
+
+lemma DEF_fstcart: "fstcart = (%u::('A, ('M, 'N) finite_sum) cart. lambda ($ u))"
by (import hollight DEF_fstcart)
-definition sndcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'N) cart" where
+definition
+ sndcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'N) cart" where
"(op ==::(('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
=> (('A::type, ('M::type, 'N::type) finite_sum) cart
@@ -7309,9 +6935,9 @@
(%i::nat.
($::('A::type, ('M::type, 'N::type) finite_sum) cart
=> nat => 'A::type)
- u ((plus::nat => nat => nat) i
+ u ((op +::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool)))))"
+ (UNIV::'M::type => bool)))))"
lemma DEF_sndcart: "(op =::(('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
@@ -7325,1974 +6951,520 @@
(%i::nat.
($::('A::type, ('M::type, 'N::type) finite_sum) cart
=> nat => 'A::type)
- u ((plus::nat => nat => nat) i
+ u ((op +::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
- (hollight.UNIV::'M::type => bool)))))"
+ (UNIV::'M::type => bool)))))"
by (import hollight DEF_sndcart)
-lemma DIMINDEX_HAS_SIZE_FINITE_SUM: "(HAS_SIZE::(('M::type, 'N::type) finite_sum => bool) => nat => bool)
- (hollight.UNIV::('M::type, 'N::type) finite_sum => bool)
- ((plus::nat => nat => nat)
- ((dimindex::('M::type => bool) => nat) (hollight.UNIV::'M::type => bool))
- ((dimindex::('N::type => bool) => nat)
- (hollight.UNIV::'N::type => bool)))"
- by (import hollight DIMINDEX_HAS_SIZE_FINITE_SUM)
-
-lemma DIMINDEX_FINITE_SUM: "(op =::nat => nat => bool)
- ((dimindex::(('M::type, 'N::type) finite_sum => bool) => nat)
- (hollight.UNIV::('M::type, 'N::type) finite_sum => bool))
- ((plus::nat => nat => nat)
- ((dimindex::('M::type => bool) => nat) (hollight.UNIV::'M::type => bool))
- ((dimindex::('N::type => bool) => nat)
- (hollight.UNIV::'N::type => bool)))"
- by (import hollight DIMINDEX_FINITE_SUM)
-
-lemma FSTCART_PASTECART: "ALL (x::('A::type, 'M::type) cart) xa::('A::type, 'N::type) cart.
- fstcart (pastecart x xa) = x"
- by (import hollight FSTCART_PASTECART)
-
-lemma SNDCART_PASTECART: "ALL (x::('A::type, 'M::type) cart) xa::('A::type, 'N::type) cart.
- sndcart (pastecart x xa) = xa"
- by (import hollight SNDCART_PASTECART)
-
-lemma PASTECART_FST_SND: "ALL x::('q_47149::type, ('q_47146::type, 'q_47144::type) finite_sum) cart.
- pastecart (fstcart x) (sndcart x) = x"
- by (import hollight PASTECART_FST_SND)
-
-lemma PASTECART_EQ: "ALL (x::('q_47187::type, ('q_47177::type, 'q_47188::type) finite_sum) cart)
- y::('q_47187::type, ('q_47177::type, 'q_47188::type) finite_sum) cart.
- (x = y) = (fstcart x = fstcart y & sndcart x = sndcart y)"
- by (import hollight PASTECART_EQ)
-
-lemma FORALL_PASTECART: "All (P::('q_47208::type, ('q_47209::type, 'q_47210::type) finite_sum) cart
- => bool) =
-(ALL (x::('q_47208::type, 'q_47209::type) cart)
- y::('q_47208::type, 'q_47210::type) cart. P (pastecart x y))"
- by (import hollight FORALL_PASTECART)
-
-lemma EXISTS_PASTECART: "Ex (P::('q_47230::type, ('q_47231::type, 'q_47232::type) finite_sum) cart
- => bool) =
-(EX (x::('q_47230::type, 'q_47231::type) cart)
- y::('q_47230::type, 'q_47232::type) cart. P (pastecart x y))"
- by (import hollight EXISTS_PASTECART)
-
-lemma SURJECTIVE_IFF_INJECTIVE_GEN: "ALL (s::'A::type => bool) (t::'B::type => bool) f::'A::type => 'B::type.
- FINITE s & FINITE t & CARD s = CARD t & SUBSET (IMAGE f s) t -->
- (ALL y::'B::type. IN y t --> (EX x::'A::type. IN x s & f x = y)) =
- (ALL (x::'A::type) y::'A::type. IN x s & IN y s & f x = f y --> x = y)"
- by (import hollight SURJECTIVE_IFF_INJECTIVE_GEN)
-
-lemma SURJECTIVE_IFF_INJECTIVE: "ALL (x::'A::type => bool) xa::'A::type => 'A::type.
- FINITE x & SUBSET (IMAGE xa x) x -->
- (ALL y::'A::type. IN y x --> (EX xb::'A::type. IN xb x & xa xb = y)) =
- (ALL (xb::'A::type) y::'A::type.
- IN xb x & IN y x & xa xb = xa y --> xb = y)"
- by (import hollight SURJECTIVE_IFF_INJECTIVE)
-
-lemma IMAGE_IMP_INJECTIVE_GEN: "ALL (s::'A::type => bool) (t::'B::type => bool) f::'A::type => 'B::type.
- FINITE s & CARD s = CARD t & IMAGE f s = t -->
- (ALL (x::'A::type) y::'A::type. IN x s & IN y s & f x = f y --> x = y)"
- by (import hollight IMAGE_IMP_INJECTIVE_GEN)
-
-lemma IMAGE_IMP_INJECTIVE: "ALL (s::'q_47557::type => bool) f::'q_47557::type => 'q_47557::type.
- FINITE s & IMAGE f s = s -->
- (ALL (x::'q_47557::type) y::'q_47557::type.
- IN x s & IN y s & f x = f y --> x = y)"
- by (import hollight IMAGE_IMP_INJECTIVE)
-
-lemma CARD_LE_INJ: "ALL (x::'A::type => bool) xa::'B::type => bool.
- FINITE x & FINITE xa & <= (CARD x) (CARD xa) -->
- (EX f::'A::type => 'B::type.
- SUBSET (IMAGE f x) xa &
- (ALL (xa::'A::type) y::'A::type.
- IN xa x & IN y x & f xa = f y --> xa = y))"
- by (import hollight CARD_LE_INJ)
-
-lemma FORALL_IN_CLAUSES: "(ALL x::'q_47663::type => bool.
- (ALL xa::'q_47663::type. IN xa EMPTY --> x xa) = True) &
-(ALL (x::'q_47703::type => bool) (xa::'q_47703::type)
- xb::'q_47703::type => bool.
- (ALL xc::'q_47703::type. IN xc (INSERT xa xb) --> x xc) =
- (x xa & (ALL xa::'q_47703::type. IN xa xb --> x xa)))"
- by (import hollight FORALL_IN_CLAUSES)
-
-lemma EXISTS_IN_CLAUSES: "(ALL x::'q_47723::type => bool.
- (EX xa::'q_47723::type. IN xa EMPTY & x xa) = False) &
-(ALL (x::'q_47763::type => bool) (xa::'q_47763::type)
- xb::'q_47763::type => bool.
- (EX xc::'q_47763::type. IN xc (INSERT xa xb) & x xc) =
- (x xa | (EX xa::'q_47763::type. IN xa xb & x xa)))"
- by (import hollight EXISTS_IN_CLAUSES)
-
-lemma SURJECTIVE_ON_RIGHT_INVERSE: "ALL (x::'q_47819::type => 'q_47820::type) xa::'q_47820::type => bool.
- (ALL xb::'q_47820::type.
- IN xb xa -->
- (EX xa::'q_47819::type.
- IN xa (s::'q_47819::type => bool) & x xa = xb)) =
- (EX g::'q_47820::type => 'q_47819::type.
- ALL y::'q_47820::type. IN y xa --> IN (g y) s & x (g y) = y)"
- by (import hollight SURJECTIVE_ON_RIGHT_INVERSE)
-
-lemma INJECTIVE_ON_LEFT_INVERSE: "ALL (x::'q_47913::type => 'q_47916::type) xa::'q_47913::type => bool.
- (ALL (xb::'q_47913::type) y::'q_47913::type.
- IN xb xa & IN y xa & x xb = x y --> xb = y) =
- (EX xb::'q_47916::type => 'q_47913::type.
- ALL xc::'q_47913::type. IN xc xa --> xb (x xc) = xc)"
- by (import hollight INJECTIVE_ON_LEFT_INVERSE)
-
-lemma SURJECTIVE_RIGHT_INVERSE: "(ALL y::'q_47941::type.
- EX x::'q_47944::type. (f::'q_47944::type => 'q_47941::type) x = y) =
-(EX g::'q_47941::type => 'q_47944::type. ALL y::'q_47941::type. f (g y) = y)"
- by (import hollight SURJECTIVE_RIGHT_INVERSE)
-
-lemma INJECTIVE_LEFT_INVERSE: "(ALL (x::'q_47978::type) xa::'q_47978::type.
- (f::'q_47978::type => 'q_47981::type) x = f xa --> x = xa) =
-(EX g::'q_47981::type => 'q_47978::type. ALL x::'q_47978::type. g (f x) = x)"
- by (import hollight INJECTIVE_LEFT_INVERSE)
-
-lemma FUNCTION_FACTORS_RIGHT: "ALL (x::'q_48017::type => 'q_48018::type)
- xa::'q_48005::type => 'q_48018::type.
- (ALL xb::'q_48017::type. EX y::'q_48005::type. xa y = x xb) =
- (EX xb::'q_48017::type => 'q_48005::type. x = xa o xb)"
- by (import hollight FUNCTION_FACTORS_RIGHT)
-
-lemma FUNCTION_FACTORS_LEFT: "ALL (x::'q_48090::type => 'q_48091::type)
- xa::'q_48090::type => 'q_48070::type.
- (ALL (xb::'q_48090::type) y::'q_48090::type.
- xa xb = xa y --> x xb = x y) =
- (EX xb::'q_48070::type => 'q_48091::type. x = xb o xa)"
- by (import hollight FUNCTION_FACTORS_LEFT)
-
-definition dotdot :: "nat => nat => nat => bool" where
- "dotdot ==
-%(u::nat) ua::nat.
- GSPEC (%ub::nat. EX x::nat. SETSPEC ub (<= u x & <= x ua) x)"
-
-lemma DEF__dot__dot_: "dotdot =
-(%(u::nat) ua::nat.
- GSPEC (%ub::nat. EX x::nat. SETSPEC ub (<= u x & <= x ua) x))"
- by (import hollight DEF__dot__dot_)
-
-lemma FINITE_NUMSEG: "ALL (m::nat) n::nat. FINITE (dotdot m n)"
- by (import hollight FINITE_NUMSEG)
-
-lemma NUMSEG_COMBINE_R: "ALL (x::'q_48166::type) (p::nat) m::nat.
- <= m p & <= p (n::nat) -->
- hollight.UNION (dotdot m p) (dotdot (p + NUMERAL_BIT1 0) n) = dotdot m n"
- by (import hollight NUMSEG_COMBINE_R)
-
-lemma NUMSEG_COMBINE_L: "ALL (x::'q_48204::type) (p::nat) m::nat.
- <= m p & <= p (n::nat) -->
- hollight.UNION (dotdot m (p - NUMERAL_BIT1 0)) (dotdot p n) = dotdot m n"
- by (import hollight NUMSEG_COMBINE_L)
-
-lemma NUMSEG_LREC: "ALL (x::nat) xa::nat.
- <= x xa --> INSERT x (dotdot (x + NUMERAL_BIT1 0) xa) = dotdot x xa"
- by (import hollight NUMSEG_LREC)
-
-lemma NUMSEG_RREC: "ALL (x::nat) xa::nat.
- <= x xa --> INSERT xa (dotdot x (xa - NUMERAL_BIT1 0)) = dotdot x xa"
- by (import hollight NUMSEG_RREC)
-
-lemma NUMSEG_REC: "ALL (x::nat) xa::nat.
- <= x (Suc xa) --> dotdot x (Suc xa) = INSERT (Suc xa) (dotdot x xa)"
- by (import hollight NUMSEG_REC)
-
-lemma IN_NUMSEG: "ALL (x::nat) (xa::nat) xb::nat. IN xb (dotdot x xa) = (<= x xb & <= xb xa)"
- by (import hollight IN_NUMSEG)
-
-lemma NUMSEG_SING: "ALL x::nat. dotdot x x = INSERT x EMPTY"
- by (import hollight NUMSEG_SING)
-
-lemma NUMSEG_EMPTY: "ALL (x::nat) xa::nat. (dotdot x xa = EMPTY) = < xa x"
- by (import hollight NUMSEG_EMPTY)
-
-lemma CARD_NUMSEG_LEMMA: "ALL (m::nat) d::nat. CARD (dotdot m (m + d)) = d + NUMERAL_BIT1 0"
- by (import hollight CARD_NUMSEG_LEMMA)
-
-lemma CARD_NUMSEG: "ALL (m::nat) n::nat. CARD (dotdot m n) = n + NUMERAL_BIT1 0 - m"
- by (import hollight CARD_NUMSEG)
-
-lemma HAS_SIZE_NUMSEG: "ALL (x::nat) xa::nat. HAS_SIZE (dotdot x xa) (xa + NUMERAL_BIT1 0 - x)"
- by (import hollight HAS_SIZE_NUMSEG)
-
-lemma CARD_NUMSEG_1: "ALL x::nat. CARD (dotdot (NUMERAL_BIT1 0) x) = x"
- by (import hollight CARD_NUMSEG_1)
-
-lemma HAS_SIZE_NUMSEG_1: "ALL x::nat. HAS_SIZE (dotdot (NUMERAL_BIT1 0) x) x"
- by (import hollight HAS_SIZE_NUMSEG_1)
-
-lemma NUMSEG_CLAUSES: "(ALL m::nat. dotdot m 0 = COND (m = 0) (INSERT 0 EMPTY) EMPTY) &
-(ALL (m::nat) n::nat.
- dotdot m (Suc n) =
- COND (<= m (Suc n)) (INSERT (Suc n) (dotdot m n)) (dotdot m n))"
- by (import hollight NUMSEG_CLAUSES)
-
-lemma FINITE_INDEX_NUMSEG: "ALL s::'A::type => bool.
- FINITE s =
- (EX f::nat => 'A::type.
- (ALL (i::nat) j::nat.
- IN i (dotdot (NUMERAL_BIT1 0) (CARD s)) &
- IN j (dotdot (NUMERAL_BIT1 0) (CARD s)) & f i = f j -->
- i = j) &
- s = IMAGE f (dotdot (NUMERAL_BIT1 0) (CARD s)))"
- by (import hollight FINITE_INDEX_NUMSEG)
-
-lemma FINITE_INDEX_NUMBERS: "ALL s::'A::type => bool.
- FINITE s =
- (EX (k::nat => bool) f::nat => 'A::type.
- (ALL (i::nat) j::nat. IN i k & IN j k & f i = f j --> i = j) &
- FINITE k & s = IMAGE f k)"
- by (import hollight FINITE_INDEX_NUMBERS)
-
-lemma DISJOINT_NUMSEG: "ALL (x::nat) (xa::nat) (xb::nat) xc::nat.
- DISJOINT (dotdot x xa) (dotdot xb xc) =
- (< xa xb | < xc x | < xa x | < xc xb)"
- by (import hollight DISJOINT_NUMSEG)
-
-lemma NUMSEG_ADD_SPLIT: "ALL (x::nat) (xa::nat) xb::nat.
- <= x (xa + NUMERAL_BIT1 0) -->
- dotdot x (xa + xb) =
- hollight.UNION (dotdot x xa) (dotdot (xa + NUMERAL_BIT1 0) (xa + xb))"
- by (import hollight NUMSEG_ADD_SPLIT)
-
-lemma NUMSEG_OFFSET_IMAGE: "ALL (x::nat) (xa::nat) xb::nat.
- dotdot (x + xb) (xa + xb) = IMAGE (%i::nat. i + xb) (dotdot x xa)"
- by (import hollight NUMSEG_OFFSET_IMAGE)
-
-lemma SUBSET_NUMSEG: "ALL (m::nat) (n::nat) (p::nat) q::nat.
- SUBSET (dotdot m n) (dotdot p q) = (< n m | <= p m & <= n q)"
- by (import hollight SUBSET_NUMSEG)
-
-definition neutral :: "('q_48985 => 'q_48985 => 'q_48985) => 'q_48985" where
- "neutral ==
-%u::'q_48985::type => 'q_48985::type => 'q_48985::type.
- SOME x::'q_48985::type. ALL y::'q_48985::type. u x y = y & u y x = y"
-
-lemma DEF_neutral: "neutral =
-(%u::'q_48985::type => 'q_48985::type => 'q_48985::type.
- SOME x::'q_48985::type. ALL y::'q_48985::type. u x y = y & u y x = y)"
- by (import hollight DEF_neutral)
-
-definition monoidal :: "('A => 'A => 'A) => bool" where
- "monoidal ==
-%u::'A::type => 'A::type => 'A::type.
- (ALL (x::'A::type) y::'A::type. u x y = u y x) &
- (ALL (x::'A::type) (y::'A::type) z::'A::type.
- u x (u y z) = u (u x y) z) &
- (ALL x::'A::type. u (neutral u) x = x)"
-
-lemma DEF_monoidal: "monoidal =
-(%u::'A::type => 'A::type => 'A::type.
- (ALL (x::'A::type) y::'A::type. u x y = u y x) &
- (ALL (x::'A::type) (y::'A::type) z::'A::type.
- u x (u y z) = u (u x y) z) &
- (ALL x::'A::type. u (neutral u) x = x))"
- by (import hollight DEF_monoidal)
-
-definition support :: "('B => 'B => 'B) => ('A => 'B) => ('A => bool) => 'A => bool" where
- "support ==
-%(u::'B::type => 'B::type => 'B::type) (ua::'A::type => 'B::type)
- ub::'A::type => bool.
- GSPEC
- (%uc::'A::type.
- EX x::'A::type. SETSPEC uc (IN x ub & ua x ~= neutral u) x)"
-
-lemma DEF_support: "support =
-(%(u::'B::type => 'B::type => 'B::type) (ua::'A::type => 'B::type)
- ub::'A::type => bool.
- GSPEC
- (%uc::'A::type.
- EX x::'A::type. SETSPEC uc (IN x ub & ua x ~= neutral u) x))"
- by (import hollight DEF_support)
-
-definition iterate :: "('q_49090 => 'q_49090 => 'q_49090)
-=> ('A => bool) => ('A => 'q_49090) => 'q_49090" where
- "iterate ==
-%(u::'q_49090::type => 'q_49090::type => 'q_49090::type)
- (ua::'A::type => bool) ub::'A::type => 'q_49090::type.
- ITSET (%x::'A::type. u (ub x)) (support u ub ua) (neutral u)"
-
-lemma DEF_iterate: "iterate =
-(%(u::'q_49090::type => 'q_49090::type => 'q_49090::type)
- (ua::'A::type => bool) ub::'A::type => 'q_49090::type.
- ITSET (%x::'A::type. u (ub x)) (support u ub ua) (neutral u))"
- by (import hollight DEF_iterate)
-
-lemma IN_SUPPORT: "ALL (x::'q_49133::type => 'q_49133::type => 'q_49133::type)
- (xa::'q_49136::type => 'q_49133::type) (xb::'q_49136::type)
- xc::'q_49136::type => bool.
- IN xb (support x xa xc) = (IN xb xc & xa xb ~= neutral x)"
- by (import hollight IN_SUPPORT)
-
-lemma SUPPORT_SUPPORT: "ALL (x::'q_49158::type => 'q_49158::type => 'q_49158::type)
- (xa::'q_49169::type => 'q_49158::type) xb::'q_49169::type => bool.
- support x xa (support x xa xb) = support x xa xb"
- by (import hollight SUPPORT_SUPPORT)
-
-lemma SUPPORT_EMPTY: "ALL (x::'q_49194::type => 'q_49194::type => 'q_49194::type)
- (xa::'q_49208::type => 'q_49194::type) xb::'q_49208::type => bool.
- (ALL xc::'q_49208::type. IN xc xb --> xa xc = neutral x) =
- (support x xa xb = EMPTY)"
- by (import hollight SUPPORT_EMPTY)
-
-lemma SUPPORT_SUBSET: "ALL (x::'q_49228::type => 'q_49228::type => 'q_49228::type)
- (xa::'q_49229::type => 'q_49228::type) xb::'q_49229::type => bool.
- SUBSET (support x xa xb) xb"
- by (import hollight SUPPORT_SUBSET)
-
-lemma FINITE_SUPPORT: "ALL (u::'q_49252::type => 'q_49252::type => 'q_49252::type)
- (f::'q_49246::type => 'q_49252::type) s::'q_49246::type => bool.
- FINITE s --> FINITE (support u f s)"
- by (import hollight FINITE_SUPPORT)
-
-lemma SUPPORT_CLAUSES: "(ALL x::'q_49270::type => 'q_49300::type.
- support (u_4247::'q_49300::type => 'q_49300::type => 'q_49300::type) x
- EMPTY =
- EMPTY) &
-(ALL (x::'q_49318::type => 'q_49300::type) (xa::'q_49318::type)
- xb::'q_49318::type => bool.
- support u_4247 x (INSERT xa xb) =
- COND (x xa = neutral u_4247) (support u_4247 x xb)
- (INSERT xa (support u_4247 x xb))) &
-(ALL (x::'q_49351::type => 'q_49300::type) (xa::'q_49351::type)
- xb::'q_49351::type => bool.
- support u_4247 x (DELETE xb xa) = DELETE (support u_4247 x xb) xa) &
-(ALL (x::'q_49389::type => 'q_49300::type) (xa::'q_49389::type => bool)
- xb::'q_49389::type => bool.
- support u_4247 x (hollight.UNION xa xb) =
- hollight.UNION (support u_4247 x xa) (support u_4247 x xb)) &
-(ALL (x::'q_49427::type => 'q_49300::type) (xa::'q_49427::type => bool)
- xb::'q_49427::type => bool.
- support u_4247 x (hollight.INTER xa xb) =
- hollight.INTER (support u_4247 x xa) (support u_4247 x xb)) &
-(ALL (x::'q_49463::type => 'q_49300::type) (xa::'q_49463::type => bool)
- xb::'q_49463::type => bool.
- support u_4247 x (DIFF xa xb) =
- DIFF (support u_4247 x xa) (support u_4247 x xb))"
- by (import hollight SUPPORT_CLAUSES)
-
-lemma ITERATE_SUPPORT: "ALL (x::'q_49484::type => 'q_49484::type => 'q_49484::type)
- (xa::'q_49485::type => 'q_49484::type) xb::'q_49485::type => bool.
- FINITE (support x xa xb) -->
- iterate x (support x xa xb) xa = iterate x xb xa"
- by (import hollight ITERATE_SUPPORT)
-
-lemma SUPPORT_DELTA: "ALL (x::'q_49529::type => 'q_49529::type => 'q_49529::type)
- (xa::'q_49557::type => bool) (xb::'q_49557::type => 'q_49529::type)
- xc::'q_49557::type.
- support x (%xa::'q_49557::type. COND (xa = xc) (xb xa) (neutral x)) xa =
- COND (IN xc xa) (support x xb (INSERT xc EMPTY)) EMPTY"
- by (import hollight SUPPORT_DELTA)
-
-lemma FINITE_SUPPORT_DELTA: "ALL (x::'q_49578::type => 'q_49578::type => 'q_49578::type)
- (xa::'q_49587::type => 'q_49578::type) xb::'q_49587::type.
- FINITE
- (support x (%xc::'q_49587::type. COND (xc = xb) (xa xc) (neutral x))
- (s::'q_49587::type => bool))"
- by (import hollight FINITE_SUPPORT_DELTA)
-
-lemma ITERATE_CLAUSES_GEN: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL f::'A::type => 'B::type. iterate u_4247 EMPTY f = neutral u_4247) &
- (ALL (f::'A::type => 'B::type) (x::'A::type) s::'A::type => bool.
- monoidal u_4247 & FINITE (support u_4247 f s) -->
- iterate u_4247 (INSERT x s) f =
- COND (IN x s) (iterate u_4247 s f)
- (u_4247 (f x) (iterate u_4247 s f)))"
- by (import hollight ITERATE_CLAUSES_GEN)
-
-lemma ITERATE_CLAUSES: "ALL x::'q_49755::type => 'q_49755::type => 'q_49755::type.
- monoidal x -->
- (ALL f::'q_49713::type => 'q_49755::type.
- iterate x EMPTY f = neutral x) &
- (ALL (f::'q_49757::type => 'q_49755::type) (xa::'q_49757::type)
- s::'q_49757::type => bool.
- FINITE s -->
- iterate x (INSERT xa s) f =
- COND (IN xa s) (iterate x s f) (x (f xa) (iterate x s f)))"
- by (import hollight ITERATE_CLAUSES)
-
-lemma ITERATE_UNION: "ALL u_4247::'q_49843::type => 'q_49843::type => 'q_49843::type.
- monoidal u_4247 -->
- (ALL (f::'q_49828::type => 'q_49843::type) (s::'q_49828::type => bool)
- x::'q_49828::type => bool.
- FINITE s & FINITE x & DISJOINT s x -->
- iterate u_4247 (hollight.UNION s x) f =
- u_4247 (iterate u_4247 s f) (iterate u_4247 x f))"
- by (import hollight ITERATE_UNION)
-
-lemma ITERATE_UNION_GEN: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'A::type => bool.
- FINITE (support u_4247 f s) &
- FINITE (support u_4247 f t) &
- DISJOINT (support u_4247 f s) (support u_4247 f t) -->
- iterate u_4247 (hollight.UNION s t) f =
- u_4247 (iterate u_4247 s f) (iterate u_4247 t f))"
- by (import hollight ITERATE_UNION_GEN)
-
-lemma ITERATE_DIFF: "ALL u::'q_49986::type => 'q_49986::type => 'q_49986::type.
- monoidal u -->
- (ALL (f::'q_49982::type => 'q_49986::type) (s::'q_49982::type => bool)
- t::'q_49982::type => bool.
- FINITE s & SUBSET t s -->
- u (iterate u (DIFF s t) f) (iterate u t f) = iterate u s f)"
- by (import hollight ITERATE_DIFF)
-
-lemma ITERATE_DIFF_GEN: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'A::type => bool.
- FINITE (support u_4247 f s) &
- SUBSET (support u_4247 f t) (support u_4247 f s) -->
- u_4247 (iterate u_4247 (DIFF s t) f) (iterate u_4247 t f) =
- iterate u_4247 s f)"
- by (import hollight ITERATE_DIFF_GEN)
-
-lemma ITERATE_CLOSED: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL P::'B::type => bool.
- P (neutral u_4247) &
- (ALL (x::'B::type) y::'B::type. P x & P y --> P (u_4247 x y)) -->
- (ALL (f::'A::type => 'B::type) x::'A::type => bool.
- FINITE x & (ALL xa::'A::type. IN xa x --> P (f xa)) -->
- P (iterate u_4247 x f)))"
- by (import hollight ITERATE_CLOSED)
-
-lemma ITERATE_CLOSED_GEN: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL P::'B::type => bool.
- P (neutral u_4247) &
- (ALL (x::'B::type) y::'B::type. P x & P y --> P (u_4247 x y)) -->
- (ALL (f::'A::type => 'B::type) s::'A::type => bool.
- FINITE (support u_4247 f s) &
- (ALL x::'A::type. IN x s & f x ~= neutral u_4247 --> P (f x)) -->
- P (iterate u_4247 s f)))"
- by (import hollight ITERATE_CLOSED_GEN)
-
-lemma ITERATE_RELATED: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL R::'B::type => 'B::type => bool.
- R (neutral u_4247) (neutral u_4247) &
- (ALL (x1::'B::type) (y1::'B::type) (x2::'B::type) y2::'B::type.
- R x1 x2 & R y1 y2 --> R (u_4247 x1 y1) (u_4247 x2 y2)) -->
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type)
- x::'A::type => bool.
- FINITE x & (ALL xa::'A::type. IN xa x --> R (f xa) (g xa)) -->
- R (iterate u_4247 x f) (iterate u_4247 x g)))"
- by (import hollight ITERATE_RELATED)
-
-lemma ITERATE_EQ_NEUTRAL: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL (f::'A::type => 'B::type) s::'A::type => bool.
- (ALL x::'A::type. IN x s --> f x = neutral u_4247) -->
- iterate u_4247 s f = neutral u_4247)"
- by (import hollight ITERATE_EQ_NEUTRAL)
-
-lemma ITERATE_SING: "ALL x::'B::type => 'B::type => 'B::type.
- monoidal x -->
- (ALL (f::'A::type => 'B::type) xa::'A::type.
- iterate x (INSERT xa EMPTY) f = f xa)"
- by (import hollight ITERATE_SING)
-
-lemma ITERATE_DELTA: "ALL u_4247::'q_50438::type => 'q_50438::type => 'q_50438::type.
- monoidal u_4247 -->
- (ALL (x::'q_50457::type => 'q_50438::type) (xa::'q_50457::type)
- xb::'q_50457::type => bool.
- iterate u_4247 xb
- (%xb::'q_50457::type. COND (xb = xa) (x xb) (neutral u_4247)) =
- COND (IN xa xb) (x xa) (neutral u_4247))"
- by (import hollight ITERATE_DELTA)
-
-lemma ITERATE_IMAGE: "ALL u_4247::'q_50536::type => 'q_50536::type => 'q_50536::type.
- monoidal u_4247 -->
- (ALL (f::'q_50535::type => 'q_50507::type)
- (g::'q_50507::type => 'q_50536::type) x::'q_50535::type => bool.
- FINITE x &
- (ALL (xa::'q_50535::type) y::'q_50535::type.
- IN xa x & IN y x & f xa = f y --> xa = y) -->
- iterate u_4247 (IMAGE f x) g = iterate u_4247 x (g o f))"
- by (import hollight ITERATE_IMAGE)
-
-lemma ITERATE_BIJECTION: "ALL u_4247::'B::type => 'B::type => 'B::type.
- monoidal u_4247 -->
- (ALL (f::'A::type => 'B::type) (p::'A::type => 'A::type)
- s::'A::type => bool.
- FINITE s &
- (ALL x::'A::type. IN x s --> IN (p x) s) &
- (ALL y::'A::type. IN y s --> (EX! x::'A::type. IN x s & p x = y)) -->
- iterate u_4247 s f = iterate u_4247 s (f o p))"
- by (import hollight ITERATE_BIJECTION)
-
-lemma ITERATE_ITERATE_PRODUCT: "ALL u_4247::'C::type => 'C::type => 'C::type.
- monoidal u_4247 -->
- (ALL (x::'A::type => bool) (xa::'A::type => 'B::type => bool)
- xb::'A::type => 'B::type => 'C::type.
- FINITE x & (ALL i::'A::type. IN i x --> FINITE (xa i)) -->
- iterate u_4247 x (%i::'A::type. iterate u_4247 (xa i) (xb i)) =
- iterate u_4247
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (i::'A::type) j::'B::type.
- SETSPEC u (IN i x & IN j (xa i)) (i, j)))
- (GABS
- (%f::'A::type * 'B::type => 'C::type.
- ALL (i::'A::type) j::'B::type. GEQ (f (i, j)) (xb i j))))"
- by (import hollight ITERATE_ITERATE_PRODUCT)
-
-lemma ITERATE_EQ: "ALL u_4247::'q_50867::type => 'q_50867::type => 'q_50867::type.
- monoidal u_4247 -->
- (ALL (f::'q_50871::type => 'q_50867::type)
- (g::'q_50871::type => 'q_50867::type) x::'q_50871::type => bool.
- FINITE x & (ALL xa::'q_50871::type. IN xa x --> f xa = g xa) -->
- iterate u_4247 x f = iterate u_4247 x g)"
- by (import hollight ITERATE_EQ)
-
-lemma ITERATE_EQ_GENERAL: "ALL u_4247::'C::type => 'C::type => 'C::type.
- monoidal u_4247 -->
- (ALL (s::'A::type => bool) (t::'B::type => bool)
- (f::'A::type => 'C::type) (g::'B::type => 'C::type)
- h::'A::type => 'B::type.
- FINITE s &
- (ALL y::'B::type. IN y t --> (EX! x::'A::type. IN x s & h x = y)) &
- (ALL x::'A::type. IN x s --> IN (h x) t & g (h x) = f x) -->
- iterate u_4247 s f = iterate u_4247 t g)"
- by (import hollight ITERATE_EQ_GENERAL)
-
-definition nsum :: "('q_51017 => bool) => ('q_51017 => nat) => nat" where
- "(op ==::(('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- => (('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- => prop)
- (nsum::('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- ((iterate::(nat => nat => nat)
- => ('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- (plus::nat => nat => nat))"
-
-lemma DEF_nsum: "(op =::(('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- => (('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- => bool)
- (nsum::('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- ((iterate::(nat => nat => nat)
- => ('q_51017::type => bool) => ('q_51017::type => nat) => nat)
- (plus::nat => nat => nat))"
- by (import hollight DEF_nsum)
-
-lemma NEUTRAL_ADD: "(op =::nat => nat => bool)
- ((neutral::(nat => nat => nat) => nat) (plus::nat => nat => nat)) (0::nat)"
- by (import hollight NEUTRAL_ADD)
-
-lemma NEUTRAL_MUL: "neutral op * = NUMERAL_BIT1 0"
- by (import hollight NEUTRAL_MUL)
-
-lemma MONOIDAL_ADD: "(monoidal::(nat => nat => nat) => bool) (plus::nat => nat => nat)"
- by (import hollight MONOIDAL_ADD)
-
-lemma MONOIDAL_MUL: "(monoidal::(nat => nat => nat) => bool) (op *::nat => nat => nat)"
- by (import hollight MONOIDAL_MUL)
-
-lemma NSUM_CLAUSES: "(ALL x::'q_51055::type => nat. nsum EMPTY x = 0) &
-(ALL (x::'q_51094::type) (xa::'q_51094::type => nat)
- xb::'q_51094::type => bool.
- FINITE xb -->
- nsum (INSERT x xb) xa = COND (IN x xb) (nsum xb xa) (xa x + nsum xb xa))"
- by (import hollight NSUM_CLAUSES)
-
-lemma NSUM_UNION: "ALL (x::'q_51120::type => nat) (xa::'q_51120::type => bool)
- xb::'q_51120::type => bool.
- FINITE xa & FINITE xb & DISJOINT xa xb -->
- nsum (hollight.UNION xa xb) x = nsum xa x + nsum xb x"
- by (import hollight NSUM_UNION)
-
-lemma NSUM_DIFF: "ALL (f::'q_51175::type => nat) (s::'q_51175::type => bool)
- t::'q_51175::type => bool.
- FINITE s & SUBSET t s --> nsum (DIFF s t) f = nsum s f - nsum t f"
- by (import hollight NSUM_DIFF)
-
-lemma NSUM_SUPPORT: "ALL (x::'q_51214::type => nat) xa::'q_51214::type => bool.
- FINITE (support plus x xa) --> nsum (support plus x xa) x = nsum xa x"
- by (import hollight NSUM_SUPPORT)
-
-lemma NSUM_ADD: "ALL (f::'q_51260::type => nat) (g::'q_51260::type => nat)
- s::'q_51260::type => bool.
- FINITE s --> nsum s (%x::'q_51260::type. f x + g x) = nsum s f + nsum s g"
- by (import hollight NSUM_ADD)
-
-lemma NSUM_CMUL: "ALL (f::'q_51298::type => nat) (c::nat) s::'q_51298::type => bool.
- FINITE s --> nsum s (%x::'q_51298::type. c * f x) = c * nsum s f"
- by (import hollight NSUM_CMUL)
-
-lemma NSUM_LE: "ALL (x::'q_51337::type => nat) (xa::'q_51337::type => nat)
- xb::'q_51337::type => bool.
- FINITE xb & (ALL xc::'q_51337::type. IN xc xb --> <= (x xc) (xa xc)) -->
- <= (nsum xb x) (nsum xb xa)"
- by (import hollight NSUM_LE)
-
-lemma NSUM_LT: "ALL (f::'A::type => nat) (g::'A::type => nat) s::'A::type => bool.
- FINITE s &
- (ALL x::'A::type. IN x s --> <= (f x) (g x)) &
- (EX x::'A::type. IN x s & < (f x) (g x)) -->
- < (nsum s f) (nsum s g)"
- by (import hollight NSUM_LT)
-
-lemma NSUM_LT_ALL: "ALL (f::'q_51459::type => nat) (g::'q_51459::type => nat)
- s::'q_51459::type => bool.
- FINITE s &
- s ~= EMPTY & (ALL x::'q_51459::type. IN x s --> < (f x) (g x)) -->
- < (nsum s f) (nsum s g)"
- by (import hollight NSUM_LT_ALL)
-
-lemma NSUM_EQ: "ALL (x::'q_51501::type => nat) (xa::'q_51501::type => nat)
- xb::'q_51501::type => bool.
- FINITE xb & (ALL xc::'q_51501::type. IN xc xb --> x xc = xa xc) -->
- nsum xb x = nsum xb xa"
- by (import hollight NSUM_EQ)
-
-lemma NSUM_CONST: "ALL (c::nat) s::'q_51531::type => bool.
- FINITE s --> nsum s (%n::'q_51531::type. c) = CARD s * c"
- by (import hollight NSUM_CONST)
-
-lemma NSUM_EQ_0: "ALL (x::'A::type => nat) xa::'A::type => bool.
- (ALL xb::'A::type. IN xb xa --> x xb = 0) --> nsum xa x = 0"
- by (import hollight NSUM_EQ_0)
-
-lemma NSUM_0: "ALL x::'A::type => bool. nsum x (%n::'A::type. 0) = 0"
- by (import hollight NSUM_0)
-
-lemma NSUM_POS_LE: "ALL (x::'q_51610::type => nat) xa::'q_51610::type => bool.
- FINITE xa & (ALL xb::'q_51610::type. IN xb xa --> <= 0 (x xb)) -->
- <= 0 (nsum xa x)"
- by (import hollight NSUM_POS_LE)
-
-lemma NSUM_POS_BOUND: "ALL (f::'A::type => nat) (b::nat) x::'A::type => bool.
- FINITE x &
- (ALL xa::'A::type. IN xa x --> <= 0 (f xa)) & <= (nsum x f) b -->
- (ALL xa::'A::type. IN xa x --> <= (f xa) b)"
- by (import hollight NSUM_POS_BOUND)
-
-lemma NSUM_POS_EQ_0: "ALL (x::'q_51745::type => nat) xa::'q_51745::type => bool.
- FINITE xa &
- (ALL xb::'q_51745::type. IN xb xa --> <= 0 (x xb)) & nsum xa x = 0 -->
- (ALL xb::'q_51745::type. IN xb xa --> x xb = 0)"
- by (import hollight NSUM_POS_EQ_0)
-
-lemma NSUM_SING: "ALL (x::'q_51765::type => nat) xa::'q_51765::type.
- nsum (INSERT xa EMPTY) x = x xa"
- by (import hollight NSUM_SING)
-
-lemma NSUM_DELTA: "ALL (x::'A::type => bool) xa::'A::type.
- nsum x (%x::'A::type. COND (x = xa) (b::nat) 0) = COND (IN xa x) b 0"
- by (import hollight NSUM_DELTA)
-
-lemma NSUM_SWAP: "ALL (f::'A::type => 'B::type => nat) (x::'A::type => bool)
- xa::'B::type => bool.
- FINITE x & FINITE xa -->
- nsum x (%i::'A::type. nsum xa (f i)) =
- nsum xa (%j::'B::type. nsum x (%i::'A::type. f i j))"
- by (import hollight NSUM_SWAP)
-
-lemma NSUM_IMAGE: "ALL (x::'q_51905::type => 'q_51881::type) (xa::'q_51881::type => nat)
- xb::'q_51905::type => bool.
- FINITE xb &
- (ALL (xa::'q_51905::type) y::'q_51905::type.
- IN xa xb & IN y xb & x xa = x y --> xa = y) -->
- nsum (IMAGE x xb) xa = nsum xb (xa o x)"
- by (import hollight NSUM_IMAGE)
-
-lemma NSUM_SUPERSET: "ALL (f::'A::type => nat) (u::'A::type => bool) v::'A::type => bool.
- FINITE u &
- SUBSET u v & (ALL x::'A::type. IN x v & ~ IN x u --> f x = 0) -->
- nsum v f = nsum u f"
- by (import hollight NSUM_SUPERSET)
-
-lemma NSUM_UNION_RZERO: "ALL (f::'A::type => nat) (u::'A::type => bool) v::'A::type => bool.
- FINITE u & (ALL x::'A::type. IN x v & ~ IN x u --> f x = 0) -->
- nsum (hollight.UNION u v) f = nsum u f"
- by (import hollight NSUM_UNION_RZERO)
-
-lemma NSUM_UNION_LZERO: "ALL (f::'A::type => nat) (u::'A::type => bool) v::'A::type => bool.
- FINITE v & (ALL x::'A::type. IN x u & ~ IN x v --> f x = 0) -->
- nsum (hollight.UNION u v) f = nsum v f"
- by (import hollight NSUM_UNION_LZERO)
-
-lemma NSUM_RESTRICT: "ALL (f::'q_52126::type => nat) s::'q_52126::type => bool.
- FINITE s -->
- nsum s (%x::'q_52126::type. COND (IN x s) (f x) 0) = nsum s f"
- by (import hollight NSUM_RESTRICT)
-
-lemma NSUM_BOUND: "ALL (x::'A::type => bool) (xa::'A::type => nat) xb::nat.
- FINITE x & (ALL xc::'A::type. IN xc x --> <= (xa xc) xb) -->
- <= (nsum x xa) (CARD x * xb)"
- by (import hollight NSUM_BOUND)
-
-lemma NSUM_BOUND_GEN: "ALL (x::'A::type => bool) (xa::'q_52201::type) b::nat.
- FINITE x &
- x ~= EMPTY &
- (ALL xa::'A::type.
- IN xa x --> <= ((f::'A::type => nat) xa) (DIV b (CARD x))) -->
- <= (nsum x f) b"
- by (import hollight NSUM_BOUND_GEN)
-
-lemma NSUM_BOUND_LT: "ALL (s::'A::type => bool) (f::'A::type => nat) b::nat.
- FINITE s &
- (ALL x::'A::type. IN x s --> <= (f x) b) &
- (EX x::'A::type. IN x s & < (f x) b) -->
- < (nsum s f) (CARD s * b)"
- by (import hollight NSUM_BOUND_LT)
-
-lemma NSUM_BOUND_LT_ALL: "ALL (s::'q_52344::type => bool) (f::'q_52344::type => nat) b::nat.
- FINITE s & s ~= EMPTY & (ALL x::'q_52344::type. IN x s --> < (f x) b) -->
- < (nsum s f) (CARD s * b)"
- by (import hollight NSUM_BOUND_LT_ALL)
-
-lemma NSUM_BOUND_LT_GEN: "ALL (s::'A::type => bool) (t::'q_52386::type) b::nat.
- FINITE s &
- s ~= EMPTY &
- (ALL x::'A::type.
- IN x s --> < ((f::'A::type => nat) x) (DIV b (CARD s))) -->
- < (nsum s f) b"
- by (import hollight NSUM_BOUND_LT_GEN)
-
-lemma NSUM_UNION_EQ: "ALL (s::'q_52445::type => bool) (t::'q_52445::type => bool)
- u::'q_52445::type => bool.
- FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
- nsum s (f::'q_52445::type => nat) + nsum t f = nsum u f"
- by (import hollight NSUM_UNION_EQ)
-
-lemma NSUM_EQ_SUPERSET: "ALL (f::'A::type => nat) (s::'A::type => bool) t::'A::type => bool.
- FINITE t &
- SUBSET t s &
- (ALL x::'A::type. IN x t --> f x = (g::'A::type => nat) x) &
- (ALL x::'A::type. IN x s & ~ IN x t --> f x = 0) -->
- nsum s f = nsum t g"
- by (import hollight NSUM_EQ_SUPERSET)
-
-lemma NSUM_RESTRICT_SET: "ALL (s::'A::type => bool) (f::'A::type => nat) r::'q_52556::type.
- FINITE s -->
- nsum
- (GSPEC
- (%u::'A::type.
- EX x::'A::type. SETSPEC u (IN x s & (P::'A::type => bool) x) x))
- f =
- nsum s (%x::'A::type. COND (P x) (f x) 0)"
- by (import hollight NSUM_RESTRICT_SET)
-
-lemma NSUM_NSUM_RESTRICT: "ALL (R::'q_52685::type => 'q_52684::type => bool)
- (f::'q_52685::type => 'q_52684::type => nat) (s::'q_52685::type => bool)
- t::'q_52684::type => bool.
- FINITE s & FINITE t -->
- nsum s
- (%x::'q_52685::type.
- nsum
- (GSPEC
- (%u::'q_52684::type.
- EX y::'q_52684::type. SETSPEC u (IN y t & R x y) y))
- (f x)) =
- nsum t
- (%y::'q_52684::type.
- nsum
- (GSPEC
- (%u::'q_52685::type.
- EX x::'q_52685::type. SETSPEC u (IN x s & R x y) x))
- (%x::'q_52685::type. f x y))"
- by (import hollight NSUM_NSUM_RESTRICT)
-
-lemma CARD_EQ_NSUM: "ALL x::'q_52704::type => bool.
- FINITE x --> CARD x = nsum x (%x::'q_52704::type. NUMERAL_BIT1 0)"
- by (import hollight CARD_EQ_NSUM)
-
-lemma NSUM_MULTICOUNT_GEN: "ALL (R::'A::type => 'B::type => bool) (s::'A::type => bool)
- (t::'B::type => bool) k::'B::type => nat.
- FINITE s &
- FINITE t &
- (ALL j::'B::type.
- IN j t -->
- CARD
- (GSPEC
- (%u::'A::type. EX i::'A::type. SETSPEC u (IN i s & R i j) i)) =
- k j) -->
- nsum s
- (%i::'A::type.
- CARD
- (GSPEC
- (%u::'B::type. EX j::'B::type. SETSPEC u (IN j t & R i j) j))) =
- nsum t k"
- by (import hollight NSUM_MULTICOUNT_GEN)
-
-lemma NSUM_MULTICOUNT: "ALL (R::'A::type => 'B::type => bool) (s::'A::type => bool)
- (t::'B::type => bool) k::nat.
- FINITE s &
- FINITE t &
- (ALL j::'B::type.
- IN j t -->
- CARD
- (GSPEC
- (%u::'A::type. EX i::'A::type. SETSPEC u (IN i s & R i j) i)) =
- k) -->
- nsum s
- (%i::'A::type.
- CARD
- (GSPEC
- (%u::'B::type. EX j::'B::type. SETSPEC u (IN j t & R i j) j))) =
- k * CARD t"
- by (import hollight NSUM_MULTICOUNT)
-
-lemma NSUM_IMAGE_GEN: "ALL (f::'A::type => 'B::type) (g::'A::type => nat) s::'A::type => bool.
- FINITE s -->
- nsum s g =
- nsum (IMAGE f s)
- (%y::'B::type.
- nsum
- (GSPEC
- (%u::'A::type. EX x::'A::type. SETSPEC u (IN x s & f x = y) x))
- g)"
- by (import hollight NSUM_IMAGE_GEN)
-
-lemma NSUM_SUBSET: "ALL (u::'A::type => bool) (v::'A::type => bool) f::'A::type => nat.
- FINITE u & FINITE v & (ALL x::'A::type. IN x (DIFF u v) --> f x = 0) -->
- <= (nsum u f) (nsum v f)"
- by (import hollight NSUM_SUBSET)
-
-lemma NSUM_SUBSET_SIMPLE: "ALL (u::'q_53164::type => bool) (v::'q_53164::type => bool)
- f::'q_53164::type => nat.
- FINITE v & SUBSET u v --> <= (nsum u f) (nsum v f)"
- by (import hollight NSUM_SUBSET_SIMPLE)
-
-lemma NSUM_ADD_NUMSEG: "ALL (x::nat => nat) (xa::nat => nat) (xb::nat) xc::nat.
- nsum (dotdot xb xc) (%i::nat. x i + xa i) =
- nsum (dotdot xb xc) x + nsum (dotdot xb xc) xa"
- by (import hollight NSUM_ADD_NUMSEG)
-
-lemma NSUM_CMUL_NUMSEG: "ALL (x::nat => nat) (xa::nat) (xb::nat) xc::nat.
- nsum (dotdot xb xc) (%i::nat. xa * x i) = xa * nsum (dotdot xb xc) x"
- by (import hollight NSUM_CMUL_NUMSEG)
-
-lemma NSUM_LE_NUMSEG: "ALL (x::nat => nat) (xa::nat => nat) (xb::nat) xc::nat.
- (ALL i::nat. <= xb i & <= i xc --> <= (x i) (xa i)) -->
- <= (nsum (dotdot xb xc) x) (nsum (dotdot xb xc) xa)"
- by (import hollight NSUM_LE_NUMSEG)
-
-lemma NSUM_EQ_NUMSEG: "ALL (f::nat => nat) (g::nat => nat) (m::nat) n::nat.
- (ALL i::nat. <= m i & <= i n --> f i = g i) -->
- nsum (dotdot m n) f = nsum (dotdot m n) g"
- by (import hollight NSUM_EQ_NUMSEG)
-
-lemma NSUM_CONST_NUMSEG: "ALL (x::nat) (xa::nat) xb::nat.
- nsum (dotdot xa xb) (%n::nat. x) = (xb + NUMERAL_BIT1 0 - xa) * x"
- by (import hollight NSUM_CONST_NUMSEG)
-
-lemma NSUM_EQ_0_NUMSEG: "ALL (x::nat => nat) xa::'q_53403::type.
- (ALL i::nat. <= (m::nat) i & <= i (n::nat) --> x i = 0) -->
- nsum (dotdot m n) x = 0"
- by (import hollight NSUM_EQ_0_NUMSEG)
-
-lemma NSUM_TRIV_NUMSEG: "ALL (f::nat => nat) (m::nat) n::nat. < n m --> nsum (dotdot m n) f = 0"
- by (import hollight NSUM_TRIV_NUMSEG)
-
-lemma NSUM_POS_LE_NUMSEG: "ALL (x::nat) (xa::nat) xb::nat => nat.
- (ALL p::nat. <= x p & <= p xa --> <= 0 (xb p)) -->
- <= 0 (nsum (dotdot x xa) xb)"
- by (import hollight NSUM_POS_LE_NUMSEG)
-
-lemma NSUM_POS_EQ_0_NUMSEG: "ALL (f::nat => nat) (m::nat) n::nat.
- (ALL p::nat. <= m p & <= p n --> <= 0 (f p)) &
- nsum (dotdot m n) f = 0 -->
- (ALL p::nat. <= m p & <= p n --> f p = 0)"
- by (import hollight NSUM_POS_EQ_0_NUMSEG)
-
-lemma NSUM_SING_NUMSEG: "ALL (x::nat => nat) xa::nat. nsum (dotdot xa xa) x = x xa"
- by (import hollight NSUM_SING_NUMSEG)
-
-lemma NSUM_CLAUSES_NUMSEG: "(ALL x::nat. nsum (dotdot x 0) (f::nat => nat) = COND (x = 0) (f 0) 0) &
-(ALL (x::nat) xa::nat.
- nsum (dotdot x (Suc xa)) f =
- COND (<= x (Suc xa)) (nsum (dotdot x xa) f + f (Suc xa))
- (nsum (dotdot x xa) f))"
- by (import hollight NSUM_CLAUSES_NUMSEG)
-
-lemma NSUM_SWAP_NUMSEG: "ALL (a::nat) (b::nat) (c::nat) (d::nat) f::nat => nat => nat.
- nsum (dotdot a b) (%i::nat. nsum (dotdot c d) (f i)) =
- nsum (dotdot c d) (%j::nat. nsum (dotdot a b) (%i::nat. f i j))"
- by (import hollight NSUM_SWAP_NUMSEG)
-
-lemma NSUM_ADD_SPLIT: "ALL (x::nat => nat) (xa::nat) (xb::nat) xc::nat.
- <= xa (xb + NUMERAL_BIT1 0) -->
- nsum (dotdot xa (xb + xc)) x =
- nsum (dotdot xa xb) x + nsum (dotdot (xb + NUMERAL_BIT1 0) (xb + xc)) x"
- by (import hollight NSUM_ADD_SPLIT)
-
-lemma NSUM_OFFSET: "ALL (x::nat => nat) (xa::nat) xb::nat.
- nsum (dotdot (xa + xb) ((n::nat) + xb)) x =
- nsum (dotdot xa n) (%i::nat. x (i + xb))"
- by (import hollight NSUM_OFFSET)
-
-lemma NSUM_OFFSET_0: "ALL (x::nat => nat) (xa::nat) xb::nat.
- <= xa xb -->
- nsum (dotdot xa xb) x = nsum (dotdot 0 (xb - xa)) (%i::nat. x (i + xa))"
- by (import hollight NSUM_OFFSET_0)
-
-lemma NSUM_CLAUSES_LEFT: "ALL (x::nat => nat) (xa::nat) xb::nat.
- <= xa xb -->
- nsum (dotdot xa xb) x = x xa + nsum (dotdot (xa + NUMERAL_BIT1 0) xb) x"
- by (import hollight NSUM_CLAUSES_LEFT)
-
-lemma NSUM_CLAUSES_RIGHT: "ALL (f::nat => nat) (m::nat) n::nat.
- < 0 n & <= m n -->
- nsum (dotdot m n) f = nsum (dotdot m (n - NUMERAL_BIT1 0)) f + f n"
- by (import hollight NSUM_CLAUSES_RIGHT)
-
-lemma NSUM_BIJECTION: "ALL (x::'A::type => nat) (xa::'A::type => 'A::type) xb::'A::type => bool.
- FINITE xb &
- (ALL x::'A::type. IN x xb --> IN (xa x) xb) &
- (ALL y::'A::type. IN y xb --> (EX! x::'A::type. IN x xb & xa x = y)) -->
- nsum xb x = nsum xb (x o xa)"
- by (import hollight NSUM_BIJECTION)
-
-lemma NSUM_NSUM_PRODUCT: "ALL (x::'A::type => bool) (xa::'A::type => 'B::type => bool)
- xb::'A::type => 'B::type => nat.
- FINITE x & (ALL i::'A::type. IN i x --> FINITE (xa i)) -->
- nsum x (%x::'A::type. nsum (xa x) (xb x)) =
- nsum
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (i::'A::type) j::'B::type.
- SETSPEC u (IN i x & IN j (xa i)) (i, j)))
- (GABS
- (%f::'A::type * 'B::type => nat.
- ALL (i::'A::type) j::'B::type. GEQ (f (i, j)) (xb i j)))"
- by (import hollight NSUM_NSUM_PRODUCT)
-
-lemma NSUM_EQ_GENERAL: "ALL (x::'A::type => bool) (xa::'B::type => bool) (xb::'A::type => nat)
- (xc::'B::type => nat) xd::'A::type => 'B::type.
- FINITE x &
- (ALL y::'B::type. IN y xa --> (EX! xa::'A::type. IN xa x & xd xa = y)) &
- (ALL xe::'A::type. IN xe x --> IN (xd xe) xa & xc (xd xe) = xb xe) -->
- nsum x xb = nsum xa xc"
- by (import hollight NSUM_EQ_GENERAL)
-
-consts
- sum :: "('q_54215 => bool) => ('q_54215 => hollight.real) => hollight.real"
-
-defs
- sum_def: "(op ==::(('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- => (('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- => prop)
- (hollight.sum::('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- ((iterate::(hollight.real => hollight.real => hollight.real)
- => ('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- (real_add::hollight.real => hollight.real => hollight.real))"
-
-lemma DEF_sum: "(op =::(('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- => (('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- => bool)
- (hollight.sum::('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- ((iterate::(hollight.real => hollight.real => hollight.real)
- => ('q_54215::type => bool)
- => ('q_54215::type => hollight.real) => hollight.real)
- (real_add::hollight.real => hollight.real => hollight.real))"
- by (import hollight DEF_sum)
-
-lemma NEUTRAL_REAL_ADD: "neutral real_add = real_of_num 0"
- by (import hollight NEUTRAL_REAL_ADD)
-
-lemma NEUTRAL_REAL_MUL: "neutral real_mul = real_of_num (NUMERAL_BIT1 0)"
- by (import hollight NEUTRAL_REAL_MUL)
-
-lemma MONOIDAL_REAL_ADD: "monoidal real_add"
- by (import hollight MONOIDAL_REAL_ADD)
-
-lemma MONOIDAL_REAL_MUL: "monoidal real_mul"
- by (import hollight MONOIDAL_REAL_MUL)
-
-lemma SUM_CLAUSES: "(ALL x::'q_54257::type => hollight.real.
- hollight.sum EMPTY x = real_of_num 0) &
-(ALL (x::'q_54298::type) (xa::'q_54298::type => hollight.real)
- xb::'q_54298::type => bool.
- FINITE xb -->
- hollight.sum (INSERT x xb) xa =
- COND (IN x xb) (hollight.sum xb xa)
- (real_add (xa x) (hollight.sum xb xa)))"
- by (import hollight SUM_CLAUSES)
-
-lemma SUM_UNION: "ALL (x::'q_54324::type => hollight.real) (xa::'q_54324::type => bool)
- xb::'q_54324::type => bool.
- FINITE xa & FINITE xb & DISJOINT xa xb -->
- hollight.sum (hollight.UNION xa xb) x =
- real_add (hollight.sum xa x) (hollight.sum xb x)"
- by (import hollight SUM_UNION)
-
-lemma SUM_DIFF: "ALL (x::'q_54364::type => hollight.real) (xa::'q_54364::type => bool)
- xb::'q_54364::type => bool.
- FINITE xa & SUBSET xb xa -->
- hollight.sum (DIFF xa xb) x =
- real_sub (hollight.sum xa x) (hollight.sum xb x)"
- by (import hollight SUM_DIFF)
-
-lemma SUM_SUPPORT: "ALL (x::'q_54403::type => hollight.real) xa::'q_54403::type => bool.
- FINITE (support real_add x xa) -->
- hollight.sum (support real_add x xa) x = hollight.sum xa x"
- by (import hollight SUM_SUPPORT)
-
-lemma SUM_ADD: "ALL (f::'q_54449::type => hollight.real)
- (g::'q_54449::type => hollight.real) s::'q_54449::type => bool.
- FINITE s -->
- hollight.sum s (%x::'q_54449::type. real_add (f x) (g x)) =
- real_add (hollight.sum s f) (hollight.sum s g)"
- by (import hollight SUM_ADD)
-
-lemma SUM_CMUL: "ALL (f::'q_54487::type => hollight.real) (c::hollight.real)
- s::'q_54487::type => bool.
- FINITE s -->
- hollight.sum s (%x::'q_54487::type. real_mul c (f x)) =
- real_mul c (hollight.sum s f)"
- by (import hollight SUM_CMUL)
-
-lemma SUM_NEG: "ALL (x::'q_54530::type => hollight.real) xa::'q_54530::type => bool.
- FINITE xa -->
- hollight.sum xa (%xa::'q_54530::type. real_neg (x xa)) =
- real_neg (hollight.sum xa x)"
- by (import hollight SUM_NEG)
-
-lemma SUM_SUB: "ALL (x::'q_54565::type => hollight.real)
- (xa::'q_54565::type => hollight.real) xb::'q_54565::type => bool.
- FINITE xb -->
- hollight.sum xb (%xb::'q_54565::type. real_sub (x xb) (xa xb)) =
- real_sub (hollight.sum xb x) (hollight.sum xb xa)"
- by (import hollight SUM_SUB)
-
-lemma SUM_LE: "ALL (x::'q_54607::type => hollight.real)
- (xa::'q_54607::type => hollight.real) xb::'q_54607::type => bool.
- FINITE xb &
- (ALL xc::'q_54607::type. IN xc xb --> real_le (x xc) (xa xc)) -->
- real_le (hollight.sum xb x) (hollight.sum xb xa)"
- by (import hollight SUM_LE)
-
-lemma SUM_LT: "ALL (f::'A::type => hollight.real) (g::'A::type => hollight.real)
- s::'A::type => bool.
- FINITE s &
- (ALL x::'A::type. IN x s --> real_le (f x) (g x)) &
- (EX x::'A::type. IN x s & real_lt (f x) (g x)) -->
- real_lt (hollight.sum s f) (hollight.sum s g)"
- by (import hollight SUM_LT)
-
-lemma SUM_LT_ALL: "ALL (f::'q_54729::type => hollight.real)
- (g::'q_54729::type => hollight.real) s::'q_54729::type => bool.
- FINITE s &
- s ~= EMPTY & (ALL x::'q_54729::type. IN x s --> real_lt (f x) (g x)) -->
- real_lt (hollight.sum s f) (hollight.sum s g)"
- by (import hollight SUM_LT_ALL)
-
-lemma SUM_EQ: "ALL (x::'q_54771::type => hollight.real)
- (xa::'q_54771::type => hollight.real) xb::'q_54771::type => bool.
- FINITE xb & (ALL xc::'q_54771::type. IN xc xb --> x xc = xa xc) -->
- hollight.sum xb x = hollight.sum xb xa"
- by (import hollight SUM_EQ)
-
-lemma SUM_ABS: "ALL (f::'q_54830::type => hollight.real) s::'q_54830::type => bool.
- FINITE s -->
- real_le (real_abs (hollight.sum s f))
- (hollight.sum s (%x::'q_54830::type. real_abs (f x)))"
- by (import hollight SUM_ABS)
-
-lemma SUM_CONST: "ALL (c::hollight.real) s::'q_54851::type => bool.
- FINITE s -->
- hollight.sum s (%n::'q_54851::type. c) =
- real_mul (real_of_num (CARD s)) c"
- by (import hollight SUM_CONST)
-
-lemma SUM_EQ_0: "ALL (x::'A::type => hollight.real) xa::'A::type => bool.
- (ALL xb::'A::type. IN xb xa --> x xb = real_of_num 0) -->
- hollight.sum xa x = real_of_num 0"
- by (import hollight SUM_EQ_0)
-
-lemma SUM_0: "ALL x::'A::type => bool.
- hollight.sum x (%n::'A::type. real_of_num 0) = real_of_num 0"
- by (import hollight SUM_0)
-
-lemma SUM_POS_LE: "ALL (x::'q_54944::type => hollight.real) xa::'q_54944::type => bool.
- FINITE xa &
- (ALL xb::'q_54944::type. IN xb xa --> real_le (real_of_num 0) (x xb)) -->
- real_le (real_of_num 0) (hollight.sum xa x)"
- by (import hollight SUM_POS_LE)
-
-lemma SUM_POS_BOUND: "ALL (f::'A::type => hollight.real) (b::hollight.real) x::'A::type => bool.
- FINITE x &
- (ALL xa::'A::type. IN xa x --> real_le (real_of_num 0) (f xa)) &
- real_le (hollight.sum x f) b -->
- (ALL xa::'A::type. IN xa x --> real_le (f xa) b)"
- by (import hollight SUM_POS_BOUND)
-
-lemma SUM_POS_EQ_0: "ALL (x::'q_55091::type => hollight.real) xa::'q_55091::type => bool.
- FINITE xa &
- (ALL xb::'q_55091::type. IN xb xa --> real_le (real_of_num 0) (x xb)) &
- hollight.sum xa x = real_of_num 0 -->
- (ALL xb::'q_55091::type. IN xb xa --> x xb = real_of_num 0)"
- by (import hollight SUM_POS_EQ_0)
-
-lemma SUM_SING: "ALL (x::'q_55113::type => hollight.real) xa::'q_55113::type.
- hollight.sum (INSERT xa EMPTY) x = x xa"
- by (import hollight SUM_SING)
-
-lemma SUM_DELTA: "ALL (x::'A::type => bool) xa::'A::type.
- hollight.sum x
- (%x::'A::type. COND (x = xa) (b::hollight.real) (real_of_num 0)) =
- COND (IN xa x) b (real_of_num 0)"
- by (import hollight SUM_DELTA)
-
-lemma SUM_SWAP: "ALL (f::'A::type => 'B::type => hollight.real) (x::'A::type => bool)
- xa::'B::type => bool.
- FINITE x & FINITE xa -->
- hollight.sum x (%i::'A::type. hollight.sum xa (f i)) =
- hollight.sum xa (%j::'B::type. hollight.sum x (%i::'A::type. f i j))"
- by (import hollight SUM_SWAP)
-
-lemma SUM_IMAGE: "ALL (x::'q_55257::type => 'q_55233::type)
- (xa::'q_55233::type => hollight.real) xb::'q_55257::type => bool.
- FINITE xb &
- (ALL (xa::'q_55257::type) y::'q_55257::type.
- IN xa xb & IN y xb & x xa = x y --> xa = y) -->
- hollight.sum (IMAGE x xb) xa = hollight.sum xb (xa o x)"
- by (import hollight SUM_IMAGE)
-
-lemma SUM_SUPERSET: "ALL (f::'A::type => hollight.real) (u::'A::type => bool)
- v::'A::type => bool.
- FINITE u &
- SUBSET u v &
- (ALL x::'A::type. IN x v & ~ IN x u --> f x = real_of_num 0) -->
- hollight.sum v f = hollight.sum u f"
- by (import hollight SUM_SUPERSET)
-
-lemma SUM_UNION_RZERO: "ALL (f::'A::type => hollight.real) (u::'A::type => bool)
- v::'A::type => bool.
- FINITE u &
- (ALL x::'A::type. IN x v & ~ IN x u --> f x = real_of_num 0) -->
- hollight.sum (hollight.UNION u v) f = hollight.sum u f"
- by (import hollight SUM_UNION_RZERO)
-
-lemma SUM_UNION_LZERO: "ALL (f::'A::type => hollight.real) (u::'A::type => bool)
- v::'A::type => bool.
- FINITE v &
- (ALL x::'A::type. IN x u & ~ IN x v --> f x = real_of_num 0) -->
- hollight.sum (hollight.UNION u v) f = hollight.sum v f"
- by (import hollight SUM_UNION_LZERO)
-
-lemma SUM_RESTRICT: "ALL (f::'q_55484::type => hollight.real) s::'q_55484::type => bool.
- FINITE s -->
- hollight.sum s
- (%x::'q_55484::type. COND (IN x s) (f x) (real_of_num 0)) =
- hollight.sum s f"
- by (import hollight SUM_RESTRICT)
-
-lemma SUM_BOUND: "ALL (x::'A::type => bool) (xa::'A::type => hollight.real) xb::hollight.real.
- FINITE x & (ALL xc::'A::type. IN xc x --> real_le (xa xc) xb) -->
- real_le (hollight.sum x xa) (real_mul (real_of_num (CARD x)) xb)"
- by (import hollight SUM_BOUND)
-
-lemma SUM_BOUND_GEN: "ALL (s::'A::type => bool) (t::'q_55543::type) b::hollight.real.
- FINITE s &
- s ~= EMPTY &
- (ALL x::'A::type.
- IN x s -->
- real_le ((f::'A::type => hollight.real) x)
- (real_div b (real_of_num (CARD s)))) -->
- real_le (hollight.sum s f) b"
- by (import hollight SUM_BOUND_GEN)
-
-lemma SUM_ABS_BOUND: "ALL (s::'A::type => bool) (f::'A::type => hollight.real) b::hollight.real.
- FINITE s & (ALL x::'A::type. IN x s --> real_le (real_abs (f x)) b) -->
- real_le (real_abs (hollight.sum s f)) (real_mul (real_of_num (CARD s)) b)"
- by (import hollight SUM_ABS_BOUND)
-
-lemma SUM_BOUND_LT: "ALL (s::'A::type => bool) (f::'A::type => hollight.real) b::hollight.real.
- FINITE s &
- (ALL x::'A::type. IN x s --> real_le (f x) b) &
- (EX x::'A::type. IN x s & real_lt (f x) b) -->
- real_lt (hollight.sum s f) (real_mul (real_of_num (CARD s)) b)"
- by (import hollight SUM_BOUND_LT)
-
-lemma SUM_BOUND_LT_ALL: "ALL (s::'q_55748::type => bool) (f::'q_55748::type => hollight.real)
- b::hollight.real.
- FINITE s &
- s ~= EMPTY & (ALL x::'q_55748::type. IN x s --> real_lt (f x) b) -->
- real_lt (hollight.sum s f) (real_mul (real_of_num (CARD s)) b)"
- by (import hollight SUM_BOUND_LT_ALL)
-
-lemma SUM_BOUND_LT_GEN: "ALL (s::'A::type => bool) (t::'q_55770::type) b::hollight.real.
- FINITE s &
- s ~= EMPTY &
- (ALL x::'A::type.
- IN x s -->
- real_lt ((f::'A::type => hollight.real) x)
- (real_div b (real_of_num (CARD s)))) -->
- real_lt (hollight.sum s f) b"
- by (import hollight SUM_BOUND_LT_GEN)
-
-lemma SUM_UNION_EQ: "ALL (s::'q_55831::type => bool) (t::'q_55831::type => bool)
- u::'q_55831::type => bool.
- FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
- real_add (hollight.sum s (f::'q_55831::type => hollight.real))
- (hollight.sum t f) =
- hollight.sum u f"
- by (import hollight SUM_UNION_EQ)
-
-lemma SUM_EQ_SUPERSET: "ALL (f::'A::type => hollight.real) (s::'A::type => bool)
- t::'A::type => bool.
- FINITE t &
- SUBSET t s &
- (ALL x::'A::type. IN x t --> f x = (g::'A::type => hollight.real) x) &
- (ALL x::'A::type. IN x s & ~ IN x t --> f x = real_of_num 0) -->
- hollight.sum s f = hollight.sum t g"
- by (import hollight SUM_EQ_SUPERSET)
-
-lemma SUM_RESTRICT_SET: "ALL (s::'A::type => bool) (f::'A::type => hollight.real) r::'q_55944::type.
- FINITE s -->
- hollight.sum
- (GSPEC
- (%u::'A::type.
- EX x::'A::type. SETSPEC u (IN x s & (P::'A::type => bool) x) x))
- f =
- hollight.sum s (%x::'A::type. COND (P x) (f x) (real_of_num 0))"
- by (import hollight SUM_RESTRICT_SET)
-
-lemma SUM_SUM_RESTRICT: "ALL (R::'q_56075::type => 'q_56074::type => bool)
- (f::'q_56075::type => 'q_56074::type => hollight.real)
- (s::'q_56075::type => bool) t::'q_56074::type => bool.
- FINITE s & FINITE t -->
- hollight.sum s
- (%x::'q_56075::type.
- hollight.sum
- (GSPEC
- (%u::'q_56074::type.
- EX y::'q_56074::type. SETSPEC u (IN y t & R x y) y))
- (f x)) =
- hollight.sum t
- (%y::'q_56074::type.
- hollight.sum
- (GSPEC
- (%u::'q_56075::type.
- EX x::'q_56075::type. SETSPEC u (IN x s & R x y) x))
- (%x::'q_56075::type. f x y))"
- by (import hollight SUM_SUM_RESTRICT)
-
-lemma CARD_EQ_SUM: "ALL x::'q_56096::type => bool.
- FINITE x -->
- real_of_num (CARD x) =
- hollight.sum x (%x::'q_56096::type. real_of_num (NUMERAL_BIT1 0))"
- by (import hollight CARD_EQ_SUM)
-
-lemma SUM_MULTICOUNT_GEN: "ALL (R::'A::type => 'B::type => bool) (s::'A::type => bool)
- (t::'B::type => bool) k::'B::type => nat.
- FINITE s &
- FINITE t &
- (ALL j::'B::type.
- IN j t -->
- CARD
- (GSPEC
- (%u::'A::type. EX i::'A::type. SETSPEC u (IN i s & R i j) i)) =
- k j) -->
- hollight.sum s
- (%i::'A::type.
- real_of_num
- (CARD
- (GSPEC
- (%u::'B::type.
- EX j::'B::type. SETSPEC u (IN j t & R i j) j)))) =
- hollight.sum t (%i::'B::type. real_of_num (k i))"
- by (import hollight SUM_MULTICOUNT_GEN)
-
-lemma SUM_MULTICOUNT: "ALL (R::'A::type => 'B::type => bool) (s::'A::type => bool)
- (t::'B::type => bool) k::nat.
- FINITE s &
- FINITE t &
- (ALL j::'B::type.
- IN j t -->
- CARD
- (GSPEC
- (%u::'A::type. EX i::'A::type. SETSPEC u (IN i s & R i j) i)) =
- k) -->
- hollight.sum s
- (%i::'A::type.
- real_of_num
- (CARD
- (GSPEC
- (%u::'B::type.
- EX j::'B::type. SETSPEC u (IN j t & R i j) j)))) =
- real_of_num (k * CARD t)"
- by (import hollight SUM_MULTICOUNT)
-
-lemma SUM_IMAGE_GEN: "ALL (f::'A::type => 'B::type) (g::'A::type => hollight.real)
- s::'A::type => bool.
- FINITE s -->
- hollight.sum s g =
- hollight.sum (IMAGE f s)
- (%y::'B::type.
- hollight.sum
- (GSPEC
- (%u::'A::type. EX x::'A::type. SETSPEC u (IN x s & f x = y) x))
- g)"
- by (import hollight SUM_IMAGE_GEN)
-
-lemma REAL_OF_NUM_SUM: "ALL (f::'q_56493::type => nat) s::'q_56493::type => bool.
- FINITE s -->
- real_of_num (nsum s f) =
- hollight.sum s (%x::'q_56493::type. real_of_num (f x))"
- by (import hollight REAL_OF_NUM_SUM)
-
-lemma SUM_SUBSET: "ALL (u::'A::type => bool) (v::'A::type => bool)
- f::'A::type => hollight.real.
- FINITE u &
- FINITE v &
- (ALL x::'A::type. IN x (DIFF u v) --> real_le (f x) (real_of_num 0)) &
- (ALL x::'A::type. IN x (DIFF v u) --> real_le (real_of_num 0) (f x)) -->
- real_le (hollight.sum u f) (hollight.sum v f)"
- by (import hollight SUM_SUBSET)
-
-lemma SUM_SUBSET_SIMPLE: "ALL (u::'A::type => bool) (v::'A::type => bool)
- f::'A::type => hollight.real.
- FINITE v &
- SUBSET u v &
- (ALL x::'A::type. IN x (DIFF v u) --> real_le (real_of_num 0) (f x)) -->
- real_le (hollight.sum u f) (hollight.sum v f)"
- by (import hollight SUM_SUBSET_SIMPLE)
-
-lemma SUM_ADD_NUMSEG: "ALL (x::nat => hollight.real) (xa::nat => hollight.real) (xb::nat) xc::nat.
- hollight.sum (dotdot xb xc) (%i::nat. real_add (x i) (xa i)) =
- real_add (hollight.sum (dotdot xb xc) x) (hollight.sum (dotdot xb xc) xa)"
- by (import hollight SUM_ADD_NUMSEG)
-
-lemma SUM_CMUL_NUMSEG: "ALL (x::nat => hollight.real) (xa::hollight.real) (xb::nat) xc::nat.
- hollight.sum (dotdot xb xc) (%i::nat. real_mul xa (x i)) =
- real_mul xa (hollight.sum (dotdot xb xc) x)"
- by (import hollight SUM_CMUL_NUMSEG)
-
-lemma SUM_NEG_NUMSEG: "ALL (x::nat => hollight.real) (xa::nat) xb::nat.
- hollight.sum (dotdot xa xb) (%i::nat. real_neg (x i)) =
- real_neg (hollight.sum (dotdot xa xb) x)"
- by (import hollight SUM_NEG_NUMSEG)
-
-lemma SUM_SUB_NUMSEG: "ALL (x::nat => hollight.real) (xa::nat => hollight.real) (xb::nat) xc::nat.
- hollight.sum (dotdot xb xc) (%i::nat. real_sub (x i) (xa i)) =
- real_sub (hollight.sum (dotdot xb xc) x) (hollight.sum (dotdot xb xc) xa)"
- by (import hollight SUM_SUB_NUMSEG)
-
-lemma SUM_LE_NUMSEG: "ALL (x::nat => hollight.real) (xa::nat => hollight.real) (xb::nat) xc::nat.
- (ALL i::nat. <= xb i & <= i xc --> real_le (x i) (xa i)) -->
- real_le (hollight.sum (dotdot xb xc) x) (hollight.sum (dotdot xb xc) xa)"
- by (import hollight SUM_LE_NUMSEG)
-
-lemma SUM_EQ_NUMSEG: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
- (ALL i::nat. <= m i & <= i n --> f i = g i) -->
- hollight.sum (dotdot m n) f = hollight.sum (dotdot m n) g"
- by (import hollight SUM_EQ_NUMSEG)
-
-lemma SUM_ABS_NUMSEG: "ALL (x::nat => hollight.real) (xa::nat) xb::nat.
- real_le (real_abs (hollight.sum (dotdot xa xb) x))
- (hollight.sum (dotdot xa xb) (%i::nat. real_abs (x i)))"
- by (import hollight SUM_ABS_NUMSEG)
-
-lemma SUM_CONST_NUMSEG: "ALL (x::hollight.real) (xa::nat) xb::nat.
- hollight.sum (dotdot xa xb) (%n::nat. x) =
- real_mul (real_of_num (xb + NUMERAL_BIT1 0 - xa)) x"
- by (import hollight SUM_CONST_NUMSEG)
-
-lemma SUM_EQ_0_NUMSEG: "ALL (x::nat => hollight.real) xa::'q_57019::type.
- (ALL i::nat. <= (m::nat) i & <= i (n::nat) --> x i = real_of_num 0) -->
- hollight.sum (dotdot m n) x = real_of_num 0"
- by (import hollight SUM_EQ_0_NUMSEG)
-
-lemma SUM_TRIV_NUMSEG: "ALL (f::nat => hollight.real) (m::nat) n::nat.
- < n m --> hollight.sum (dotdot m n) f = real_of_num 0"
- by (import hollight SUM_TRIV_NUMSEG)
-
-lemma SUM_POS_LE_NUMSEG: "ALL (x::nat) (xa::nat) xb::nat => hollight.real.
- (ALL p::nat. <= x p & <= p xa --> real_le (real_of_num 0) (xb p)) -->
- real_le (real_of_num 0) (hollight.sum (dotdot x xa) xb)"
- by (import hollight SUM_POS_LE_NUMSEG)
-
-lemma SUM_POS_EQ_0_NUMSEG: "ALL (f::nat => hollight.real) (m::nat) n::nat.
- (ALL p::nat. <= m p & <= p n --> real_le (real_of_num 0) (f p)) &
- hollight.sum (dotdot m n) f = real_of_num 0 -->
- (ALL p::nat. <= m p & <= p n --> f p = real_of_num 0)"
- by (import hollight SUM_POS_EQ_0_NUMSEG)
-
-lemma SUM_SING_NUMSEG: "ALL (x::nat => hollight.real) xa::nat. hollight.sum (dotdot xa xa) x = x xa"
- by (import hollight SUM_SING_NUMSEG)
-
-lemma SUM_CLAUSES_NUMSEG: "(ALL x::nat.
- hollight.sum (dotdot x 0) (f::nat => hollight.real) =
- COND (x = 0) (f 0) (real_of_num 0)) &
-(ALL (x::nat) xa::nat.
- hollight.sum (dotdot x (Suc xa)) f =
- COND (<= x (Suc xa))
- (real_add (hollight.sum (dotdot x xa) f) (f (Suc xa)))
- (hollight.sum (dotdot x xa) f))"
- by (import hollight SUM_CLAUSES_NUMSEG)
-
-lemma SUM_SWAP_NUMSEG: "ALL (a::nat) (b::nat) (c::nat) (d::nat) f::nat => nat => hollight.real.
- hollight.sum (dotdot a b) (%i::nat. hollight.sum (dotdot c d) (f i)) =
- hollight.sum (dotdot c d)
- (%j::nat. hollight.sum (dotdot a b) (%i::nat. f i j))"
- by (import hollight SUM_SWAP_NUMSEG)
-
-lemma SUM_ADD_SPLIT: "ALL (x::nat => hollight.real) (xa::nat) (xb::nat) xc::nat.
- <= xa (xb + NUMERAL_BIT1 0) -->
- hollight.sum (dotdot xa (xb + xc)) x =
- real_add (hollight.sum (dotdot xa xb) x)
- (hollight.sum (dotdot (xb + NUMERAL_BIT1 0) (xb + xc)) x)"
- by (import hollight SUM_ADD_SPLIT)
-
-lemma SUM_OFFSET: "ALL (x::nat => hollight.real) (xa::nat) xb::nat.
- hollight.sum (dotdot (xa + xb) ((n::nat) + xb)) x =
- hollight.sum (dotdot xa n) (%i::nat. x (i + xb))"
- by (import hollight SUM_OFFSET)
-
-lemma SUM_OFFSET_0: "ALL (x::nat => hollight.real) (xa::nat) xb::nat.
- <= xa xb -->
- hollight.sum (dotdot xa xb) x =
- hollight.sum (dotdot 0 (xb - xa)) (%i::nat. x (i + xa))"
- by (import hollight SUM_OFFSET_0)
-
-lemma SUM_CLAUSES_LEFT: "ALL (x::nat => hollight.real) (xa::nat) xb::nat.
- <= xa xb -->
- hollight.sum (dotdot xa xb) x =
- real_add (x xa) (hollight.sum (dotdot (xa + NUMERAL_BIT1 0) xb) x)"
- by (import hollight SUM_CLAUSES_LEFT)
-
-lemma SUM_CLAUSES_RIGHT: "ALL (f::nat => hollight.real) (m::nat) n::nat.
- < 0 n & <= m n -->
- hollight.sum (dotdot m n) f =
- real_add (hollight.sum (dotdot m (n - NUMERAL_BIT1 0)) f) (f n)"
- by (import hollight SUM_CLAUSES_RIGHT)
-
-lemma REAL_OF_NUM_SUM_NUMSEG: "ALL (x::nat => nat) (xa::nat) xb::nat.
- real_of_num (nsum (dotdot xa xb) x) =
- hollight.sum (dotdot xa xb) (%i::nat. real_of_num (x i))"
- by (import hollight REAL_OF_NUM_SUM_NUMSEG)
-
-lemma SUM_BIJECTION: "ALL (x::'A::type => hollight.real) (xa::'A::type => 'A::type)
- xb::'A::type => bool.
- FINITE xb &
- (ALL x::'A::type. IN x xb --> IN (xa x) xb) &
- (ALL y::'A::type. IN y xb --> (EX! x::'A::type. IN x xb & xa x = y)) -->
- hollight.sum xb x = hollight.sum xb (x o xa)"
- by (import hollight SUM_BIJECTION)
-
-lemma SUM_SUM_PRODUCT: "ALL (x::'A::type => bool) (xa::'A::type => 'B::type => bool)
- xb::'A::type => 'B::type => hollight.real.
- FINITE x & (ALL i::'A::type. IN i x --> FINITE (xa i)) -->
- hollight.sum x (%x::'A::type. hollight.sum (xa x) (xb x)) =
- hollight.sum
- (GSPEC
- (%u::'A::type * 'B::type.
- EX (i::'A::type) j::'B::type.
- SETSPEC u (IN i x & IN j (xa i)) (i, j)))
- (GABS
- (%f::'A::type * 'B::type => hollight.real.
- ALL (i::'A::type) j::'B::type. GEQ (f (i, j)) (xb i j)))"
- by (import hollight SUM_SUM_PRODUCT)
-
-lemma SUM_EQ_GENERAL: "ALL (x::'A::type => bool) (xa::'B::type => bool)
- (xb::'A::type => hollight.real) (xc::'B::type => hollight.real)
- xd::'A::type => 'B::type.
- FINITE x &
- (ALL y::'B::type. IN y xa --> (EX! xa::'A::type. IN xa x & xd xa = y)) &
- (ALL xe::'A::type. IN xe x --> IN (xd xe) xa & xc (xd xe) = xb xe) -->
- hollight.sum x xb = hollight.sum xa xc"
- by (import hollight SUM_EQ_GENERAL)
-
-definition CASEWISE :: "(('q_57926 => 'q_57930) * ('q_57931 => 'q_57926 => 'q_57890)) hollight.list
-=> 'q_57931 => 'q_57930 => 'q_57890" where
+lemma FINITE_SUM_IMAGE: "(op =::(('A::type, 'B::type) finite_sum => bool)
+ => (('A::type, 'B::type) finite_sum => bool) => bool)
+ (UNIV::('A::type, 'B::type) finite_sum => bool)
+ ((op `::(nat => ('A::type, 'B::type) finite_sum)
+ => (nat => bool) => ('A::type, 'B::type) finite_sum => bool)
+ (mk_finite_sum::nat => ('A::type, 'B::type) finite_sum)
+ ((atLeastAtMost::nat => nat => nat => bool) (1::nat)
+ ((op +::nat => nat => nat)
+ ((dimindex::('A::type => bool) => nat) (UNIV::'A::type => bool))
+ ((dimindex::('B::type => bool) => nat) (UNIV::'B::type => bool)))))"
+ by (import hollight FINITE_SUM_IMAGE)
+
+lemma HAS_SIZE_1: "(HAS_SIZE::(unit => bool) => nat => bool) (UNIV::unit => bool) (1::nat)"
+ by (import hollight HAS_SIZE_1)
+
+typedef (open) N_2 = "{x. x : dotdot (NUMERAL (NUMERAL_BIT1 0))
+ (NUMERAL (NUMERAL_BIT0 (NUMERAL_BIT1 0)))}" morphisms "dest_auto_define_finite_type_2" "mk_auto_define_finite_type_2"
+ apply (rule light_ex_imp_nonempty[where t="SOME x.
+ x : dotdot (NUMERAL (NUMERAL_BIT1 0))
+ (NUMERAL (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"])
+ by (import hollight TYDEF_2)
+
+syntax
+ dest_auto_define_finite_type_2 :: _
+
+syntax
+ mk_auto_define_finite_type_2 :: _
+
+lemmas "TYDEF_2_@intern" = typedef_hol2hollight
+ [where a="a :: N_2" and r=r ,
+ OF type_definition_N_2]
+
+typedef (open) N_3 = "{x. x : dotdot (NUMERAL (NUMERAL_BIT1 0))
+ (NUMERAL (NUMERAL_BIT1 (NUMERAL_BIT1 0)))}" morphisms "dest_auto_define_finite_type_3" "mk_auto_define_finite_type_3"
+ apply (rule light_ex_imp_nonempty[where t="SOME x.
+ x : dotdot (NUMERAL (NUMERAL_BIT1 0))
+ (NUMERAL (NUMERAL_BIT1 (NUMERAL_BIT1 0)))"])
+ by (import hollight TYDEF_3)
+
+syntax
+ dest_auto_define_finite_type_3 :: _
+
+syntax
+ mk_auto_define_finite_type_3 :: _
+
+lemmas "TYDEF_3_@intern" = typedef_hol2hollight
+ [where a="a :: N_3" and r=r ,
+ OF type_definition_N_3]
+
+lemma FINITE_CART: "(op ==>::prop => prop => prop)
+ ((all::(nat => prop) => prop)
+ (%i::nat.
+ (op ==>::prop => prop => prop)
+ ((Trueprop::bool => prop)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
+ ((dimindex::('N::type => bool) => nat)
+ (UNIV::'N::type => bool)))))
+ ((Trueprop::bool => prop)
+ ((finite::('A::type => bool) => bool)
+ ((Collect::('A::type => bool) => 'A::type => bool)
+ (%u::'A::type.
+ (Ex::('A::type => bool) => bool)
+ (%x::'A::type.
+ (op &::bool => bool => bool)
+ ((P::nat => 'A::type => bool) i x)
+ ((op =::'A::type => 'A::type => bool) u x))))))))
+ ((Trueprop::bool => prop)
+ ((finite::(('A::type, 'N::type) cart => bool) => bool)
+ ((Collect::(('A::type, 'N::type) cart => bool)
+ => ('A::type, 'N::type) cart => bool)
+ (%u::('A::type, 'N::type) cart.
+ (Ex::(('A::type, 'N::type) cart => bool) => bool)
+ (%v::('A::type, 'N::type) cart.
+ (op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%i::nat.
+ (op -->::bool => bool => bool)
+ ((op &::bool => bool => bool)
+ ((op <=::nat => nat => bool) (1::nat) i)
+ ((op <=::nat => nat => bool) i
+ ((dimindex::('N::type => bool) => nat)
+ (UNIV::'N::type => bool))))
+ (P i (($::('A::type, 'N::type) cart
+ => nat => 'A::type)
+ v i))))
+ ((op =::('A::type, 'N::type) cart
+ => ('A::type, 'N::type) cart => bool)
+ u v))))))"
+ by (import hollight FINITE_CART)
+
+definition
+ vector :: "'A list => ('A, 'N) cart" where
+ "(op ==::('A::type list => ('A::type, 'N::type) cart)
+ => ('A::type list => ('A::type, 'N::type) cart) => prop)
+ (vector::'A::type list => ('A::type, 'N::type) cart)
+ (%u::'A::type list.
+ (lambda::(nat => 'A::type) => ('A::type, 'N::type) cart)
+ (%i::nat.
+ (op !::'A::type list => nat => 'A::type) u
+ ((op -::nat => nat => nat) i (1::nat))))"
+
+lemma DEF_vector: "(op =::('A::type list => ('A::type, 'N::type) cart)
+ => ('A::type list => ('A::type, 'N::type) cart) => bool)
+ (vector::'A::type list => ('A::type, 'N::type) cart)
+ (%u::'A::type list.
+ (lambda::(nat => 'A::type) => ('A::type, 'N::type) cart)
+ (%i::nat.
+ (op !::'A::type list => nat => 'A::type) u
+ ((op -::nat => nat => nat) i (1::nat))))"
+ by (import hollight DEF_vector)
+
+definition
+ CASEWISE :: "(('q_74835 => 'q_74839) * ('q_74840 => 'q_74835 => 'q_74799)) list
+=> 'q_74840 => 'q_74839 => 'q_74799" where
"CASEWISE ==
-SOME CASEWISE::(('q_57926::type => 'q_57930::type) *
- ('q_57931::type
- => 'q_57926::type => 'q_57890::type)) hollight.list
- => 'q_57931::type => 'q_57930::type => 'q_57890::type.
- (ALL (f::'q_57931::type) x::'q_57930::type.
- CASEWISE NIL f x = (SOME y::'q_57890::type. True)) &
- (ALL (h::('q_57926::type => 'q_57930::type) *
- ('q_57931::type => 'q_57926::type => 'q_57890::type))
- (t::(('q_57926::type => 'q_57930::type) *
- ('q_57931::type
- => 'q_57926::type => 'q_57890::type)) hollight.list)
- (f::'q_57931::type) x::'q_57930::type.
- CASEWISE (CONS h t) f x =
- COND (EX y::'q_57926::type. fst h y = x)
- (snd h f (SOME y::'q_57926::type. fst h y = x)) (CASEWISE t f x))"
+SOME CASEWISE::(('q_74835 => 'q_74839) *
+ ('q_74840 => 'q_74835 => 'q_74799)) list
+ => 'q_74840 => 'q_74839 => 'q_74799.
+ (ALL (f::'q_74840) x::'q_74839.
+ CASEWISE [] f x = (SOME y::'q_74799. True)) &
+ (ALL (h::('q_74835 => 'q_74839) * ('q_74840 => 'q_74835 => 'q_74799))
+ (t::(('q_74835 => 'q_74839) *
+ ('q_74840 => 'q_74835 => 'q_74799)) list)
+ (f::'q_74840) x::'q_74839.
+ CASEWISE (h # t) f x =
+ (if EX y::'q_74835. fst h y = x
+ then snd h f (SOME y::'q_74835. fst h y = x) else CASEWISE t f x))"
lemma DEF_CASEWISE: "CASEWISE =
-(SOME CASEWISE::(('q_57926::type => 'q_57930::type) *
- ('q_57931::type
- => 'q_57926::type => 'q_57890::type)) hollight.list
- => 'q_57931::type => 'q_57930::type => 'q_57890::type.
- (ALL (f::'q_57931::type) x::'q_57930::type.
- CASEWISE NIL f x = (SOME y::'q_57890::type. True)) &
- (ALL (h::('q_57926::type => 'q_57930::type) *
- ('q_57931::type => 'q_57926::type => 'q_57890::type))
- (t::(('q_57926::type => 'q_57930::type) *
- ('q_57931::type
- => 'q_57926::type => 'q_57890::type)) hollight.list)
- (f::'q_57931::type) x::'q_57930::type.
- CASEWISE (CONS h t) f x =
- COND (EX y::'q_57926::type. fst h y = x)
- (snd h f (SOME y::'q_57926::type. fst h y = x)) (CASEWISE t f x)))"
+(SOME CASEWISE::(('q_74835 => 'q_74839) *
+ ('q_74840 => 'q_74835 => 'q_74799)) list
+ => 'q_74840 => 'q_74839 => 'q_74799.
+ (ALL (f::'q_74840) x::'q_74839.
+ CASEWISE [] f x = (SOME y::'q_74799. True)) &
+ (ALL (h::('q_74835 => 'q_74839) * ('q_74840 => 'q_74835 => 'q_74799))
+ (t::(('q_74835 => 'q_74839) *
+ ('q_74840 => 'q_74835 => 'q_74799)) list)
+ (f::'q_74840) x::'q_74839.
+ CASEWISE (h # t) f x =
+ (if EX y::'q_74835. fst h y = x
+ then snd h f (SOME y::'q_74835. fst h y = x) else CASEWISE t f x)))"
by (import hollight DEF_CASEWISE)
-lemma CASEWISE: "(op &::bool => bool => bool)
- ((op =::'q_57950::type => 'q_57950::type => bool)
- ((CASEWISE::(('q_57942::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57942::type => 'q_57950::type)) hollight.list
- => 'q_57991::type => 'q_57990::type => 'q_57950::type)
- (NIL::(('q_57942::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57942::type => 'q_57950::type)) hollight.list)
- (f::'q_57991::type) (x::'q_57990::type))
- ((Eps::('q_57950::type => bool) => 'q_57950::type)
- (%y::'q_57950::type. True::bool)))
- ((op =::'q_57951::type => 'q_57951::type => bool)
- ((CASEWISE::(('q_57993::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57993::type => 'q_57951::type)) hollight.list
- => 'q_57991::type => 'q_57990::type => 'q_57951::type)
- ((CONS::('q_57993::type => 'q_57990::type) *
- ('q_57991::type => 'q_57993::type => 'q_57951::type)
- => (('q_57993::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57993::type => 'q_57951::type)) hollight.list
- => (('q_57993::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57993::type => 'q_57951::type)) hollight.list)
- ((Pair::('q_57993::type => 'q_57990::type)
- => ('q_57991::type => 'q_57993::type => 'q_57951::type)
- => ('q_57993::type => 'q_57990::type) *
- ('q_57991::type => 'q_57993::type => 'q_57951::type))
- (s::'q_57993::type => 'q_57990::type)
- (t::'q_57991::type => 'q_57993::type => 'q_57951::type))
- (clauses::(('q_57993::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57993::type => 'q_57951::type)) hollight.list))
- f x)
- ((COND::bool => 'q_57951::type => 'q_57951::type => 'q_57951::type)
- ((Ex::('q_57993::type => bool) => bool)
- (%y::'q_57993::type.
- (op =::'q_57990::type => 'q_57990::type => bool) (s y) x))
- (t f ((Eps::('q_57993::type => bool) => 'q_57993::type)
- (%y::'q_57993::type.
- (op =::'q_57990::type => 'q_57990::type => bool) (s y) x)))
- ((CASEWISE::(('q_57993::type => 'q_57990::type) *
- ('q_57991::type
- => 'q_57993::type => 'q_57951::type)) hollight.list
- => 'q_57991::type => 'q_57990::type => 'q_57951::type)
- clauses f x)))"
- by (import hollight CASEWISE)
-
-lemma CASEWISE_CASES: "ALL (clauses::(('q_58085::type => 'q_58082::type) *
- ('q_58083::type
- => 'q_58085::type => 'q_58092::type)) hollight.list)
- (c::'q_58083::type) x::'q_58082::type.
- (EX (s::'q_58085::type => 'q_58082::type)
- (t::'q_58083::type => 'q_58085::type => 'q_58092::type)
- a::'q_58085::type.
- MEM (s, t) clauses & s a = x & CASEWISE clauses c x = t c a) |
- ~ (EX (s::'q_58085::type => 'q_58082::type)
- (t::'q_58083::type => 'q_58085::type => 'q_58092::type)
- a::'q_58085::type. MEM (s, t) clauses & s a = x) &
- CASEWISE clauses c x = (SOME y::'q_58092::type. True)"
- by (import hollight CASEWISE_CASES)
-
-lemma CASEWISE_WORKS: "ALL (x::(('P::type => 'A::type) *
- ('C::type => 'P::type => 'B::type)) hollight.list)
- xa::'C::type.
- (ALL (s::'P::type => 'A::type) (t::'C::type => 'P::type => 'B::type)
- (s'::'P::type => 'A::type) (t'::'C::type => 'P::type => 'B::type)
- (xb::'P::type) y::'P::type.
- MEM (s, t) x & MEM (s', t') x & s xb = s' y -->
- t xa xb = t' xa y) -->
- ALL_list
- (GABS
- (%f::('P::type => 'A::type) * ('C::type => 'P::type => 'B::type)
- => bool.
- ALL (s::'P::type => 'A::type) t::'C::type => 'P::type => 'B::type.
- GEQ (f (s, t))
- (ALL xb::'P::type. CASEWISE x xa (s xb) = t xa xb)))
- x"
- by (import hollight CASEWISE_WORKS)
-
-definition admissible :: "('q_58228 => 'q_58221 => bool)
-=> (('q_58228 => 'q_58224) => 'q_58234 => bool)
- => ('q_58234 => 'q_58221)
- => (('q_58228 => 'q_58224) => 'q_58234 => 'q_58229) => bool" where
+definition
+ admissible :: "('q_75137 => 'q_75130 => bool)
+=> (('q_75137 => 'q_75133) => 'q_75143 => bool)
+ => ('q_75143 => 'q_75130)
+ => (('q_75137 => 'q_75133) => 'q_75143 => 'q_75138) => bool" where
"admissible ==
-%(u::'q_58228::type => 'q_58221::type => bool)
- (ua::('q_58228::type => 'q_58224::type) => 'q_58234::type => bool)
- (ub::'q_58234::type => 'q_58221::type)
- uc::('q_58228::type => 'q_58224::type)
- => 'q_58234::type => 'q_58229::type.
- ALL (f::'q_58228::type => 'q_58224::type)
- (g::'q_58228::type => 'q_58224::type) a::'q_58234::type.
- ua f a &
- ua g a & (ALL z::'q_58228::type. u z (ub a) --> f z = g z) -->
+%(u::'q_75137 => 'q_75130 => bool)
+ (ua::('q_75137 => 'q_75133) => 'q_75143 => bool)
+ (ub::'q_75143 => 'q_75130)
+ uc::('q_75137 => 'q_75133) => 'q_75143 => 'q_75138.
+ ALL (f::'q_75137 => 'q_75133) (g::'q_75137 => 'q_75133) a::'q_75143.
+ ua f a & ua g a & (ALL z::'q_75137. u z (ub a) --> f z = g z) -->
uc f a = uc g a"
lemma DEF_admissible: "admissible =
-(%(u::'q_58228::type => 'q_58221::type => bool)
- (ua::('q_58228::type => 'q_58224::type) => 'q_58234::type => bool)
- (ub::'q_58234::type => 'q_58221::type)
- uc::('q_58228::type => 'q_58224::type)
- => 'q_58234::type => 'q_58229::type.
- ALL (f::'q_58228::type => 'q_58224::type)
- (g::'q_58228::type => 'q_58224::type) a::'q_58234::type.
- ua f a &
- ua g a & (ALL z::'q_58228::type. u z (ub a) --> f z = g z) -->
+(%(u::'q_75137 => 'q_75130 => bool)
+ (ua::('q_75137 => 'q_75133) => 'q_75143 => bool)
+ (ub::'q_75143 => 'q_75130)
+ uc::('q_75137 => 'q_75133) => 'q_75143 => 'q_75138.
+ ALL (f::'q_75137 => 'q_75133) (g::'q_75137 => 'q_75133) a::'q_75143.
+ ua f a & ua g a & (ALL z::'q_75137. u z (ub a) --> f z = g z) -->
uc f a = uc g a)"
by (import hollight DEF_admissible)
-definition tailadmissible :: "('A => 'A => bool)
+definition
+ tailadmissible :: "('A => 'A => bool)
=> (('A => 'B) => 'P => bool)
- => ('P => 'A) => (('A => 'B) => 'P => 'B) => bool" where
+ => ('P => 'A) => (('A => 'B) => 'P => 'B) => bool" where
"tailadmissible ==
-%(u::'A::type => 'A::type => bool)
- (ua::('A::type => 'B::type) => 'P::type => bool)
- (ub::'P::type => 'A::type)
- uc::('A::type => 'B::type) => 'P::type => 'B::type.
- EX (P::('A::type => 'B::type) => 'P::type => bool)
- (G::('A::type => 'B::type) => 'P::type => 'A::type)
- H::('A::type => 'B::type) => 'P::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (a::'P::type) y::'A::type.
+%(u::'A => 'A => bool) (ua::('A => 'B) => 'P => bool) (ub::'P => 'A)
+ uc::('A => 'B) => 'P => 'B.
+ EX (P::('A => 'B) => 'P => bool) (G::('A => 'B) => 'P => 'A)
+ H::('A => 'B) => 'P => 'B.
+ (ALL (f::'A => 'B) (a::'P) y::'A.
P f a & u y (G f a) --> u y (ub a)) &
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) a::'P::type.
- (ALL z::'A::type. u z (ub a) --> f z = g z) -->
+ (ALL (f::'A => 'B) (g::'A => 'B) a::'P.
+ (ALL z::'A. u z (ub a) --> f z = g z) -->
P f a = P g a & G f a = G g a & H f a = H g a) &
- (ALL (f::'A::type => 'B::type) a::'P::type.
- ua f a --> uc f a = COND (P f a) (f (G f a)) (H f a))"
+ (ALL (f::'A => 'B) a::'P.
+ ua f a --> uc f a = (if P f a then f (G f a) else H f a))"
lemma DEF_tailadmissible: "tailadmissible =
-(%(u::'A::type => 'A::type => bool)
- (ua::('A::type => 'B::type) => 'P::type => bool)
- (ub::'P::type => 'A::type)
- uc::('A::type => 'B::type) => 'P::type => 'B::type.
- EX (P::('A::type => 'B::type) => 'P::type => bool)
- (G::('A::type => 'B::type) => 'P::type => 'A::type)
- H::('A::type => 'B::type) => 'P::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (a::'P::type) y::'A::type.
+(%(u::'A => 'A => bool) (ua::('A => 'B) => 'P => bool) (ub::'P => 'A)
+ uc::('A => 'B) => 'P => 'B.
+ EX (P::('A => 'B) => 'P => bool) (G::('A => 'B) => 'P => 'A)
+ H::('A => 'B) => 'P => 'B.
+ (ALL (f::'A => 'B) (a::'P) y::'A.
P f a & u y (G f a) --> u y (ub a)) &
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) a::'P::type.
- (ALL z::'A::type. u z (ub a) --> f z = g z) -->
+ (ALL (f::'A => 'B) (g::'A => 'B) a::'P.
+ (ALL z::'A. u z (ub a) --> f z = g z) -->
P f a = P g a & G f a = G g a & H f a = H g a) &
- (ALL (f::'A::type => 'B::type) a::'P::type.
- ua f a --> uc f a = COND (P f a) (f (G f a)) (H f a)))"
+ (ALL (f::'A => 'B) a::'P.
+ ua f a --> uc f a = (if P f a then f (G f a) else H f a)))"
by (import hollight DEF_tailadmissible)
-definition superadmissible :: "('q_58378 => 'q_58378 => bool)
-=> (('q_58378 => 'q_58380) => 'q_58386 => bool)
- => ('q_58386 => 'q_58378)
- => (('q_58378 => 'q_58380) => 'q_58386 => 'q_58380) => bool" where
+definition
+ superadmissible :: "('q_75287 => 'q_75287 => bool)
+=> (('q_75287 => 'q_75289) => 'q_75295 => bool)
+ => ('q_75295 => 'q_75287)
+ => (('q_75287 => 'q_75289) => 'q_75295 => 'q_75289) => bool" where
"superadmissible ==
-%(u::'q_58378::type => 'q_58378::type => bool)
- (ua::('q_58378::type => 'q_58380::type) => 'q_58386::type => bool)
- (ub::'q_58386::type => 'q_58378::type)
- uc::('q_58378::type => 'q_58380::type)
- => 'q_58386::type => 'q_58380::type.
- admissible u
- (%(f::'q_58378::type => 'q_58380::type) a::'q_58386::type. True) ub
- ua -->
+%(u::'q_75287 => 'q_75287 => bool)
+ (ua::('q_75287 => 'q_75289) => 'q_75295 => bool)
+ (ub::'q_75295 => 'q_75287)
+ uc::('q_75287 => 'q_75289) => 'q_75295 => 'q_75289.
+ admissible u (%(f::'q_75287 => 'q_75289) a::'q_75295. True) ub ua -->
tailadmissible u ua ub uc"
lemma DEF_superadmissible: "superadmissible =
-(%(u::'q_58378::type => 'q_58378::type => bool)
- (ua::('q_58378::type => 'q_58380::type) => 'q_58386::type => bool)
- (ub::'q_58386::type => 'q_58378::type)
- uc::('q_58378::type => 'q_58380::type)
- => 'q_58386::type => 'q_58380::type.
- admissible u
- (%(f::'q_58378::type => 'q_58380::type) a::'q_58386::type. True) ub
- ua -->
+(%(u::'q_75287 => 'q_75287 => bool)
+ (ua::('q_75287 => 'q_75289) => 'q_75295 => bool)
+ (ub::'q_75295 => 'q_75287)
+ uc::('q_75287 => 'q_75289) => 'q_75295 => 'q_75289.
+ admissible u (%(f::'q_75287 => 'q_75289) a::'q_75295. True) ub ua -->
tailadmissible u ua ub uc)"
by (import hollight DEF_superadmissible)
-lemma SUPERADMISSIBLE_COND: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::('A::type => 'B::type) => 'P::type => bool)
- (xc::'P::type => 'A::type)
- (xd::('A::type => 'B::type) => 'P::type => 'B::type)
- xe::('A::type => 'B::type) => 'P::type => 'B::type.
- admissible x xa xc xb &
- superadmissible x
- (%(f::'A::type => 'B::type) x::'P::type. xa f x & xb f x) xc xd &
- superadmissible x
- (%(f::'A::type => 'B::type) x::'P::type. xa f x & ~ xb f x) xc xe -->
- superadmissible x xa xc
- (%(f::'A::type => 'B::type) x::'P::type.
- COND (xb f x) (xd f x) (xe f x))"
- by (import hollight SUPERADMISSIBLE_COND)
-
-lemma ADMISSIBLE_IMP_SUPERADMISSIBLE: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- xc::('A::type => 'B::type) => 'P::type => 'B::type.
- admissible x xa xb xc --> superadmissible x xa xb xc"
- by (import hollight ADMISSIBLE_IMP_SUPERADMISSIBLE)
-
-lemma TAIL_IMP_SUPERADMISSIBLE: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- xc::('A::type => 'B::type) => 'P::type => 'A::type.
- admissible x xa xb xc &
- (ALL (f::'A::type => 'B::type) a::'P::type.
- xa f a --> (ALL y::'A::type. x y (xc f a) --> x y (xb a))) -->
- superadmissible x xa xb
- (%(f::'A::type => 'B::type) x::'P::type. f (xc f x))"
- by (import hollight TAIL_IMP_SUPERADMISSIBLE)
-
-lemma ADMISSIBLE_COND: "ALL (u_353::'A::type => 'q_58766::type => bool)
- (p::('A::type => 'B::type) => 'P::type => bool)
- (P::('A::type => 'B::type) => 'P::type => bool)
- (s::'P::type => 'q_58766::type)
- (h::('A::type => 'B::type) => 'P::type => 'B::type)
- k::('A::type => 'B::type) => 'P::type => 'B::type.
- admissible u_353 p s P &
- admissible u_353 (%(f::'A::type => 'B::type) x::'P::type. p f x & P f x)
- s h &
- admissible u_353
- (%(f::'A::type => 'B::type) x::'P::type. p f x & ~ P f x) s k -->
- admissible u_353 p s
- (%(f::'A::type => 'B::type) x::'P::type. COND (P f x) (h f x) (k f x))"
- by (import hollight ADMISSIBLE_COND)
-
-lemma ADMISSIBLE_CONST: "admissible (u_353::'q_58841::type => 'q_58840::type => bool)
- (p::('q_58841::type => 'q_58842::type) => 'q_58843::type => bool)
- (s::'q_58843::type => 'q_58840::type)
- (%f::'q_58841::type => 'q_58842::type. c::'q_58843::type => 'q_58844::type)"
+lemma MATCH_SEQPATTERN: "_MATCH (x::'q_75330)
+ (_SEQPATTERN (r::'q_75330 => 'q_75323 => bool)
+ (s::'q_75330 => 'q_75323 => bool)) =
+(if Ex (r x) then _MATCH x r else _MATCH x s)"
+ by (import hollight MATCH_SEQPATTERN)
+
+lemma ADMISSIBLE_CONST: "admissible (u_556::'q_75351 => 'q_75350 => bool)
+ (x::('q_75351 => 'q_75352) => 'q_75353 => bool) (xa::'q_75353 => 'q_75350)
+ (%f::'q_75351 => 'q_75352. xb::'q_75353 => 'q_75354)"
by (import hollight ADMISSIBLE_CONST)
-lemma ADMISSIBLE_COMB: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- (xc::('A::type => 'B::type) => 'P::type => 'C::type => 'D::type)
- xd::('A::type => 'B::type) => 'P::type => 'C::type.
- admissible x xa xb xc & admissible x xa xb xd -->
- admissible x xa xb
- (%(f::'A::type => 'B::type) x::'P::type. xc f x (xd f x))"
+lemma ADMISSIBLE_BASE: "(!!(f::'A => 'B) a::'P.
+ (xa::('A => 'B) => 'P => bool) f a
+ ==> (x::'A => 'A => bool) ((xc::'P => 'A) a) ((xb::'P => 'A) a))
+==> admissible x xa xb (%(f::'A => 'B) x::'P. f (xc x))"
+ by (import hollight ADMISSIBLE_BASE)
+
+lemma ADMISSIBLE_COMB: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'A) (xc::('A => 'B) => 'P => 'C => 'D) &
+admissible x xa xb (xd::('A => 'B) => 'P => 'C)
+==> admissible x xa xb (%(f::'A => 'B) x::'P. xc f x (xd f x))"
by (import hollight ADMISSIBLE_COMB)
-lemma ADMISSIBLE_BASE: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type) xc::'P::type => 'A::type.
- (ALL (f::'A::type => 'B::type) a::'P::type.
- xa f a --> x (xc a) (xb a)) -->
- admissible x xa xb (%(f::'A::type => 'B::type) x::'P::type. f (xc x))"
- by (import hollight ADMISSIBLE_BASE)
-
-lemma ADMISSIBLE_NEST: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- xc::('A::type => 'B::type) => 'P::type => 'A::type.
- admissible x xa xb xc &
- (ALL (f::'A::type => 'B::type) a::'P::type.
- xa f a --> x (xc f a) (xb a)) -->
- admissible x xa xb (%(f::'A::type => 'B::type) x::'P::type. f (xc f x))"
+lemma ADMISSIBLE_RAND: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'A) (xd::('A => 'B) => 'P => 'C)
+==> admissible x xa xb
+ (%(f::'A => 'B) x::'P. (xc::'P => 'C => 'D) x (xd f x))"
+ by (import hollight ADMISSIBLE_RAND)
+
+lemma ADMISSIBLE_LAMBDA: "admissible (x::'A => 'A => bool)
+ (%f::'A => 'B.
+ SOME fa::'C * 'P => bool.
+ ALL (u::'C) x::'P. fa (u, x) = (xa::('A => 'B) => 'P => bool) f x)
+ (SOME f::'C * 'P => 'A. ALL (u::'C) x::'P. f (u, x) = (xb::'P => 'A) x)
+ (%f::'A => 'B.
+ SOME fa::'C * 'P => bool.
+ ALL (u::'C) x::'P.
+ fa (u, x) = (xc::('A => 'B) => 'C => 'P => bool) f u x)
+==> admissible x xa xb (%(f::'A => 'B) (x::'P) u::'C. xc f u x)"
+ by (import hollight ADMISSIBLE_LAMBDA)
+
+lemma ADMISSIBLE_NEST: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'A) (xc::('A => 'B) => 'P => 'A) &
+(ALL (f::'A => 'B) a::'P. xa f a --> x (xc f a) (xb a))
+==> admissible x xa xb (%(f::'A => 'B) x::'P. f (xc f x))"
by (import hollight ADMISSIBLE_NEST)
-lemma ADMISSIBLE_LAMBDA: "ALL (x::'A::type => 'A::type => bool)
- (xa::('A::type => 'B::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- xc::'C::type => ('A::type => 'B::type) => 'P::type => 'D::type.
- (ALL xd::'C::type. admissible x xa xb (xc xd)) -->
- admissible x xa xb
- (%(f::'A::type => 'B::type) (x::'P::type) u::'C::type. xc u f x)"
- by (import hollight ADMISSIBLE_LAMBDA)
-
-lemma ADMISSIBLE_NSUM: "ALL (x::'B::type => 'A::type => bool)
- (xa::('B::type => 'C::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- (xc::('B::type => 'C::type) => 'P::type => nat => nat)
- (xd::'P::type => nat) xe::'P::type => nat.
- (ALL xf::nat.
- admissible x
- (%(f::'B::type => 'C::type) x::'P::type.
- <= (xd x) xf & <= xf (xe x) & xa f x)
- xb (%(f::'B::type => 'C::type) x::'P::type. xc f x xf)) -->
- admissible x xa xb
- (%(f::'B::type => 'C::type) x::'P::type.
- nsum (dotdot (xd x) (xe x)) (xc f x))"
+lemma ADMISSIBLE_COND: "admissible (u_556::'q_75688 => 'q_75687 => bool)
+ (p::('q_75688 => 'q_75719) => 'P => bool) (s::'P => 'q_75687)
+ (P::('q_75688 => 'q_75719) => 'P => bool) &
+admissible u_556 (%(f::'q_75688 => 'q_75719) x::'P. p f x & P f x) s
+ (h::('q_75688 => 'q_75719) => 'P => 'q_75744) &
+admissible u_556 (%(f::'q_75688 => 'q_75719) x::'P. p f x & ~ P f x) s
+ (k::('q_75688 => 'q_75719) => 'P => 'q_75744)
+==> admissible u_556 p s
+ (%(f::'q_75688 => 'q_75719) x::'P. if P f x then h f x else k f x)"
+ by (import hollight ADMISSIBLE_COND)
+
+lemma ADMISSIBLE_MATCH: "admissible (x::'q_75790 => 'q_75789 => bool)
+ (xa::('q_75790 => 'q_75791) => 'P => bool) (xb::'P => 'q_75789)
+ (xc::('q_75790 => 'q_75791) => 'P => 'q_75826) &
+admissible x xa xb
+ (%(f::'q_75790 => 'q_75791) x::'P.
+ (c::('q_75790 => 'q_75791) => 'P => 'q_75826 => 'q_75823 => bool) f x
+ (xc f x))
+==> admissible x xa xb
+ (%(f::'q_75790 => 'q_75791) x::'P. _MATCH (xc f x) (c f x))"
+ by (import hollight ADMISSIBLE_MATCH)
+
+lemma ADMISSIBLE_SEQPATTERN: "admissible (x::'q_75867 => 'q_75866 => bool)
+ (xa::('q_75867 => 'q_75929) => 'P => bool) (xb::'P => 'q_75866)
+ (%(f::'q_75867 => 'q_75929) x::'P.
+ Ex ((xc::('q_75867 => 'q_75929) => 'P => 'q_75955 => 'q_75945 => bool)
+ f x ((xe::('q_75867 => 'q_75929) => 'P => 'q_75955) f x))) &
+admissible x
+ (%(f::'q_75867 => 'q_75929) x::'P. xa f x & Ex (xc f x (xe f x))) xb
+ (%(f::'q_75867 => 'q_75929) x::'P. xc f x (xe f x)) &
+admissible x
+ (%(f::'q_75867 => 'q_75929) x::'P. xa f x & ~ Ex (xc f x (xe f x))) xb
+ (%(f::'q_75867 => 'q_75929) x::'P.
+ (xd::('q_75867 => 'q_75929) => 'P => 'q_75955 => 'q_75945 => bool) f x
+ (xe f x))
+==> admissible x xa xb
+ (%(f::'q_75867 => 'q_75929) x::'P.
+ _SEQPATTERN (xc f x) (xd f x) (xe f x))"
+ by (import hollight ADMISSIBLE_SEQPATTERN)
+
+lemma ADMISSIBLE_UNGUARDED_PATTERN: "admissible (u_556::'q_76041 => 'q_76040 => bool)
+ (p::('q_76041 => 'q_76088) => 'P => bool) (s::'P => 'q_76040)
+ (pat::('q_76041 => 'q_76088) => 'P => 'q_76121) &
+admissible u_556 p s (e::('q_76041 => 'q_76088) => 'P => 'q_76121) &
+admissible u_556 (%(f::'q_76041 => 'q_76088) x::'P. p f x & pat f x = e f x)
+ s (t::('q_76041 => 'q_76088) => 'P => 'q_76128) &
+admissible u_556 (%(f::'q_76041 => 'q_76088) x::'P. p f x & pat f x = e f x)
+ s (y::('q_76041 => 'q_76088) => 'P => 'q_76128)
+==> admissible u_556 p s
+ (%(f::'q_76041 => 'q_76088) x::'P.
+ _UNGUARDED_PATTERN (pat f x = e f x) (t f x = y f x))"
+ by (import hollight ADMISSIBLE_UNGUARDED_PATTERN)
+
+lemma ADMISSIBLE_GUARDED_PATTERN: "admissible (u_556::'q_76215 => 'q_76214 => bool)
+ (p::('q_76215 => 'q_76292) => 'P => bool) (s::'P => 'q_76214)
+ (pat::('q_76215 => 'q_76292) => 'P => 'q_76330) &
+admissible u_556 p s (e::('q_76215 => 'q_76292) => 'P => 'q_76330) &
+admissible u_556
+ (%(f::'q_76215 => 'q_76292) x::'P.
+ p f x &
+ pat f x = e f x & (q::('q_76215 => 'q_76292) => 'P => bool) f x)
+ s (t::('q_76215 => 'q_76292) => 'P => 'q_76339) &
+admissible u_556 (%(f::'q_76215 => 'q_76292) x::'P. p f x & pat f x = e f x)
+ s q &
+admissible u_556
+ (%(f::'q_76215 => 'q_76292) x::'P. p f x & pat f x = e f x & q f x) s
+ (y::('q_76215 => 'q_76292) => 'P => 'q_76339)
+==> admissible u_556 p s
+ (%(f::'q_76215 => 'q_76292) x::'P.
+ _GUARDED_PATTERN (pat f x = e f x) (q f x) (t f x = y f x))"
+ by (import hollight ADMISSIBLE_GUARDED_PATTERN)
+
+lemma ADMISSIBLE_NSUM: "admissible (x::'B => 'A => bool)
+ (%f::'B => 'C.
+ SOME fa::nat * 'P => bool.
+ ALL (k::nat) x::'P.
+ fa (k, x) =
+ ((xd::'P => nat) x <= k &
+ k <= (xe::'P => nat) x & (xa::('B => 'C) => 'P => bool) f x))
+ (SOME f::nat * 'P => 'A. ALL (k::nat) x::'P. f (k, x) = (xb::'P => 'A) x)
+ (%f::'B => 'C.
+ SOME fa::nat * 'P => nat.
+ ALL (k::nat) x::'P.
+ fa (k, x) = (xc::('B => 'C) => 'P => nat => nat) f x k)
+==> admissible x xa xb (%(f::'B => 'C) x::'P. nsum {xd x..xe x} (xc f x))"
by (import hollight ADMISSIBLE_NSUM)
-lemma ADMISSIBLE_SUM: "ALL (x::'B::type => 'A::type => bool)
- (xa::('B::type => 'C::type) => 'P::type => bool)
- (xb::'P::type => 'A::type)
- (xc::('B::type => 'C::type) => 'P::type => nat => hollight.real)
- (xd::'P::type => nat) xe::'P::type => nat.
- (ALL xf::nat.
- admissible x
- (%(f::'B::type => 'C::type) x::'P::type.
- <= (xd x) xf & <= xf (xe x) & xa f x)
- xb (%(f::'B::type => 'C::type) x::'P::type. xc f x xf)) -->
- admissible x xa xb
- (%(f::'B::type => 'C::type) x::'P::type.
- hollight.sum (dotdot (xd x) (xe x)) (xc f x))"
+lemma ADMISSIBLE_SUM: "admissible (x::'B => 'A => bool)
+ (%f::'B => 'C.
+ SOME fa::nat * 'P => bool.
+ ALL (k::nat) x::'P.
+ fa (k, x) =
+ ((xd::'P => nat) x <= k &
+ k <= (xe::'P => nat) x & (xa::('B => 'C) => 'P => bool) f x))
+ (SOME f::nat * 'P => 'A. ALL (k::nat) x::'P. f (k, x) = (xb::'P => 'A) x)
+ (%f::'B => 'C.
+ SOME fa::nat * 'P => hollight.real.
+ ALL (k::nat) x::'P.
+ fa (k, x) = (xc::('B => 'C) => 'P => nat => hollight.real) f x k)
+==> admissible x xa xb
+ (%(f::'B => 'C) x::'P. hollight.sum {xd x..xe x} (xc f x))"
by (import hollight ADMISSIBLE_SUM)
-lemma WF_REC_CASES: "ALL (u_353::'A::type => 'A::type => bool)
- clauses::(('P::type => 'A::type) *
- (('A::type => 'B::type)
- => 'P::type => 'B::type)) hollight.list.
- WF u_353 &
- ALL_list
- (GABS
- (%f::('P::type => 'A::type) *
- (('A::type => 'B::type) => 'P::type => 'B::type)
- => bool.
- ALL (s::'P::type => 'A::type)
- t::('A::type => 'B::type) => 'P::type => 'B::type.
- GEQ (f (s, t))
- (EX (P::('A::type => 'B::type) => 'P::type => bool)
- (G::('A::type => 'B::type) => 'P::type => 'A::type)
- H::('A::type => 'B::type) => 'P::type => 'B::type.
- (ALL (f::'A::type => 'B::type) (a::'P::type) y::'A::type.
- P f a & u_353 y (G f a) --> u_353 y (s a)) &
- (ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type)
- a::'P::type.
- (ALL z::'A::type. u_353 z (s a) --> f z = g z) -->
- P f a = P g a & G f a = G g a & H f a = H g a) &
- (ALL (f::'A::type => 'B::type) a::'P::type.
- t f a = COND (P f a) (f (G f a)) (H f a)))))
- clauses -->
- (EX f::'A::type => 'B::type. ALL x::'A::type. f x = CASEWISE clauses f x)"
- by (import hollight WF_REC_CASES)
-
-lemma RECURSION_CASEWISE: "ALL clauses::(('P::type => 'A::type) *
- (('A::type => 'B::type)
- => 'P::type => 'B::type)) hollight.list.
- (EX u::'A::type => 'A::type => bool.
- WF u &
- ALL_list
- (GABS
- (%f::('P::type => 'A::type) *
- (('A::type => 'B::type) => 'P::type => 'B::type)
- => bool.
- ALL (s::'P::type => 'A::type)
- t::('A::type => 'B::type) => 'P::type => 'B::type.
- GEQ (f (s, t))
- (tailadmissible u
- (%(f::'A::type => 'B::type) a::'P::type. True) s t)))
- clauses) &
- (ALL (x::'P::type => 'A::type)
- (xa::('A::type => 'B::type) => 'P::type => 'B::type)
- (xb::'P::type => 'A::type)
- (xc::('A::type => 'B::type) => 'P::type => 'B::type)
- (xd::'A::type => 'B::type) (xe::'P::type) xf::'P::type.
- MEM (x, xa) clauses & MEM (xb, xc) clauses -->
- x xe = xb xf --> xa xd xe = xc xd xf) -->
- (EX f::'A::type => 'B::type.
- ALL_list
- (GABS
- (%fa::('P::type => 'A::type) *
- (('A::type => 'B::type) => 'P::type => 'B::type)
- => bool.
- ALL (s::'P::type => 'A::type)
- t::('A::type => 'B::type) => 'P::type => 'B::type.
- GEQ (fa (s, t)) (ALL x::'P::type. f (s x) = t f x)))
- clauses)"
- by (import hollight RECURSION_CASEWISE)
-
-lemma cth: "ALL (p1::'A::type => 'q_59947::type)
- (p2::'q_59958::type => 'A::type => 'q_59952::type)
- (p1'::'A::type => 'q_59947::type)
- p2'::'q_59958::type => 'A::type => 'q_59952::type.
- (ALL (c::'q_59958::type) (x::'A::type) y::'A::type.
- p1 x = p1' y --> p2 c x = p2' c y) -->
- (ALL (c::'q_59958::type) (x::'A::type) y::'A::type.
- p1' x = p1 y --> p2' c x = p2 c y)"
+lemma ADMISSIBLE_MAP: "admissible (x::'A => 'q_76632 => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'q_76632) (xd::('A => 'B) => 'P => 'q_76647 list) &
+admissible x
+ (%f::'A => 'B.
+ SOME fa::'q_76647 * 'P => bool.
+ ALL (y::'q_76647) x::'P. fa (y, x) = (xa f x & y : set (xd f x)))
+ (SOME f::'q_76647 * 'P => 'q_76632.
+ ALL (y::'q_76647) x::'P. f (y, x) = xb x)
+ (%f::'A => 'B.
+ SOME fa::'q_76647 * 'P => 'q_76641.
+ ALL (y::'q_76647) x::'P.
+ fa (y, x) = (xc::('A => 'B) => 'P => 'q_76647 => 'q_76641) f x y)
+==> admissible x xa xb (%(f::'A => 'B) x::'P. map (xc f x) (xd f x))"
+ by (import hollight ADMISSIBLE_MAP)
+
+lemma ADMISSIBLE_MATCH_SEQPATTERN: "admissible (x::'q_76705 => 'q_76704 => bool)
+ (xa::('q_76705 => 'q_76770) => 'P => bool) (xb::'P => 'q_76704)
+ (%(f::'q_76705 => 'q_76770) x::'P.
+ Ex ((xc::('q_76705 => 'q_76770) => 'P => 'q_76825 => 'q_76794 => bool)
+ f x ((xe::('q_76705 => 'q_76770) => 'P => 'q_76825) f x))) &
+admissible x
+ (%(f::'q_76705 => 'q_76770) x::'P. xa f x & Ex (xc f x (xe f x))) xb
+ (%(f::'q_76705 => 'q_76770) x::'P. _MATCH (xe f x) (xc f x)) &
+admissible x
+ (%(f::'q_76705 => 'q_76770) x::'P. xa f x & ~ Ex (xc f x (xe f x))) xb
+ (%(f::'q_76705 => 'q_76770) x::'P.
+ _MATCH (xe f x)
+ ((xd::('q_76705 => 'q_76770) => 'P => 'q_76825 => 'q_76794 => bool) f
+ x))
+==> admissible x xa xb
+ (%(x::'q_76705 => 'q_76770) xa::'P.
+ _MATCH (xe x xa) (_SEQPATTERN (xc x xa) (xd x xa)))"
+ by (import hollight ADMISSIBLE_MATCH_SEQPATTERN)
+
+lemma ADMISSIBLE_IMP_SUPERADMISSIBLE: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'A) (xc::('A => 'B) => 'P => 'B)
+==> superadmissible x xa xb xc"
+ by (import hollight ADMISSIBLE_IMP_SUPERADMISSIBLE)
+
+lemma SUPERADMISSIBLE_CONST: "superadmissible (u_556::'q_76904 => 'q_76904 => bool)
+ (p::('q_76904 => 'q_76906) => 'q_76905 => bool) (s::'q_76905 => 'q_76904)
+ (%f::'q_76904 => 'q_76906. c::'q_76905 => 'q_76906)"
+ by (import hollight SUPERADMISSIBLE_CONST)
+
+lemma SUPERADMISSIBLE_TAIL: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xb::'P => 'A) (xc::('A => 'B) => 'P => 'A) &
+(ALL (f::'A => 'B) a::'P.
+ xa f a --> (ALL y::'A. x y (xc f a) --> x y (xb a)))
+==> superadmissible x xa xb (%(f::'A => 'B) x::'P. f (xc f x))"
+ by (import hollight SUPERADMISSIBLE_TAIL)
+
+lemma SUPERADMISSIBLE_COND: "admissible (x::'A => 'A => bool) (xa::('A => 'B) => 'P => bool)
+ (xc::'P => 'A) (xb::('A => 'B) => 'P => bool) &
+superadmissible x (%(f::'A => 'B) x::'P. xa f x & xb f x) xc
+ (xd::('A => 'B) => 'P => 'B) &
+superadmissible x (%(f::'A => 'B) x::'P. xa f x & ~ xb f x) xc
+ (xe::('A => 'B) => 'P => 'B)
+==> superadmissible x xa xc
+ (%(f::'A => 'B) x::'P. if xb f x then xd f x else xe f x)"
+ by (import hollight SUPERADMISSIBLE_COND)
+
+lemma SUPERADMISSIBLE_MATCH_SEQPATTERN: "admissible (x::'q_77225 => 'q_77225 => bool)
+ (xa::('q_77225 => 'q_77341) => 'P => bool) (xb::'P => 'q_77225)
+ (%(f::'q_77225 => 'q_77341) x::'P.
+ Ex ((xc::('q_77225 => 'q_77341) => 'P => 'q_77340 => 'q_77341 => bool)
+ f x ((xe::('q_77225 => 'q_77341) => 'P => 'q_77340) f x))) &
+superadmissible x
+ (%(f::'q_77225 => 'q_77341) x::'P. xa f x & Ex (xc f x (xe f x))) xb
+ (%(f::'q_77225 => 'q_77341) x::'P. _MATCH (xe f x) (xc f x)) &
+superadmissible x
+ (%(f::'q_77225 => 'q_77341) x::'P. xa f x & ~ Ex (xc f x (xe f x))) xb
+ (%(f::'q_77225 => 'q_77341) x::'P.
+ _MATCH (xe f x)
+ ((xd::('q_77225 => 'q_77341) => 'P => 'q_77340 => 'q_77341 => bool) f
+ x))
+==> superadmissible x xa xb
+ (%(x::'q_77225 => 'q_77341) xa::'P.
+ _MATCH (xe x xa) (_SEQPATTERN (xc x xa) (xd x xa)))"
+ by (import hollight SUPERADMISSIBLE_MATCH_SEQPATTERN)
+
+lemma SUPERADMISSIBLE_MATCH_UNGUARDED_PATTERN: "(ALL (f::'A => 'B) (a::'P) (t::'Q) u::'Q.
+ (p::('A => 'B) => 'P => bool) f a &
+ (pat::'Q => 'D) t = (e::'P => 'D) a & pat u = e a -->
+ (arg::'P => 'Q => 'A) a t = arg a u) &
+(ALL (f::'A => 'B) (a::'P) t::'Q.
+ p f a & pat t = e a -->
+ (ALL y::'A.
+ (u_556::'A => 'A => bool) y (arg a t) -->
+ u_556 y ((s::'P => 'A) a)))
+==> superadmissible u_556 p s
+ (%(f::'A => 'B) x::'P.
+ _MATCH (e x)
+ (%(u::'D) v::'B.
+ EX t::'Q. _UNGUARDED_PATTERN (pat t = u) (f (arg x t) = v)))"
+ by (import hollight SUPERADMISSIBLE_MATCH_UNGUARDED_PATTERN)
+
+lemma SUPERADMISSIBLE_MATCH_GUARDED_PATTERN: "(ALL (f::'A => 'B) (a::'P) (t::'Q) u::'Q.
+ (p::('A => 'B) => 'P => bool) f a &
+ (pat::'Q => 'D) t = (e::'P => 'D) a &
+ (q::'P => 'Q => bool) a t & pat u = e a & q a u -->
+ (arg::'P => 'Q => 'A) a t = arg a u) &
+(ALL (f::'A => 'B) (a::'P) t::'Q.
+ p f a & q a t & pat t = e a -->
+ (ALL y::'A.
+ (u_556::'A => 'A => bool) y (arg a t) -->
+ u_556 y ((s::'P => 'A) a)))
+==> superadmissible u_556 p s
+ (%(f::'A => 'B) x::'P.
+ _MATCH (e x)
+ (%(u::'D) v::'B.
+ EX t::'Q.
+ _GUARDED_PATTERN (pat t = u) (q x t) (f (arg x t) = v)))"
+ by (import hollight SUPERADMISSIBLE_MATCH_GUARDED_PATTERN)
+
+lemma cth: "[| !!(c::'q_78547) (x::'A) y::'A.
+ (p1::'A => 'q_78536) x = (p1'::'A => 'q_78536) y
+ ==> (p2::'q_78547 => 'A => 'q_78541) c x =
+ (p2'::'q_78547 => 'A => 'q_78541) c y;
+ p1' (x::'A) = p1 (y::'A) |]
+==> p2' (c::'q_78547) x = p2 c y"
by (import hollight cth)
-lemma RECURSION_CASEWISE_PAIRWISE: "ALL x::(('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)) hollight.list.
- (EX u::'q_59975::type => 'q_59975::type => bool.
- WF u &
- ALL_list
- (GABS
- (%f::('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => bool.
- ALL (s::'q_59995::type => 'q_59975::type)
- t::('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type.
- GEQ (f (s, t))
- (tailadmissible u
- (%(f::'q_59975::type => 'q_59991::type)
- a::'q_59995::type. True)
- s t)))
- x) &
- ALL_list
- (GABS
- (%f::('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => bool.
- ALL (a::'q_59995::type => 'q_59975::type)
- b::('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type.
- GEQ (f (a, b))
- (ALL (c::'q_59975::type => 'q_59991::type) (x::'q_59995::type)
- y::'q_59995::type. a x = a y --> b c x = b c y)))
- x &
- PAIRWISE
- (GABS
- (%f::('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => ('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => bool.
- ALL (s::'q_59995::type => 'q_59975::type)
- t::('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type.
- GEQ (f (s, t))
- (GABS
- (%f::('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => bool.
- ALL (s'::'q_59995::type => 'q_59975::type)
- t'::('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type.
- GEQ (f (s', t'))
- (ALL (c::'q_59975::type => 'q_59991::type)
- (x::'q_59995::type) y::'q_59995::type.
- s x = s' y --> t c x = t' c y)))))
- x -->
- (EX f::'q_59975::type => 'q_59991::type.
- ALL_list
- (GABS
- (%fa::('q_59995::type => 'q_59975::type) *
- (('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type)
- => bool.
- ALL (s::'q_59995::type => 'q_59975::type)
- t::('q_59975::type => 'q_59991::type)
- => 'q_59995::type => 'q_59991::type.
- GEQ (fa (s, t)) (ALL x::'q_59995::type. f (s x) = t f x)))
- x)"
- by (import hollight RECURSION_CASEWISE_PAIRWISE)
-
-lemma SUPERADMISSIBLE_T: "superadmissible (u_353::'q_60105::type => 'q_60105::type => bool)
- (%(f::'q_60105::type => 'q_60107::type) x::'q_60111::type. True)
- (s::'q_60111::type => 'q_60105::type)
- (t::('q_60105::type => 'q_60107::type)
- => 'q_60111::type => 'q_60107::type) =
-tailadmissible u_353
- (%(f::'q_60105::type => 'q_60107::type) x::'q_60111::type. True) s t"
+lemma SUPERADMISSIBLE_T: "superadmissible (u_556::'q_78694 => 'q_78694 => bool)
+ (%(f::'q_78694 => 'q_78696) x::'q_78700. True) (s::'q_78700 => 'q_78694)
+ (t::('q_78694 => 'q_78696) => 'q_78700 => 'q_78696) =
+tailadmissible u_556 (%(f::'q_78694 => 'q_78696) x::'q_78700. True) s t"
by (import hollight SUPERADMISSIBLE_T)
+lemma elemma0: "(ALL z::'q_78985 * 'q_78984.
+ (f::'q_78985 * 'q_78984 => 'q_78975) z =
+ (g::'q_78985 * 'q_78984 => 'q_78975) z) =
+(ALL (x::'q_78985) y::'q_78984. f (x, y) = g (x, y)) &
+(P::'q_79002 * 'q_79001 => 'q_78994) =
+(SOME f::'q_79002 * 'q_79001 => 'q_78994.
+ ALL (x::'q_79002) y::'q_79001. f (x, y) = P (x, y))"
+ by (import hollight elemma0)
+
;end_setup
end
--- a/src/HOL/Import/HOLLight/hollight.imp Wed Jul 13 00:23:24 2011 +0900
+++ b/src/HOL/Import/HOLLight/hollight.imp Wed Jul 13 00:29:33 2011 +0900
@@ -2,21 +2,158 @@
import_segment "hollight"
+def_maps
+ "vector" > "vector_def"
+ "treal_of_num" > "treal_of_num_def"
+ "treal_neg" > "treal_neg_def"
+ "treal_mul" > "treal_mul_def"
+ "treal_le" > "treal_le_def"
+ "treal_inv" > "treal_inv_def"
+ "treal_eq" > "treal_eq_def"
+ "treal_add" > "treal_add_def"
+ "tailadmissible" > "tailadmissible_def"
+ "support" > "support_def"
+ "superadmissible" > "superadmissible_def"
+ "sum" > "sum_def"
+ "sndcart" > "sndcart_def"
+ "rem" > "rem_def"
+ "real_sub" > "real_sub_def"
+ "real_sgn" > "real_sgn_def"
+ "real_pow" > "real_pow_def"
+ "real_of_num" > "real_of_num_def"
+ "real_neg" > "real_neg_def"
+ "real_mul" > "real_mul_def"
+ "real_mod" > "real_mod_def"
+ "real_min" > "real_min_def"
+ "real_max" > "real_max_def"
+ "real_lt" > "real_lt_def"
+ "real_le" > "real_le_def"
+ "real_inv" > "real_inv_def"
+ "real_gt" > "real_gt_def"
+ "real_ge" > "real_ge_def"
+ "real_div" > "real_div_def"
+ "real_add" > "real_add_def"
+ "real_abs" > "real_abs_def"
+ "pastecart" > "pastecart_def"
+ "pairwise" > "pairwise_def"
+ "num_of_int" > "num_of_int_def"
+ "num_mod" > "num_mod_def"
+ "num_gcd" > "num_gcd_def"
+ "num_divides" > "num_divides_def"
+ "num_coprime" > "num_coprime_def"
+ "nsum" > "nsum_def"
+ "neutral" > "neutral_def"
+ "nadd_rinv" > "nadd_rinv_def"
+ "nadd_of_num" > "nadd_of_num_def"
+ "nadd_mul" > "nadd_mul_def"
+ "nadd_le" > "nadd_le_def"
+ "nadd_inv" > "nadd_inv_def"
+ "nadd_eq" > "nadd_eq_def"
+ "nadd_add" > "nadd_add_def"
+ "monoidal" > "monoidal_def"
+ "minimal" > "minimal_def"
+ "lambda" > "lambda_def"
+ "iterate" > "iterate_def"
+ "is_nadd" > "is_nadd_def"
+ "integer" > "integer_def"
+ "int_sub" > "int_sub_def"
+ "int_sgn" > "int_sgn_def"
+ "int_pow" > "int_pow_def"
+ "int_of_num" > "int_of_num_def"
+ "int_neg" > "int_neg_def"
+ "int_mul" > "int_mul_def"
+ "int_mod" > "int_mod_def"
+ "int_min" > "int_min_def"
+ "int_max" > "int_max_def"
+ "int_lt" > "int_lt_def"
+ "int_le" > "int_le_def"
+ "int_gt" > "int_gt_def"
+ "int_ge" > "int_ge_def"
+ "int_gcd" > "int_gcd_def"
+ "int_divides" > "int_divides_def"
+ "int_coprime" > "int_coprime_def"
+ "int_add" > "int_add_def"
+ "int_abs" > "int_abs_def"
+ "hreal_of_num" > "hreal_of_num_def"
+ "hreal_mul" > "hreal_mul_def"
+ "hreal_le" > "hreal_le_def"
+ "hreal_inv" > "hreal_inv_def"
+ "hreal_add" > "hreal_add_def"
+ "fstcart" > "fstcart_def"
+ "eqeq" > "eqeq_def"
+ "div" > "div_def"
+ "dist" > "dist_def"
+ "dimindex" > "dimindex_def"
+ "admissible" > "admissible_def"
+ "_UNGUARDED_PATTERN" > "_UNGUARDED_PATTERN_def"
+ "_SEQPATTERN" > "_SEQPATTERN_def"
+ "_MATCH" > "_MATCH_def"
+ "_GUARDED_PATTERN" > "_GUARDED_PATTERN_def"
+ "_FUNCTION" > "_FUNCTION_def"
+ "_FALSITY_" > "_FALSITY__def"
+ "_11937" > "_11937_def"
+ "ZRECSPACE" > "ZRECSPACE_def"
+ "ZCONSTR" > "ZCONSTR_def"
+ "ZBOT" > "ZBOT_def"
+ "UNCURRY" > "UNCURRY_def"
+ "SURJ" > "SURJ_def"
+ "SING" > "SING_def"
+ "REST" > "REST_def"
+ "PASSOC" > "PASSOC_def"
+ "PAIRWISE" > "PAIRWISE_def"
+ "NUM_REP" > "NUM_REP_def"
+ "NUMSUM" > "NUMSUM_def"
+ "NUMSND" > "NUMSND_def"
+ "NUMRIGHT" > "NUMRIGHT_def"
+ "NUMPAIR" > "NUMPAIR_def"
+ "NUMLEFT" > "NUMLEFT_def"
+ "NUMFST" > "NUMFST_def"
+ "LET_END" > "LET_END_def"
+ "ITSET" > "ITSET_def"
+ "ISO" > "ISO_def"
+ "INJP" > "INJP_def"
+ "INJN" > "INJN_def"
+ "INJF" > "INJF_def"
+ "INJA" > "INJA_def"
+ "INJ" > "INJ_def"
+ "IND_SUC" > "IND_SUC_def"
+ "IND_0" > "IND_0_def"
+ "HAS_SIZE" > "HAS_SIZE_def"
+ "FNIL" > "FNIL_def"
+ "FINREC" > "FINREC_def"
+ "FCONS" > "FCONS_def"
+ "DECIMAL" > "DECIMAL_def"
+ "CROSS" > "CROSS_def"
+ "COUNTABLE" > "COUNTABLE_def"
+ "CONSTR" > "CONSTR_def"
+ "CASEWISE" > "CASEWISE_def"
+ "CARD" > "CARD_def"
+ "BOTTOM" > "BOTTOM_def"
+ "BIJ" > "BIJ_def"
+ "ASCII" > "ASCII_def"
+ ">_c" > ">_c_def"
+ ">=_c" > ">=_c_def"
+ "=_c" > "=_c_def"
+ "<_c" > "<_c_def"
+ "<=_c" > "<=_c_def"
+ "$" > "$_def"
+
type_maps
- "sum" > "+"
+ "sum" > "Sum_Type.sum"
"recspace" > "HOLLight.hollight.recspace"
"real" > "HOLLight.hollight.real"
"prod" > "Product_Type.prod"
- "option" > "HOLLight.hollight.option"
+ "option" > "Datatype.option"
"num" > "Nat.nat"
"nadd" > "HOLLight.hollight.nadd"
- "list" > "HOLLight.hollight.list"
+ "list" > "List.list"
"int" > "HOLLight.hollight.int"
"ind" > "Nat.ind"
"hreal" > "HOLLight.hollight.hreal"
"fun" > "fun"
"finite_sum" > "HOLLight.hollight.finite_sum"
"finite_image" > "HOLLight.hollight.finite_image"
+ "char" > "HOLLight.hollight.char"
"cart" > "HOLLight.hollight.cart"
"bool" > "HOL.bool"
"N_3" > "HOLLight.hollight.N_3"
@@ -25,8 +162,7 @@
const_maps
"~" > "HOL.Not"
- "two_2" > "HOLLight.hollight.two_2"
- "two_1" > "HOLLight.hollight.two_1"
+ "vector" > "HOLLight.hollight.vector"
"treal_of_num" > "HOLLight.hollight.treal_of_num"
"treal_neg" > "HOLLight.hollight.treal_neg"
"treal_mul" > "HOLLight.hollight.treal_mul"
@@ -34,20 +170,20 @@
"treal_inv" > "HOLLight.hollight.treal_inv"
"treal_eq" > "HOLLight.hollight.treal_eq"
"treal_add" > "HOLLight.hollight.treal_add"
- "three_3" > "HOLLight.hollight.three_3"
- "three_2" > "HOLLight.hollight.three_2"
- "three_1" > "HOLLight.hollight.three_1"
"tailadmissible" > "HOLLight.hollight.tailadmissible"
"support" > "HOLLight.hollight.support"
"superadmissible" > "HOLLight.hollight.superadmissible"
"sum" > "HOLLight.hollight.sum"
"sndcart" > "HOLLight.hollight.sndcart"
- "set_of_list" > "HOLLight.hollight.set_of_list"
+ "set_of_list" > "List.set"
+ "rem" > "HOLLight.hollight.rem"
"real_sub" > "HOLLight.hollight.real_sub"
+ "real_sgn" > "HOLLight.hollight.real_sgn"
"real_pow" > "HOLLight.hollight.real_pow"
"real_of_num" > "HOLLight.hollight.real_of_num"
"real_neg" > "HOLLight.hollight.real_neg"
"real_mul" > "HOLLight.hollight.real_mul"
+ "real_mod" > "HOLLight.hollight.real_mod"
"real_min" > "HOLLight.hollight.real_min"
"real_max" > "HOLLight.hollight.real_max"
"real_lt" > "HOLLight.hollight.real_lt"
@@ -62,6 +198,11 @@
"pairwise" > "HOLLight.hollight.pairwise"
"one" > "Product_Type.Unity"
"o" > "Fun.comp"
+ "num_of_int" > "HOLLight.hollight.num_of_int"
+ "num_mod" > "HOLLight.hollight.num_mod"
+ "num_gcd" > "HOLLight.hollight.num_gcd"
+ "num_divides" > "HOLLight.hollight.num_divides"
+ "num_coprime" > "HOLLight.hollight.num_coprime"
"nsum" > "HOLLight.hollight.nsum"
"neutral" > "HOLLight.hollight.neutral"
"nadd_rinv" > "HOLLight.hollight.nadd_rinv"
@@ -72,28 +213,31 @@
"nadd_eq" > "HOLLight.hollight.nadd_eq"
"nadd_add" > "HOLLight.hollight.nadd_add"
"monoidal" > "HOLLight.hollight.monoidal"
- "mod_real" > "HOLLight.hollight.mod_real"
- "mod_nat" > "HOLLight.hollight.mod_nat"
- "mod_int" > "HOLLight.hollight.mod_int"
"mk_pair" > "Product_Type.Pair_Rep"
+ "mk_num" > "Fun.id"
"minimal" > "HOLLight.hollight.minimal"
- "measure" > "HOLLight.hollight.measure"
- "list_of_set" > "HOLLight.hollight.list_of_set"
+ "list_EX" > "List.list_ex"
+ "list_ALL" > "List.list_all"
"lambda" > "HOLLight.hollight.lambda"
"iterate" > "HOLLight.hollight.iterate"
"is_nadd" > "HOLLight.hollight.is_nadd"
- "is_int" > "HOLLight.hollight.is_int"
+ "integer" > "HOLLight.hollight.integer"
"int_sub" > "HOLLight.hollight.int_sub"
+ "int_sgn" > "HOLLight.hollight.int_sgn"
"int_pow" > "HOLLight.hollight.int_pow"
"int_of_num" > "HOLLight.hollight.int_of_num"
"int_neg" > "HOLLight.hollight.int_neg"
"int_mul" > "HOLLight.hollight.int_mul"
+ "int_mod" > "HOLLight.hollight.int_mod"
"int_min" > "HOLLight.hollight.int_min"
"int_max" > "HOLLight.hollight.int_max"
"int_lt" > "HOLLight.hollight.int_lt"
"int_le" > "HOLLight.hollight.int_le"
"int_gt" > "HOLLight.hollight.int_gt"
"int_ge" > "HOLLight.hollight.int_ge"
+ "int_gcd" > "HOLLight.hollight.int_gcd"
+ "int_divides" > "HOLLight.hollight.int_divides"
+ "int_coprime" > "HOLLight.hollight.int_coprime"
"int_add" > "HOLLight.hollight.int_add"
"int_abs" > "HOLLight.hollight.int_abs"
"hreal_of_num" > "HOLLight.hollight.hreal_of_num"
@@ -102,76 +246,79 @@
"hreal_inv" > "HOLLight.hollight.hreal_inv"
"hreal_add" > "HOLLight.hollight.hreal_add"
"fstcart" > "HOLLight.hollight.fstcart"
- "finite_index" > "HOLLight.hollight.finite_index"
"eqeq" > "HOLLight.hollight.eqeq"
- "dotdot" > "HOLLight.hollight.dotdot"
+ "div" > "HOLLight.hollight.div"
"dist" > "HOLLight.hollight.dist"
"dimindex" > "HOLLight.hollight.dimindex"
"admissible" > "HOLLight.hollight.admissible"
+ "_UNGUARDED_PATTERN" > "HOLLight.hollight._UNGUARDED_PATTERN"
+ "_SEQPATTERN" > "HOLLight.hollight._SEQPATTERN"
+ "_MATCH" > "HOLLight.hollight._MATCH"
+ "_GUARDED_PATTERN" > "HOLLight.hollight._GUARDED_PATTERN"
+ "_FUNCTION" > "HOLLight.hollight._FUNCTION"
"_FALSITY_" > "HOLLight.hollight._FALSITY_"
- "_10328" > "HOLLight.hollight._10328"
- "_10327" > "HOLLight.hollight._10327"
- "_10326" > "HOLLight.hollight._10326"
- "_10303" > "HOLLight.hollight._10303"
- "_10302" > "HOLLight.hollight._10302"
- "_0" > "0" :: "nat"
- "\\/" > HOL.disj
+ "_11937" > "HOLLight.hollight._11937"
+ "_0" > "Groups.zero_class.zero" :: "nat"
+ "\\/" > "HOL.disj"
"ZRECSPACE" > "HOLLight.hollight.ZRECSPACE"
- "ZIP" > "HOLLight.hollight.ZIP"
+ "ZIP" > "List.zip"
"ZCONSTR" > "HOLLight.hollight.ZCONSTR"
"ZBOT" > "HOLLight.hollight.ZBOT"
- "WF" > "HOLLight.hollight.WF"
- "UNIV" > "HOLLight.hollight.UNIV"
- "UNIONS" > "HOLLight.hollight.UNIONS"
- "UNION" > "HOLLight.hollight.UNION"
+ "WF" > "Wellfounded.wfP"
+ "UNIV" > "Orderings.top_class.top" :: "'a => bool"
+ "UNIONS" > "Complete_Lattice.Sup_class.Sup" :: "(('a => bool) => bool) => 'a => bool"
+ "UNION" > "Lattices.semilattice_sup_class.sup" :: "('a => bool) => ('a => bool) => 'a => bool"
"UNCURRY" > "HOLLight.hollight.UNCURRY"
- "TL" > "HOLLight.hollight.TL"
- "T" > "True"
+ "TL" > "List.tl"
+ "T" > "HOL.True"
"SURJ" > "HOLLight.hollight.SURJ"
"SUC" > "Nat.Suc"
- "SUBSET" > "HOLLight.hollight.SUBSET"
- "SOME" > "HOLLight.hollight.SOME"
+ "SUBSET" > "Orderings.ord_class.less_eq" :: "('a => bool) => ('a => bool) => bool"
+ "SOME" > "Datatype.Some"
"SND" > "Product_Type.snd"
"SING" > "HOLLight.hollight.SING"
- "SETSPEC" > "HOLLight.hollight.SETSPEC"
- "REVERSE" > "HOLLight.hollight.REVERSE"
+ "SETSPEC" > "HOLLightCompat.SETSPEC"
+ "REVERSE" > "List.rev"
"REST" > "HOLLight.hollight.REST"
- "REP_prod" > "Rep_Prod"
- "REPLICATE" > "HOLLight.hollight.REPLICATE"
- "PSUBSET" > "HOLLight.hollight.PSUBSET"
+ "REPLICATE" > "List.replicate"
+ "PSUBSET" > "Orderings.ord_class.less" :: "('a => bool) => ('a => bool) => bool"
"PRE" > "HOLLightCompat.Pred"
"PASSOC" > "HOLLight.hollight.PASSOC"
"PAIRWISE" > "HOLLight.hollight.PAIRWISE"
- "OUTR" > "HOLLight.hollight.OUTR"
- "OUTL" > "HOLLight.hollight.OUTL"
+ "OUTR" > "HOLLightCompat.OUTR"
+ "OUTL" > "HOLLightCompat.OUTL"
"ONTO" > "Fun.surj"
- "ONE_ONE" > "HOL4Setup.ONE_ONE"
- "ODD" > "HOLLight.hollight.ODD"
+ "ONE_ONE" > "Fun.inj"
+ "ODD" > "HOLLightCompat.ODD"
+ "NUM_REP" > "HOLLight.hollight.NUM_REP"
"NUMSUM" > "HOLLight.hollight.NUMSUM"
"NUMSND" > "HOLLight.hollight.NUMSND"
"NUMRIGHT" > "HOLLight.hollight.NUMRIGHT"
"NUMPAIR" > "HOLLight.hollight.NUMPAIR"
"NUMLEFT" > "HOLLight.hollight.NUMLEFT"
"NUMFST" > "HOLLight.hollight.NUMFST"
- "NUMERAL" > "HOL4Compat.NUMERAL"
- "NULL" > "HOLLight.hollight.NULL"
- "NONE" > "HOLLight.hollight.NONE"
- "NIL" > "HOLLight.hollight.NIL"
- "MOD" > "HOLLight.hollight.MOD"
- "MEM" > "HOLLight.hollight.MEM"
- "MAP2" > "HOLLight.hollight.MAP2"
- "MAP" > "HOLLight.hollight.MAP"
+ "NUMERAL" > "HOLLightCompat.NUMERAL"
+ "NULL" > "List.null"
+ "NONE" > "Datatype.None"
+ "NIL" > "List.list.Nil"
+ "MOD" > "Divides.div_class.mod" :: "nat => nat => nat"
+ "MIN" > "Orderings.ord_class.min" :: "nat => nat => nat"
+ "MEM" > "HOLLightList.list_mem"
+ "MEASURE" > "HOLLightCompat.MEASURE"
+ "MAX" > "Orderings.ord_class.max" :: "nat => nat => nat"
+ "MAP2" > "HOLLightList.map2"
+ "MAP" > "List.map"
"LET_END" > "HOLLight.hollight.LET_END"
- "LET" > "HOL4Compat.LET"
- "LENGTH" > "HOLLight.hollight.LENGTH"
- "LAST" > "HOLLight.hollight.LAST"
+ "LET" > "HOLLightCompat.LET"
+ "LENGTH" > "List.length"
+ "LAST" > "List.last"
"ITSET" > "HOLLight.hollight.ITSET"
- "ITLIST2" > "HOLLight.hollight.ITLIST2"
- "ITLIST" > "HOLLight.hollight.ITLIST"
+ "ITLIST2" > "HOLLightList.fold2"
+ "ITLIST" > "List.foldr"
"ISO" > "HOLLight.hollight.ISO"
- "INTERS" > "HOLLight.hollight.INTERS"
- "INTER" > "HOLLight.hollight.INTER"
- "INSERT" > "HOLLight.hollight.INSERT"
+ "INTERS" > "Complete_Lattice.Inf_class.Inf" :: "(('a => bool) => bool) => 'a => bool"
+ "INTER" > "Lattices.semilattice_inf_class.inf" :: "('a => bool) => ('a => bool) => 'a => bool"
+ "INSERT" > "Set.insert"
"INR" > "Sum_Type.Inr"
"INL" > "Sum_Type.Inl"
"INJP" > "HOLLight.hollight.INJP"
@@ -179,80 +326,81 @@
"INJF" > "HOLLight.hollight.INJF"
"INJA" > "HOLLight.hollight.INJA"
"INJ" > "HOLLight.hollight.INJ"
- "INFINITE" > "HOLLight.hollight.INFINITE"
- "IN" > "HOLLight.hollight.IN"
- "IMAGE" > "HOLLight.hollight.IMAGE"
+ "INFINITE" > "HOLLightCompat.INFINITE"
+ "IND_SUC" > "HOLLight.hollight.IND_SUC"
+ "IND_0" > "HOLLight.hollight.IND_0"
+ "IN" > "Set.member"
+ "IMAGE" > "Set.image"
"I" > "Fun.id"
- "HD" > "HOLLight.hollight.HD"
+ "HD" > "List.hd"
"HAS_SIZE" > "HOLLight.hollight.HAS_SIZE"
- "GSPEC" > "HOLLight.hollight.GSPEC"
- "GEQ" > "HOLLight.hollight.GEQ"
- "GABS" > "HOLLight.hollight.GABS"
+ "GSPEC" > "Set.Collect"
+ "GEQ" > "HOL.eq"
+ "GABS" > "Hilbert_Choice.Eps"
"FST" > "Product_Type.fst"
"FNIL" > "HOLLight.hollight.FNIL"
"FINREC" > "HOLLight.hollight.FINREC"
- "FINITE" > "HOLLight.hollight.FINITE"
- "FILTER" > "HOLLight.hollight.FILTER"
+ "FINITE" > "Finite_Set.finite"
+ "FILTER" > "List.filter"
"FCONS" > "HOLLight.hollight.FCONS"
- "FACT" > "HOLLight.hollight.FACT"
- "F" > "False"
- "EXP" > "HOLLight.hollight.EXP"
- "EX" > "HOLLight.hollight.EX"
- "EVEN" > "HOLLight.hollight.EVEN"
- "EMPTY" > "HOLLight.hollight.EMPTY"
- "EL" > "HOLLight.hollight.EL"
- "DIV" > "HOLLight.hollight.DIV"
- "DISJOINT" > "HOLLight.hollight.DISJOINT"
- "DIFF" > "HOLLight.hollight.DIFF"
- "DELETE" > "HOLLight.hollight.DELETE"
+ "FACT" > "Fact.fact_class.fact" :: "nat => nat"
+ "F" > "HOL.False"
+ "EXP" > "Power.power_class.power" :: "nat => nat => nat"
+ "EVEN" > "Parity.even_odd_class.even" :: "nat => bool"
+ "EMPTY" > "Orderings.bot_class.bot" :: "'a => bool"
+ "EL" > "HOLLightList.list_el"
+ "DIV" > "Divides.div_class.div" :: "nat => nat => nat"
+ "DISJOINT" > "HOLLightCompat.DISJOINT"
+ "DIFF" > "Groups.minus_class.minus" :: "('a => bool) => ('a => bool) => 'a => bool"
+ "DELETE" > "HOLLightCompat.DELETE"
"DECIMAL" > "HOLLight.hollight.DECIMAL"
- "CURRY" > "HOLLight.hollight.CURRY"
+ "CURRY" > "Product_Type.curry"
+ "CROSS" > "HOLLight.hollight.CROSS"
"COUNTABLE" > "HOLLight.hollight.COUNTABLE"
"CONSTR" > "HOLLight.hollight.CONSTR"
- "CONS" > "HOLLight.hollight.CONS"
- "COND" > "HOLLight.hollight.COND"
- "CHOICE" > "HOLLight.hollight.CHOICE"
+ "CONS" > "List.list.Cons"
+ "COND" > "HOL.If"
+ "CHOICE" > "Hilbert_Choice.Eps"
"CASEWISE" > "HOLLight.hollight.CASEWISE"
- "CARD_LT" > "HOLLight.hollight.CARD_LT"
- "CARD_LE" > "HOLLight.hollight.CARD_LE"
- "CARD_GT" > "HOLLight.hollight.CARD_GT"
- "CARD_GE" > "HOLLight.hollight.CARD_GE"
- "CARD_EQ" > "HOLLight.hollight.CARD_EQ"
"CARD" > "HOLLight.hollight.CARD"
+ "BUTLAST" > "List.butlast"
"BOTTOM" > "HOLLight.hollight.BOTTOM"
- "BIT1" > "HOL4Compat.NUMERAL_BIT1"
+ "BIT1" > "HOLLightCompat.NUMERAL_BIT1"
"BIT0" > "HOLLightCompat.NUMERAL_BIT0"
"BIJ" > "HOLLight.hollight.BIJ"
- "ASSOC" > "HOLLight.hollight.ASSOC"
- "APPEND" > "HOLLight.hollight.APPEND"
- "ALL_list" > "HOLLight.hollight.ALL_list"
- "ALL2" > "HOLLight.hollight.ALL2"
- "ABS_prod" > "Abs_Prod"
+ "ASCII" > "HOLLight.hollight.ASCII"
+ "APPEND" > "List.append"
+ "ALL2" > "List.list_all2"
"@" > "Hilbert_Choice.Eps"
"?!" > "HOL.Ex1"
"?" > "HOL.Ex"
- ">=" > "HOLLight.hollight.>="
- ">" > "HOLLight.hollight.>"
- "==>" > HOL.implies
- "=" > HOL.eq
- "<=" > "HOLLight.hollight.<="
- "<" > "HOLLight.hollight.<"
- "/\\" > HOL.conj
- "-" > "Groups.minus" :: "nat => nat => nat"
+ ">_c" > "HOLLight.hollight.>_c"
+ ">=_c" > "HOLLight.hollight.>=_c"
+ ">=" > "Orderings.ord_class.greater_eq" :: "nat => nat => bool"
+ ">" > "Orderings.ord_class.greater" :: "nat => nat => bool"
+ "=_c" > "HOLLight.hollight.=_c"
+ "==>" > "HOL.implies"
+ "=" > "HOL.eq"
+ "<_c" > "HOLLight.hollight.<_c"
+ "<=_c" > "HOLLight.hollight.<=_c"
+ "<=" > "Orderings.ord_class.less_eq" :: "nat => nat => bool"
+ "<" > "Orderings.ord_class.less" :: "nat => nat => bool"
+ "/\\" > "HOL.conj"
+ ".." > "HOLLightCompat.dotdot"
+ "-" > "Groups.minus_class.minus" :: "nat => nat => nat"
"," > "Product_Type.Pair"
- "+" > "Groups.plus" :: "nat => nat => nat"
- "*" > "Groups.times" :: "nat => nat => nat"
+ "+" > "Groups.plus_class.plus" :: "nat => nat => nat"
+ "*" > "Groups.times_class.times" :: "nat => nat => nat"
"$" > "HOLLight.hollight.$"
"!" > "HOL.All"
const_renames
- "ALL" > "ALL_list"
+ "EX" > "list_EX"
+ "ALL" > "list_ALL"
"==" > "eqeq"
- ".." > "dotdot"
thm_maps
- "two_2_def" > "HOLLight.hollight.two_2_def"
- "two_1_def" > "HOLLight.hollight.two_1_def"
+ "vector_def" > "HOLLight.hollight.vector_def"
"treal_of_num_def" > "HOLLight.hollight.treal_of_num_def"
"treal_neg_def" > "HOLLight.hollight.treal_neg_def"
"treal_mul_def" > "HOLLight.hollight.treal_mul_def"
@@ -260,26 +408,24 @@
"treal_inv_def" > "HOLLight.hollight.treal_inv_def"
"treal_eq_def" > "HOLLight.hollight.treal_eq_def"
"treal_add_def" > "HOLLight.hollight.treal_add_def"
- "three_3_def" > "HOLLight.hollight.three_3_def"
- "three_2_def" > "HOLLight.hollight.three_2_def"
- "three_1_def" > "HOLLight.hollight.three_1_def"
"th_cond" > "HOLLight.hollight.th_cond"
- "th" > "Fun.id_apply"
+ "th" > "HOL.eta_contract_eq"
"tailadmissible_def" > "HOLLight.hollight.tailadmissible_def"
"support_def" > "HOLLight.hollight.support_def"
"superadmissible_def" > "HOLLight.hollight.superadmissible_def"
"sum_def" > "HOLLight.hollight.sum_def"
- "sum_RECURSION" > "HOLLightCompat.sum_Recursion"
- "sum_INDUCT" > "HOLLightCompat.sumlift.induct"
+ "string_INFINITE" > "List.infinite_UNIV_listI"
"sth" > "HOLLight.hollight.sth"
"sndcart_def" > "HOLLight.hollight.sndcart_def"
- "set_of_list_def" > "HOLLight.hollight.set_of_list_def"
"right_th" > "HOLLight.hollight.right_th"
+ "rem_def" > "HOLLight.hollight.rem_def"
"real_sub_def" > "HOLLight.hollight.real_sub_def"
+ "real_sgn_def" > "HOLLight.hollight.real_sgn_def"
"real_pow_def" > "HOLLight.hollight.real_pow_def"
"real_of_num_def" > "HOLLight.hollight.real_of_num_def"
"real_neg_def" > "HOLLight.hollight.real_neg_def"
"real_mul_def" > "HOLLight.hollight.real_mul_def"
+ "real_mod_def" > "HOLLight.hollight.real_mod_def"
"real_min_def" > "HOLLight.hollight.real_min_def"
"real_max_def" > "HOLLight.hollight.real_max_def"
"real_lt_def" > "HOLLight.hollight.real_lt_def"
@@ -290,27 +436,34 @@
"real_div_def" > "HOLLight.hollight.real_div_def"
"real_add_def" > "HOLLight.hollight.real_add_def"
"real_abs_def" > "HOLLight.hollight.real_abs_def"
+ "real_INFINITE" > "HOLLight.hollight.real_INFINITE"
"pastecart_def" > "HOLLight.hollight.pastecart_def"
"pairwise_def" > "HOLLight.hollight.pairwise_def"
"pair_RECURSION" > "HOLLight.hollight.pair_RECURSION"
- "pair_INDUCT" > "Datatype.prod.inducts"
+ "pair_INDUCT" > "Product_Type.prod.induct"
"one_axiom" > "HOLLight.hollight.one_axiom"
"one_RECURSION" > "HOLLight.hollight.one_RECURSION"
- "one_INDUCT" > "Datatype.unit.inducts"
+ "one_INDUCT" > "Product_Type.unit.induct"
"one_Axiom" > "HOLLight.hollight.one_Axiom"
- "one" > "HOL4Compat.one"
- "o_THM" > "Fun.o_apply"
+ "one" > "HOLLightCompat.one"
+ "o_THM" > "Fun.comp_def"
"o_ASSOC" > "HOLLight.hollight.o_ASSOC"
+ "num_of_int_def" > "HOLLight.hollight.num_of_int_def"
+ "num_mod_def" > "HOLLight.hollight.num_mod_def"
+ "num_gcd_def" > "HOLLight.hollight.num_gcd_def"
+ "num_divides_def" > "HOLLight.hollight.num_divides_def"
+ "num_coprime_def" > "HOLLight.hollight.num_coprime_def"
+ "num_congruent" > "HOLLight.hollight.num_congruent"
"num_WOP" > "HOLLight.hollight.num_WOP"
- "num_WF" > "HOLLight.hollight.num_WF"
+ "num_WF" > "Nat.nat_less_induct"
"num_RECURSION_STD" > "HOLLight.hollight.num_RECURSION_STD"
"num_MAX" > "HOLLight.hollight.num_MAX"
- "num_INFINITE" > "HOLLight.hollight.num_INFINITE"
- "num_INDUCTION" > "List.lexn.induct"
+ "num_INFINITE" > "Finite_Set.infinite_UNIV_char_0"
+ "num_INDUCTION" > "Fact.fact_nat.induct"
"num_FINITE_AVOID" > "HOLLight.hollight.num_FINITE_AVOID"
"num_FINITE" > "HOLLight.hollight.num_FINITE"
"num_CASES" > "Nat.nat.nchotomy"
- "num_Axiom" > "HOLLight.hollight.num_Axiom"
+ "num_Axiom" > "HOLLightCompat.num_Axiom"
"nsum_def" > "HOLLight.hollight.nsum_def"
"neutral_def" > "HOLLight.hollight.neutral_def"
"nadd_rinv_def" > "HOLLight.hollight.nadd_rinv_def"
@@ -321,21 +474,19 @@
"nadd_eq_def" > "HOLLight.hollight.nadd_eq_def"
"nadd_add_def" > "HOLLight.hollight.nadd_add_def"
"monoidal_def" > "HOLLight.hollight.monoidal_def"
- "mod_real_def" > "HOLLight.hollight.mod_real_def"
- "mod_nat_def" > "HOLLight.hollight.mod_nat_def"
- "mod_int_def" > "HOLLight.hollight.mod_int_def"
"minimal_def" > "HOLLight.hollight.minimal_def"
- "measure_def" > "HOLLight.hollight.measure_def"
- "list_of_set_def" > "HOLLight.hollight.list_of_set_def"
- "list_INDUCT" > "HOLLight.hollight.list_INDUCT"
- "list_CASES" > "HOLLight.hollight.list_CASES"
+ "list_INDUCT" > "List.list.induct"
+ "list_CASES" > "List.list.nchotomy"
"lambda_def" > "HOLLight.hollight.lambda_def"
"iterate_def" > "HOLLight.hollight.iterate_def"
"is_nadd_def" > "HOLLight.hollight.is_nadd_def"
"is_nadd_0" > "HOLLight.hollight.is_nadd_0"
- "is_int_def" > "HOLLight.hollight.is_int_def"
+ "is_int" > "HOLLight.hollight.is_int"
+ "integer_def" > "HOLLight.hollight.integer_def"
"int_sub_th" > "HOLLight.hollight.int_sub_th"
"int_sub_def" > "HOLLight.hollight.int_sub_def"
+ "int_sgn_th" > "HOLLight.hollight.int_sgn_th"
+ "int_sgn_def" > "HOLLight.hollight.int_sgn_def"
"int_pow_th" > "HOLLight.hollight.int_pow_th"
"int_pow_def" > "HOLLight.hollight.int_pow_def"
"int_of_num_th" > "HOLLight.hollight.int_of_num_th"
@@ -344,6 +495,7 @@
"int_neg_def" > "HOLLight.hollight.int_neg_def"
"int_mul_th" > "HOLLight.hollight.int_mul_th"
"int_mul_def" > "HOLLight.hollight.int_mul_def"
+ "int_mod_def" > "HOLLight.hollight.int_mod_def"
"int_min_th" > "HOLLight.hollight.int_min_th"
"int_min_def" > "HOLLight.hollight.int_min_def"
"int_max_th" > "HOLLight.hollight.int_max_th"
@@ -352,7 +504,11 @@
"int_le_def" > "HOLLight.hollight.int_le_def"
"int_gt_def" > "HOLLight.hollight.int_gt_def"
"int_ge_def" > "HOLLight.hollight.int_ge_def"
- "int_eq" > "HOLLight.hollight.int.dest_int_inject"
+ "int_gcd_def" > "HOLLight.hollight.int_gcd_def"
+ "int_eq" > "HOLLight.hollight.int.real_of_int_inject"
+ "int_divides_def" > "HOLLight.hollight.int_divides_def"
+ "int_coprime_def" > "HOLLight.hollight.int_coprime_def"
+ "int_congruent" > "HOLLight.hollight.int_congruent"
"int_add_th" > "HOLLight.hollight.int_add_th"
"int_add_def" > "HOLLight.hollight.int_add_def"
"int_abs_th" > "HOLLight.hollight.int_abs_th"
@@ -363,34 +519,34 @@
"hreal_inv_def" > "HOLLight.hollight.hreal_inv_def"
"hreal_add_def" > "HOLLight.hollight.hreal_add_def"
"fstcart_def" > "HOLLight.hollight.fstcart_def"
- "finite_index_def" > "HOLLight.hollight.finite_index_def"
"eqeq_def" > "HOLLight.hollight.eqeq_def"
- "dotdot_def" > "HOLLight.hollight.dotdot_def"
+ "elemma0" > "HOLLight.hollight.elemma0"
+ "div_def" > "HOLLight.hollight.div_def"
"dist_def" > "HOLLight.hollight.dist_def"
"dimindex_def" > "HOLLight.hollight.dimindex_def"
"dest_int_rep" > "HOLLight.hollight.dest_int_rep"
"cth" > "HOLLight.hollight.cth"
"bool_RECURSION" > "HOLLight.hollight.bool_RECURSION"
+ "bool_INDUCT" > "Product_Type.bool.induct"
"ax__3" > "HOL4Setup.INFINITY_AX"
- "ax__2" > "Hilbert_Choice.tfl_some"
- "ax__1" > "Presburger.fm_modd_pinf"
+ "ax__2" > "Hilbert_Choice.someI"
+ "ax__1" > "HOL.eta_contract_eq"
"admissible_def" > "HOLLight.hollight.admissible_def"
+ "_UNGUARDED_PATTERN_def" > "HOLLight.hollight._UNGUARDED_PATTERN_def"
+ "_SEQPATTERN_def" > "HOLLight.hollight._SEQPATTERN_def"
+ "_MATCH_def" > "HOLLight.hollight._MATCH_def"
+ "_GUARDED_PATTERN_def" > "HOLLight.hollight._GUARDED_PATTERN_def"
+ "_FUNCTION_def" > "HOLLight.hollight._FUNCTION_def"
"_FALSITY__def" > "HOLLight.hollight._FALSITY__def"
- "_10328_def" > "HOLLight.hollight._10328_def"
- "_10327_def" > "HOLLight.hollight._10327_def"
- "_10326_def" > "HOLLight.hollight._10326_def"
- "_10303_def" > "HOLLight.hollight._10303_def"
- "_10302_def" > "HOLLight.hollight._10302_def"
+ "_11937_def" > "HOLLight.hollight._11937_def"
"ZRECSPACE_def" > "HOLLight.hollight.ZRECSPACE_def"
- "ZIP_def" > "HOLLight.hollight.ZIP_def"
- "ZIP" > "HOLLight.hollight.ZIP"
+ "ZIP" > "HOLLightList.ZIP"
"ZCONSTR_def" > "HOLLight.hollight.ZCONSTR_def"
"ZCONSTR_ZBOT" > "HOLLight.hollight.ZCONSTR_ZBOT"
"ZBOT_def" > "HOLLight.hollight.ZBOT_def"
"WLOG_LT" > "HOLLight.hollight.WLOG_LT"
"WLOG_LE" > "HOLLight.hollight.WLOG_LE"
"WF_num" > "HOLLight.hollight.WF_num"
- "WF_def" > "HOLLight.hollight.WF_def"
"WF_UREC_WF" > "HOLLight.hollight.WF_UREC_WF"
"WF_UREC" > "HOLLight.hollight.WF_UREC"
"WF_SUBSET" > "HOLLight.hollight.WF_SUBSET"
@@ -400,49 +556,52 @@
"WF_REC_TAIL_GENERAL" > "HOLLight.hollight.WF_REC_TAIL_GENERAL"
"WF_REC_TAIL" > "HOLLight.hollight.WF_REC_TAIL"
"WF_REC_INVARIANT" > "HOLLight.hollight.WF_REC_INVARIANT"
- "WF_REC_CASES" > "HOLLight.hollight.WF_REC_CASES"
"WF_REC" > "HOLLight.hollight.WF_REC"
"WF_POINTWISE" > "HOLLight.hollight.WF_POINTWISE"
"WF_MEASURE_GEN" > "HOLLight.hollight.WF_MEASURE_GEN"
"WF_MEASURE" > "HOLLight.hollight.WF_MEASURE"
"WF_LEX_DEPENDENT" > "HOLLight.hollight.WF_LEX_DEPENDENT"
"WF_LEX" > "HOLLight.hollight.WF_LEX"
+ "WF_INT_MEASURE_2" > "HOLLight.hollight.WF_INT_MEASURE_2"
+ "WF_INT_MEASURE" > "HOLLight.hollight.WF_INT_MEASURE"
"WF_IND" > "HOLLight.hollight.WF_IND"
- "WF_FALSE" > "HOLLight.hollight.WF_FALSE"
+ "WF_FINITE" > "HOLLight.hollight.WF_FINITE"
+ "WF_FALSE" > "Wellfounded.wfP_empty"
"WF_EREC" > "HOLLight.hollight.WF_EREC"
"WF_EQ" > "HOLLight.hollight.WF_EQ"
"WF_DCHAIN" > "HOLLight.hollight.WF_DCHAIN"
"UNWIND_THM2" > "HOL.simp_thms_39"
"UNWIND_THM1" > "HOL.simp_thms_40"
- "UNIV_def" > "HOLLight.hollight.UNIV_def"
"UNIV_SUBSET" > "HOLLight.hollight.UNIV_SUBSET"
- "UNIV_NOT_EMPTY" > "HOLLight.hollight.UNIV_NOT_EMPTY"
+ "UNIV_NOT_EMPTY" > "Set.UNIV_not_empty"
"UNIQUE_SKOLEM_THM" > "HOL.choice_eq"
"UNIQUE_SKOLEM_ALT" > "HOLLight.hollight.UNIQUE_SKOLEM_ALT"
- "UNION_def" > "HOLLight.hollight.UNION_def"
"UNION_UNIV" > "HOLLight.hollight.UNION_UNIV"
- "UNION_SUBSET" > "HOLLight.hollight.UNION_SUBSET"
- "UNION_OVER_INTER" > "HOLLight.hollight.UNION_OVER_INTER"
- "UNION_IDEMPOT" > "HOLLight.hollight.UNION_IDEMPOT"
+ "UNION_SUBSET" > "Lattices.semilattice_sup_class.le_sup_iff"
+ "UNION_OVER_INTER" > "Lattices.distrib_lattice_class.distrib_3"
+ "UNION_IDEMPOT" > "Big_Operators.lattice_class.Sup_fin.idem"
"UNION_EMPTY" > "HOLLight.hollight.UNION_EMPTY"
- "UNION_COMM" > "HOLLight.hollight.UNION_COMM"
- "UNION_ASSOC" > "HOLLight.hollight.UNION_ASSOC"
+ "UNION_COMM" > "Lattices.lattice_class.inf_sup_aci_5"
+ "UNION_ASSOC" > "Lattices.lattice_class.inf_sup_aci_6"
"UNION_ACI" > "HOLLight.hollight.UNION_ACI"
- "UNIONS_def" > "HOLLight.hollight.UNIONS_def"
- "UNIONS_INSERT" > "HOLLight.hollight.UNIONS_INSERT"
- "UNIONS_2" > "HOLLight.hollight.UNIONS_2"
- "UNIONS_1" > "HOLLight.hollight.UNIONS_1"
- "UNIONS_0" > "HOLLight.hollight.UNIONS_0"
+ "UNIONS_UNION" > "Complete_Lattice.Union_Un_distrib"
+ "UNIONS_SUBSET" > "HOLLight.hollight.UNIONS_SUBSET"
+ "UNIONS_INTERS" > "HOLLight.hollight.UNIONS_INTERS"
+ "UNIONS_INSERT" > "Complete_Lattice.Union_insert"
+ "UNIONS_IMAGE" > "HOLLight.hollight.UNIONS_IMAGE"
+ "UNIONS_GSPEC" > "HOLLight.hollight.UNIONS_GSPEC"
+ "UNIONS_2" > "Complete_Lattice.Un_eq_Union"
+ "UNIONS_1" > "Complete_Lattice.complete_lattice_class.Sup_singleton"
+ "UNIONS_0" > "Complete_Lattice.Union_empty"
"UNCURRY_def" > "HOLLight.hollight.UNCURRY_def"
"TYDEF_recspace" > "HOLLight.hollight.TYDEF_recspace"
"TYDEF_real" > "HOLLight.hollight.TYDEF_real"
- "TYDEF_option" > "HOLLight.hollight.TYDEF_option"
"TYDEF_nadd" > "HOLLight.hollight.TYDEF_nadd"
- "TYDEF_list" > "HOLLight.hollight.TYDEF_list"
"TYDEF_int" > "HOLLight.hollight.TYDEF_int"
"TYDEF_hreal" > "HOLLight.hollight.TYDEF_hreal"
"TYDEF_finite_sum" > "HOLLight.hollight.TYDEF_finite_sum"
"TYDEF_finite_image" > "HOLLight.hollight.TYDEF_finite_image"
+ "TYDEF_char" > "HOLLight.hollight.TYDEF_char"
"TYDEF_cart" > "HOLLight.hollight.TYDEF_cart"
"TYDEF_3" > "HOLLight.hollight.TYDEF_3"
"TYDEF_2" > "HOLLight.hollight.TYDEF_2"
@@ -488,25 +647,43 @@
"TREAL_ADD_LID" > "HOLLight.hollight.TREAL_ADD_LID"
"TREAL_ADD_LDISTRIB" > "HOLLight.hollight.TREAL_ADD_LDISTRIB"
"TREAL_ADD_ASSOC" > "HOLLight.hollight.TREAL_ADD_ASSOC"
- "TL_def" > "HOLLight.hollight.TL_def"
- "TAIL_IMP_SUPERADMISSIBLE" > "HOLLight.hollight.TAIL_IMP_SUPERADMISSIBLE"
- "SWAP_FORALL_THM" > "HOLLight.hollight.SWAP_FORALL_THM"
- "SWAP_EXISTS_THM" > "HOLLight.hollight.SWAP_EXISTS_THM"
+ "TRANSITIVE_STEPWISE_LT_EQ" > "HOLLight.hollight.TRANSITIVE_STEPWISE_LT_EQ"
+ "TRANSITIVE_STEPWISE_LT" > "HOLLight.hollight.TRANSITIVE_STEPWISE_LT"
+ "TRANSITIVE_STEPWISE_LE_EQ" > "HOLLight.hollight.TRANSITIVE_STEPWISE_LE_EQ"
+ "TRANSITIVE_STEPWISE_LE" > "HOLLight.hollight.TRANSITIVE_STEPWISE_LE"
+ "TOPOLOGICAL_SORT" > "HOLLight.hollight.TOPOLOGICAL_SORT"
+ "SWAP_FORALL_THM" > "HOL.all_comm"
+ "SWAP_EXISTS_THM" > "HOL.ex_comm"
"SURJ_def" > "HOLLight.hollight.SURJ_def"
"SURJECTIVE_RIGHT_INVERSE" > "HOLLight.hollight.SURJECTIVE_RIGHT_INVERSE"
"SURJECTIVE_ON_RIGHT_INVERSE" > "HOLLight.hollight.SURJECTIVE_ON_RIGHT_INVERSE"
+ "SURJECTIVE_ON_IMAGE" > "HOLLight.hollight.SURJECTIVE_ON_IMAGE"
+ "SURJECTIVE_MAP" > "HOLLightList.SURJECTIVE_MAP"
+ "SURJECTIVE_IMAGE_THM" > "HOLLight.hollight.SURJECTIVE_IMAGE_THM"
+ "SURJECTIVE_IMAGE_EQ" > "HOLLight.hollight.SURJECTIVE_IMAGE_EQ"
+ "SURJECTIVE_IMAGE" > "HOLLight.hollight.SURJECTIVE_IMAGE"
"SURJECTIVE_IFF_INJECTIVE_GEN" > "HOLLight.hollight.SURJECTIVE_IFF_INJECTIVE_GEN"
"SURJECTIVE_IFF_INJECTIVE" > "HOLLight.hollight.SURJECTIVE_IFF_INJECTIVE"
+ "SURJECTIVE_FORALL_THM" > "HOLLight.hollight.SURJECTIVE_FORALL_THM"
+ "SURJECTIVE_EXISTS_THM" > "HOLLight.hollight.SURJECTIVE_EXISTS_THM"
"SUPPORT_SUPPORT" > "HOLLight.hollight.SUPPORT_SUPPORT"
"SUPPORT_SUBSET" > "HOLLight.hollight.SUPPORT_SUBSET"
"SUPPORT_EMPTY" > "HOLLight.hollight.SUPPORT_EMPTY"
"SUPPORT_DELTA" > "HOLLight.hollight.SUPPORT_DELTA"
"SUPPORT_CLAUSES" > "HOLLight.hollight.SUPPORT_CLAUSES"
+ "SUPERADMISSIBLE_TAIL" > "HOLLight.hollight.SUPERADMISSIBLE_TAIL"
"SUPERADMISSIBLE_T" > "HOLLight.hollight.SUPERADMISSIBLE_T"
+ "SUPERADMISSIBLE_MATCH_UNGUARDED_PATTERN" > "HOLLight.hollight.SUPERADMISSIBLE_MATCH_UNGUARDED_PATTERN"
+ "SUPERADMISSIBLE_MATCH_SEQPATTERN" > "HOLLight.hollight.SUPERADMISSIBLE_MATCH_SEQPATTERN"
+ "SUPERADMISSIBLE_MATCH_GUARDED_PATTERN" > "HOLLight.hollight.SUPERADMISSIBLE_MATCH_GUARDED_PATTERN"
+ "SUPERADMISSIBLE_CONST" > "HOLLight.hollight.SUPERADMISSIBLE_CONST"
"SUPERADMISSIBLE_COND" > "HOLLight.hollight.SUPERADMISSIBLE_COND"
+ "SUM_ZERO_EXISTS" > "HOLLight.hollight.SUM_ZERO_EXISTS"
"SUM_UNION_RZERO" > "HOLLight.hollight.SUM_UNION_RZERO"
+ "SUM_UNION_NONZERO" > "HOLLight.hollight.SUM_UNION_NONZERO"
"SUM_UNION_LZERO" > "HOLLight.hollight.SUM_UNION_LZERO"
"SUM_UNION_EQ" > "HOLLight.hollight.SUM_UNION_EQ"
+ "SUM_UNIONS_NONZERO" > "HOLLight.hollight.SUM_UNIONS_NONZERO"
"SUM_UNION" > "HOLLight.hollight.SUM_UNION"
"SUM_TRIV_NUMSEG" > "HOLLight.hollight.SUM_TRIV_NUMSEG"
"SUM_SWAP_NUMSEG" > "HOLLight.hollight.SUM_SWAP_NUMSEG"
@@ -521,6 +698,7 @@
"SUM_SUB" > "HOLLight.hollight.SUM_SUB"
"SUM_SING_NUMSEG" > "HOLLight.hollight.SUM_SING_NUMSEG"
"SUM_SING" > "HOLLight.hollight.SUM_SING"
+ "SUM_RMUL" > "HOLLight.hollight.SUM_RMUL"
"SUM_RESTRICT_SET" > "HOLLight.hollight.SUM_RESTRICT_SET"
"SUM_RESTRICT" > "HOLLight.hollight.SUM_RESTRICT"
"SUM_POS_LE_NUMSEG" > "HOLLight.hollight.SUM_POS_LE_NUMSEG"
@@ -528,34 +706,51 @@
"SUM_POS_EQ_0_NUMSEG" > "HOLLight.hollight.SUM_POS_EQ_0_NUMSEG"
"SUM_POS_EQ_0" > "HOLLight.hollight.SUM_POS_EQ_0"
"SUM_POS_BOUND" > "HOLLight.hollight.SUM_POS_BOUND"
+ "SUM_PARTIAL_SUC" > "HOLLight.hollight.SUM_PARTIAL_SUC"
+ "SUM_PARTIAL_PRE" > "HOLLight.hollight.SUM_PARTIAL_PRE"
+ "SUM_PAIR" > "HOLLight.hollight.SUM_PAIR"
"SUM_OFFSET_0" > "HOLLight.hollight.SUM_OFFSET_0"
"SUM_OFFSET" > "HOLLight.hollight.SUM_OFFSET"
- "SUM_NEG_NUMSEG" > "HOLLight.hollight.SUM_NEG_NUMSEG"
"SUM_NEG" > "HOLLight.hollight.SUM_NEG"
"SUM_MULTICOUNT_GEN" > "HOLLight.hollight.SUM_MULTICOUNT_GEN"
"SUM_MULTICOUNT" > "HOLLight.hollight.SUM_MULTICOUNT"
"SUM_LT_ALL" > "HOLLight.hollight.SUM_LT_ALL"
"SUM_LT" > "HOLLight.hollight.SUM_LT"
+ "SUM_LMUL" > "HOLLight.hollight.SUM_LMUL"
"SUM_LE_NUMSEG" > "HOLLight.hollight.SUM_LE_NUMSEG"
+ "SUM_LE_INCLUDED" > "HOLLight.hollight.SUM_LE_INCLUDED"
"SUM_LE" > "HOLLight.hollight.SUM_LE"
+ "SUM_INJECTION" > "HOLLight.hollight.SUM_INJECTION"
+ "SUM_INCL_EXCL" > "HOLLight.hollight.SUM_INCL_EXCL"
+ "SUM_IMAGE_NONZERO" > "HOLLight.hollight.SUM_IMAGE_NONZERO"
+ "SUM_IMAGE_LE" > "HOLLight.hollight.SUM_IMAGE_LE"
"SUM_IMAGE_GEN" > "HOLLight.hollight.SUM_IMAGE_GEN"
"SUM_IMAGE" > "HOLLight.hollight.SUM_IMAGE"
+ "SUM_GROUP" > "HOLLight.hollight.SUM_GROUP"
"SUM_EQ_SUPERSET" > "HOLLight.hollight.SUM_EQ_SUPERSET"
"SUM_EQ_NUMSEG" > "HOLLight.hollight.SUM_EQ_NUMSEG"
+ "SUM_EQ_GENERAL_INVERSES" > "HOLLight.hollight.SUM_EQ_GENERAL_INVERSES"
"SUM_EQ_GENERAL" > "HOLLight.hollight.SUM_EQ_GENERAL"
"SUM_EQ_0_NUMSEG" > "HOLLight.hollight.SUM_EQ_0_NUMSEG"
"SUM_EQ_0" > "HOLLight.hollight.SUM_EQ_0"
"SUM_EQ" > "HOLLight.hollight.SUM_EQ"
+ "SUM_DIFFS_ALT" > "HOLLight.hollight.SUM_DIFFS_ALT"
+ "SUM_DIFFS" > "HOLLight.hollight.SUM_DIFFS"
"SUM_DIFF" > "HOLLight.hollight.SUM_DIFF"
"SUM_DELTA" > "HOLLight.hollight.SUM_DELTA"
+ "SUM_DELETE_CASES" > "HOLLight.hollight.SUM_DELETE_CASES"
+ "SUM_DELETE" > "HOLLight.hollight.SUM_DELETE"
"SUM_CONST_NUMSEG" > "HOLLight.hollight.SUM_CONST_NUMSEG"
"SUM_CONST" > "HOLLight.hollight.SUM_CONST"
- "SUM_CMUL_NUMSEG" > "HOLLight.hollight.SUM_CMUL_NUMSEG"
- "SUM_CMUL" > "HOLLight.hollight.SUM_CMUL"
+ "SUM_COMBINE_R" > "HOLLight.hollight.SUM_COMBINE_R"
+ "SUM_COMBINE_L" > "HOLLight.hollight.SUM_COMBINE_L"
+ "SUM_CLOSED" > "HOLLight.hollight.SUM_CLOSED"
"SUM_CLAUSES_RIGHT" > "HOLLight.hollight.SUM_CLAUSES_RIGHT"
"SUM_CLAUSES_NUMSEG" > "HOLLight.hollight.SUM_CLAUSES_NUMSEG"
"SUM_CLAUSES_LEFT" > "HOLLight.hollight.SUM_CLAUSES_LEFT"
"SUM_CLAUSES" > "HOLLight.hollight.SUM_CLAUSES"
+ "SUM_CASES_1" > "HOLLight.hollight.SUM_CASES_1"
+ "SUM_CASES" > "HOLLight.hollight.SUM_CASES"
"SUM_BOUND_LT_GEN" > "HOLLight.hollight.SUM_BOUND_LT_GEN"
"SUM_BOUND_LT_ALL" > "HOLLight.hollight.SUM_BOUND_LT_ALL"
"SUM_BOUND_LT" > "HOLLight.hollight.SUM_BOUND_LT"
@@ -564,58 +759,66 @@
"SUM_BIJECTION" > "HOLLight.hollight.SUM_BIJECTION"
"SUM_ADD_SPLIT" > "HOLLight.hollight.SUM_ADD_SPLIT"
"SUM_ADD_NUMSEG" > "HOLLight.hollight.SUM_ADD_NUMSEG"
+ "SUM_ADD_GEN" > "HOLLight.hollight.SUM_ADD_GEN"
"SUM_ADD" > "HOLLight.hollight.SUM_ADD"
"SUM_ABS_NUMSEG" > "HOLLight.hollight.SUM_ABS_NUMSEG"
+ "SUM_ABS_LE" > "HOLLight.hollight.SUM_ABS_LE"
"SUM_ABS_BOUND" > "HOLLight.hollight.SUM_ABS_BOUND"
"SUM_ABS" > "HOLLight.hollight.SUM_ABS"
"SUM_0" > "HOLLight.hollight.SUM_0"
- "SUC_SUB1" > "HOLLight.hollight.SUC_SUB1"
- "SUC_INJ" > "Nat.nat.simps_3"
+ "SUC_SUB1" > "Nat.diff_Suc_1"
+ "SUC_INJ" > "HOLLightCompat.SUC_INJ"
"SUB_SUC" > "Nat.diff_Suc_Suc"
"SUB_REFL" > "Nat.diff_self_eq_0"
"SUB_PRESUC" > "HOLLight.hollight.SUB_PRESUC"
- "SUB_EQ_0" > "HOLLight.hollight.SUB_EQ_0"
+ "SUB_EQ_0" > "Nat.diff_is_0_eq"
"SUB_ELIM_THM" > "HOLLight.hollight.SUB_ELIM_THM"
"SUB_ADD_RCANCEL" > "Nat.diff_cancel2"
"SUB_ADD_LCANCEL" > "Nat.diff_cancel"
- "SUB_ADD" > "HOLLight.hollight.SUB_ADD"
+ "SUB_ADD" > "Nat.le_add_diff_inverse2"
"SUB_0" > "HOLLight.hollight.SUB_0"
- "SUBSET_def" > "HOLLight.hollight.SUBSET_def"
- "SUBSET_UNIV" > "HOLLight.hollight.SUBSET_UNIV"
- "SUBSET_UNION_ABSORPTION" > "HOLLight.hollight.SUBSET_UNION_ABSORPTION"
+ "SUBSET_UNIV" > "Orderings.top_class.top_greatest"
+ "SUBSET_UNION_ABSORPTION" > "Lattices.semilattice_sup_class.le_iff_sup"
+ "SUBSET_UNIONS" > "Complete_Lattice.Union_mono"
"SUBSET_UNION" > "HOLLight.hollight.SUBSET_UNION"
- "SUBSET_TRANS" > "HOLLight.hollight.SUBSET_TRANS"
+ "SUBSET_TRANS" > "Orderings.order_trans_rules_23"
"SUBSET_RESTRICT" > "HOLLight.hollight.SUBSET_RESTRICT"
- "SUBSET_REFL" > "HOLLight.hollight.SUBSET_REFL"
- "SUBSET_PSUBSET_TRANS" > "HOLLight.hollight.SUBSET_PSUBSET_TRANS"
+ "SUBSET_REFL" > "Inductive.basic_monos_1"
+ "SUBSET_PSUBSET_TRANS" > "Orderings.order_le_less_trans"
"SUBSET_NUMSEG" > "HOLLight.hollight.SUBSET_NUMSEG"
- "SUBSET_INTER_ABSORPTION" > "HOLLight.hollight.SUBSET_INTER_ABSORPTION"
+ "SUBSET_INTER_ABSORPTION" > "Lattices.semilattice_inf_class.le_iff_inf"
+ "SUBSET_INTER" > "Lattices.semilattice_inf_class.le_inf_iff"
"SUBSET_INSERT_DELETE" > "HOLLight.hollight.SUBSET_INSERT_DELETE"
- "SUBSET_INSERT" > "HOLLight.hollight.SUBSET_INSERT"
- "SUBSET_IMAGE" > "HOLLight.hollight.SUBSET_IMAGE"
- "SUBSET_EMPTY" > "HOLLight.hollight.SUBSET_EMPTY"
- "SUBSET_DIFF" > "HOLLight.hollight.SUBSET_DIFF"
+ "SUBSET_INSERT" > "Set.subset_insert"
+ "SUBSET_IMAGE" > "Set.subset_image_iff"
+ "SUBSET_EMPTY" > "Set.subset_empty"
+ "SUBSET_DIFF" > "Set.Diff_subset"
"SUBSET_DELETE" > "HOLLight.hollight.SUBSET_DELETE"
- "SUBSET_ANTISYM" > "HOLLight.hollight.SUBSET_ANTISYM"
- "SOME_def" > "HOLLight.hollight.SOME_def"
- "SNDCART_PASTECART" > "HOLLight.hollight.SNDCART_PASTECART"
+ "SUBSET_CARD_EQ" > "HOLLight.hollight.SUBSET_CARD_EQ"
+ "SUBSET_ANTISYM_EQ" > "Orderings.order_class.eq_iff"
+ "SUBSET_ANTISYM" > "Orderings.order_antisym"
"SND" > "Product_Type.snd_conv"
"SKOLEM_THM" > "HOLLight.hollight.SKOLEM_THM"
"SING_def" > "HOLLight.hollight.SING_def"
+ "SING_SUBSET" > "HOLLight.hollight.SING_SUBSET"
+ "SING_GSPEC" > "HOLLight.hollight.SING_GSPEC"
+ "SIMPLE_IMAGE_GEN" > "HOLLight.hollight.SIMPLE_IMAGE_GEN"
"SIMPLE_IMAGE" > "HOLLight.hollight.SIMPLE_IMAGE"
"SET_RECURSION_LEMMA" > "HOLLight.hollight.SET_RECURSION_LEMMA"
- "SET_OF_LIST_OF_SET" > "HOLLight.hollight.SET_OF_LIST_OF_SET"
- "SET_OF_LIST_APPEND" > "HOLLight.hollight.SET_OF_LIST_APPEND"
+ "SET_PROVE_CASES" > "HOLLight.hollight.SET_PROVE_CASES"
+ "SET_PAIR_THM" > "HOLLight.hollight.SET_PAIR_THM"
+ "SET_OF_LIST_MAP" > "List.set_map"
+ "SET_OF_LIST_EQ_EMPTY" > "List.set_empty"
+ "SET_OF_LIST_APPEND" > "List.set_append"
"SET_CASES" > "HOLLight.hollight.SET_CASES"
- "SETSPEC_def" > "HOLLight.hollight.SETSPEC_def"
"SEMIRING_PTHS" > "HOLLight.hollight.SEMIRING_PTHS"
"SELECT_UNIQUE" > "HOLLight.hollight.SELECT_UNIQUE"
"SELECT_REFL" > "Hilbert_Choice.some_eq_trivial"
"SELECT_LEMMA" > "Hilbert_Choice.some_sym_eq_trivial"
- "RIGHT_SUB_DISTRIB" > "Nat.nat_distrib_3"
+ "RIGHT_SUB_DISTRIB" > "Nat.diff_mult_distrib"
"RIGHT_OR_FORALL_THM" > "HOL.all_simps_4"
"RIGHT_OR_EXISTS_THM" > "HOL.ex_simps_4"
- "RIGHT_OR_DISTRIB" > "HOL.conj_disj_distribR"
+ "RIGHT_OR_DISTRIB" > "Groebner_Basis.dnf_2"
"RIGHT_IMP_FORALL_THM" > "HOL.all_simps_6"
"RIGHT_IMP_EXISTS_THM" > "HOL.ex_simps_6"
"RIGHT_FORALL_OR_THM" > "HOL.all_simps_4"
@@ -624,16 +827,11 @@
"RIGHT_EXISTS_AND_THM" > "HOL.ex_simps_2"
"RIGHT_AND_FORALL_THM" > "HOL.all_simps_2"
"RIGHT_AND_EXISTS_THM" > "HOL.ex_simps_2"
- "RIGHT_ADD_DISTRIB" > "Nat.nat_distrib_1"
- "REVERSE_def" > "HOLLight.hollight.REVERSE_def"
- "REVERSE_REVERSE" > "HOLLight.hollight.REVERSE_REVERSE"
- "REVERSE_APPEND" > "HOLLight.hollight.REVERSE_APPEND"
+ "RIGHT_ADD_DISTRIB" > "Fields.linordered_field_class.sign_simps_26"
+ "REVERSE_REVERSE" > "List.rev_rev_ident"
+ "REVERSE_APPEND" > "List.rev_append"
"REST_def" > "HOLLight.hollight.REST_def"
- "REP_ABS_PAIR" > "HOLLightCompat.REP_ABS_PAIR"
- "REPLICATE_def" > "HOLLight.hollight.REPLICATE_def"
- "REFL_CLAUSE" > "HOL.simp_thms_6"
- "RECURSION_CASEWISE_PAIRWISE" > "HOLLight.hollight.RECURSION_CASEWISE_PAIRWISE"
- "RECURSION_CASEWISE" > "HOLLight.hollight.RECURSION_CASEWISE"
+ "REFL_CLAUSE" > "Groebner_Basis.bool_simps_6"
"REAL_WLOG_LT" > "HOLLight.hollight.REAL_WLOG_LT"
"REAL_WLOG_LE" > "HOLLight.hollight.REAL_WLOG_LE"
"REAL_SUB_TRIANGLE" > "HOLLight.hollight.REAL_SUB_TRIANGLE"
@@ -643,6 +841,9 @@
"REAL_SUB_RNEG" > "HOLLight.hollight.REAL_SUB_RNEG"
"REAL_SUB_REFL" > "HOLLight.hollight.REAL_SUB_REFL"
"REAL_SUB_RDISTRIB" > "HOLLight.hollight.REAL_SUB_RDISTRIB"
+ "REAL_SUB_POW_R1" > "HOLLight.hollight.REAL_SUB_POW_R1"
+ "REAL_SUB_POW_L1" > "HOLLight.hollight.REAL_SUB_POW_L1"
+ "REAL_SUB_POW" > "HOLLight.hollight.REAL_SUB_POW"
"REAL_SUB_NEG2" > "HOLLight.hollight.REAL_SUB_NEG2"
"REAL_SUB_LZERO" > "HOLLight.hollight.REAL_SUB_LZERO"
"REAL_SUB_LT" > "HOLLight.hollight.REAL_SUB_LT"
@@ -655,7 +856,15 @@
"REAL_SUB_ABS" > "HOLLight.hollight.REAL_SUB_ABS"
"REAL_SUB_0" > "HOLLight.hollight.REAL_SUB_0"
"REAL_SOS_EQ_0" > "HOLLight.hollight.REAL_SOS_EQ_0"
+ "REAL_SGN_NEG" > "HOLLight.hollight.REAL_SGN_NEG"
+ "REAL_SGN_MUL" > "HOLLight.hollight.REAL_SGN_MUL"
+ "REAL_SGN_INV" > "HOLLight.hollight.REAL_SGN_INV"
+ "REAL_SGN_DIV" > "HOLLight.hollight.REAL_SGN_DIV"
+ "REAL_SGN_ABS" > "HOLLight.hollight.REAL_SGN_ABS"
+ "REAL_SGN_0" > "HOLLight.hollight.REAL_SGN_0"
+ "REAL_SGN" > "HOLLight.hollight.REAL_SGN"
"REAL_RNEG_UNIQ" > "HOLLight.hollight.REAL_RNEG_UNIQ"
+ "REAL_POW_ZERO" > "HOLLight.hollight.REAL_POW_ZERO"
"REAL_POW_SUB" > "HOLLight.hollight.REAL_POW_SUB"
"REAL_POW_POW" > "HOLLight.hollight.REAL_POW_POW"
"REAL_POW_ONE" > "HOLLight.hollight.REAL_POW_ONE"
@@ -663,17 +872,33 @@
"REAL_POW_NEG" > "HOLLight.hollight.REAL_POW_NEG"
"REAL_POW_MUL" > "HOLLight.hollight.REAL_POW_MUL"
"REAL_POW_MONO_LT" > "HOLLight.hollight.REAL_POW_MONO_LT"
+ "REAL_POW_MONO_INV" > "HOLLight.hollight.REAL_POW_MONO_INV"
"REAL_POW_MONO" > "HOLLight.hollight.REAL_POW_MONO"
+ "REAL_POW_LT_1" > "HOLLight.hollight.REAL_POW_LT_1"
+ "REAL_POW_LT2_REV" > "HOLLight.hollight.REAL_POW_LT2_REV"
+ "REAL_POW_LT2_ODD_EQ" > "HOLLight.hollight.REAL_POW_LT2_ODD_EQ"
+ "REAL_POW_LT2_ODD" > "HOLLight.hollight.REAL_POW_LT2_ODD"
"REAL_POW_LT2" > "HOLLight.hollight.REAL_POW_LT2"
"REAL_POW_LT" > "HOLLight.hollight.REAL_POW_LT"
"REAL_POW_LE_1" > "HOLLight.hollight.REAL_POW_LE_1"
+ "REAL_POW_LE2_REV" > "HOLLight.hollight.REAL_POW_LE2_REV"
+ "REAL_POW_LE2_ODD_EQ" > "HOLLight.hollight.REAL_POW_LE2_ODD_EQ"
+ "REAL_POW_LE2_ODD" > "HOLLight.hollight.REAL_POW_LE2_ODD"
"REAL_POW_LE2" > "HOLLight.hollight.REAL_POW_LE2"
"REAL_POW_LE" > "HOLLight.hollight.REAL_POW_LE"
"REAL_POW_INV" > "HOLLight.hollight.REAL_POW_INV"
+ "REAL_POW_EQ_ODD_EQ" > "HOLLight.hollight.REAL_POW_EQ_ODD_EQ"
+ "REAL_POW_EQ_ODD" > "HOLLight.hollight.REAL_POW_EQ_ODD"
+ "REAL_POW_EQ_EQ" > "HOLLight.hollight.REAL_POW_EQ_EQ"
+ "REAL_POW_EQ_ABS" > "HOLLight.hollight.REAL_POW_EQ_ABS"
+ "REAL_POW_EQ_1_IMP" > "HOLLight.hollight.REAL_POW_EQ_1_IMP"
+ "REAL_POW_EQ_1" > "HOLLight.hollight.REAL_POW_EQ_1"
"REAL_POW_EQ_0" > "HOLLight.hollight.REAL_POW_EQ_0"
+ "REAL_POW_EQ" > "HOLLight.hollight.REAL_POW_EQ"
"REAL_POW_DIV" > "HOLLight.hollight.REAL_POW_DIV"
"REAL_POW_ADD" > "HOLLight.hollight.REAL_POW_ADD"
"REAL_POW_2" > "HOLLight.hollight.REAL_POW_2"
+ "REAL_POW_1_LT" > "HOLLight.hollight.REAL_POW_1_LT"
"REAL_POW_1_LE" > "HOLLight.hollight.REAL_POW_1_LE"
"REAL_POW_1" > "HOLLight.hollight.REAL_POW_1"
"REAL_POW2_ABS" > "HOLLight.hollight.REAL_POW2_ABS"
@@ -686,6 +911,8 @@
"REAL_OF_NUM_SUC" > "HOLLight.hollight.REAL_OF_NUM_SUC"
"REAL_OF_NUM_SUB" > "HOLLight.hollight.REAL_OF_NUM_SUB"
"REAL_OF_NUM_POW" > "HOLLight.hollight.REAL_OF_NUM_POW"
+ "REAL_OF_NUM_MIN" > "HOLLight.hollight.REAL_OF_NUM_MIN"
+ "REAL_OF_NUM_MAX" > "HOLLight.hollight.REAL_OF_NUM_MAX"
"REAL_OF_NUM_LT" > "HOLLight.hollight.REAL_OF_NUM_LT"
"REAL_OF_NUM_GT" > "HOLLight.hollight.REAL_OF_NUM_GT"
"REAL_OF_NUM_GE" > "HOLLight.hollight.REAL_OF_NUM_GE"
@@ -712,6 +939,8 @@
"REAL_MUL_RINV_UNIQ" > "HOLLight.hollight.REAL_MUL_RINV_UNIQ"
"REAL_MUL_RINV" > "HOLLight.hollight.REAL_MUL_RINV"
"REAL_MUL_RID" > "HOLLight.hollight.REAL_MUL_RID"
+ "REAL_MUL_POS_LT" > "HOLLight.hollight.REAL_MUL_POS_LT"
+ "REAL_MUL_POS_LE" > "HOLLight.hollight.REAL_MUL_POS_LE"
"REAL_MUL_LZERO" > "HOLLight.hollight.REAL_MUL_LZERO"
"REAL_MUL_LNEG" > "HOLLight.hollight.REAL_MUL_LNEG"
"REAL_MUL_LINV_UNIQ" > "HOLLight.hollight.REAL_MUL_LINV_UNIQ"
@@ -735,10 +964,12 @@
"REAL_LT_TOTAL" > "HOLLight.hollight.REAL_LT_TOTAL"
"REAL_LT_SUB_RADD" > "HOLLight.hollight.REAL_LT_SUB_RADD"
"REAL_LT_SUB_LADD" > "HOLLight.hollight.REAL_LT_SUB_LADD"
+ "REAL_LT_SQUARE_ABS" > "HOLLight.hollight.REAL_LT_SQUARE_ABS"
"REAL_LT_SQUARE" > "HOLLight.hollight.REAL_LT_SQUARE"
"REAL_LT_RNEG" > "HOLLight.hollight.REAL_LT_RNEG"
"REAL_LT_RMUL_EQ" > "HOLLight.hollight.REAL_LT_RMUL_EQ"
"REAL_LT_RMUL" > "HOLLight.hollight.REAL_LT_RMUL"
+ "REAL_LT_RINV" > "HOLLight.hollight.REAL_LT_RINV"
"REAL_LT_REFL" > "HOLLight.hollight.REAL_LT_REFL"
"REAL_LT_RDIV_EQ" > "HOLLight.hollight.REAL_LT_RDIV_EQ"
"REAL_LT_RCANCEL_IMP" > "HOLLight.hollight.REAL_LT_RCANCEL_IMP"
@@ -747,6 +978,7 @@
"REAL_LT_NEGTOTAL" > "HOLLight.hollight.REAL_LT_NEGTOTAL"
"REAL_LT_NEG2" > "HOLLight.hollight.REAL_LT_NEG2"
"REAL_LT_NEG" > "HOLLight.hollight.REAL_LT_NEG"
+ "REAL_LT_MUL_EQ" > "HOLLight.hollight.REAL_LT_MUL_EQ"
"REAL_LT_MUL2" > "HOLLight.hollight.REAL_LT_MUL2"
"REAL_LT_MUL" > "HOLLight.hollight.REAL_LT_MUL"
"REAL_LT_MIN" > "HOLLight.hollight.REAL_LT_MIN"
@@ -754,6 +986,7 @@
"REAL_LT_LNEG" > "HOLLight.hollight.REAL_LT_LNEG"
"REAL_LT_LMUL_EQ" > "HOLLight.hollight.REAL_LT_LMUL_EQ"
"REAL_LT_LMUL" > "HOLLight.hollight.REAL_LT_LMUL"
+ "REAL_LT_LINV" > "HOLLight.hollight.REAL_LT_LINV"
"REAL_LT_LE" > "HOLLight.hollight.REAL_LT_LE"
"REAL_LT_LDIV_EQ" > "HOLLight.hollight.REAL_LT_LDIV_EQ"
"REAL_LT_LCANCEL_IMP" > "HOLLight.hollight.REAL_LT_LCANCEL_IMP"
@@ -784,7 +1017,6 @@
"REAL_LTE_ADD2" > "HOLLight.hollight.REAL_LTE_ADD2"
"REAL_LTE_ADD" > "HOLLight.hollight.REAL_LTE_ADD"
"REAL_LNEG_UNIQ" > "HOLLight.hollight.REAL_LNEG_UNIQ"
- "REAL_LE_TOTAL" > "HOLLight.hollight.REAL_LE_TOTAL"
"REAL_LE_SUB_RADD" > "HOLLight.hollight.REAL_LE_SUB_RADD"
"REAL_LE_SUB_LADD" > "HOLLight.hollight.REAL_LE_SUB_LADD"
"REAL_LE_SQUARE_ABS" > "HOLLight.hollight.REAL_LE_SQUARE_ABS"
@@ -792,16 +1024,18 @@
"REAL_LE_RNEG" > "HOLLight.hollight.REAL_LE_RNEG"
"REAL_LE_RMUL_EQ" > "HOLLight.hollight.REAL_LE_RMUL_EQ"
"REAL_LE_RMUL" > "HOLLight.hollight.REAL_LE_RMUL"
- "REAL_LE_REFL" > "HOLLight.hollight.REAL_LE_REFL"
+ "REAL_LE_RINV" > "HOLLight.hollight.REAL_LE_RINV"
"REAL_LE_RDIV_EQ" > "HOLLight.hollight.REAL_LE_RDIV_EQ"
"REAL_LE_RCANCEL_IMP" > "HOLLight.hollight.REAL_LE_RCANCEL_IMP"
"REAL_LE_RADD" > "HOLLight.hollight.REAL_LE_RADD"
+ "REAL_LE_POW_2" > "HOLLight.hollight.REAL_LE_POW_2"
"REAL_LE_POW2" > "HOLLight.hollight.REAL_LE_POW2"
"REAL_LE_NEGTOTAL" > "HOLLight.hollight.REAL_LE_NEGTOTAL"
"REAL_LE_NEGR" > "HOLLight.hollight.REAL_LE_NEGR"
"REAL_LE_NEGL" > "HOLLight.hollight.REAL_LE_NEGL"
"REAL_LE_NEG2" > "HOLLight.hollight.REAL_LE_NEG2"
"REAL_LE_NEG" > "HOLLight.hollight.REAL_LE_NEG"
+ "REAL_LE_MUL_EQ" > "HOLLight.hollight.REAL_LE_MUL_EQ"
"REAL_LE_MUL2" > "HOLLight.hollight.REAL_LE_MUL2"
"REAL_LE_MIN" > "HOLLight.hollight.REAL_LE_MIN"
"REAL_LE_MAX" > "HOLLight.hollight.REAL_LE_MAX"
@@ -809,6 +1043,7 @@
"REAL_LE_LNEG" > "HOLLight.hollight.REAL_LE_LNEG"
"REAL_LE_LMUL_EQ" > "HOLLight.hollight.REAL_LE_LMUL_EQ"
"REAL_LE_LMUL" > "HOLLight.hollight.REAL_LE_LMUL"
+ "REAL_LE_LINV" > "HOLLight.hollight.REAL_LE_LINV"
"REAL_LE_LDIV_EQ" > "HOLLight.hollight.REAL_LE_LDIV_EQ"
"REAL_LE_LCANCEL_IMP" > "HOLLight.hollight.REAL_LE_LCANCEL_IMP"
"REAL_LE_LADD" > "HOLLight.hollight.REAL_LE_LADD"
@@ -818,7 +1053,6 @@
"REAL_LE_DOUBLE" > "HOLLight.hollight.REAL_LE_DOUBLE"
"REAL_LE_DIV2_EQ" > "HOLLight.hollight.REAL_LE_DIV2_EQ"
"REAL_LE_DIV" > "HOLLight.hollight.REAL_LE_DIV"
- "REAL_LE_ANTISYM" > "HOLLight.hollight.REAL_LE_ANTISYM"
"REAL_LE_ADDR" > "HOLLight.hollight.REAL_LE_ADDR"
"REAL_LE_ADDL" > "HOLLight.hollight.REAL_LE_ADDL"
"REAL_LE_ADD2" > "HOLLight.hollight.REAL_LE_ADD2"
@@ -829,18 +1063,24 @@
"REAL_LET_ANTISYM" > "HOLLight.hollight.REAL_LET_ANTISYM"
"REAL_LET_ADD2" > "HOLLight.hollight.REAL_LET_ADD2"
"REAL_LET_ADD" > "HOLLight.hollight.REAL_LET_ADD"
+ "REAL_INV_POW" > "HOLLight.hollight.REAL_INV_POW"
"REAL_INV_NEG" > "HOLLight.hollight.REAL_INV_NEG"
"REAL_INV_MUL" > "HOLLight.hollight.REAL_INV_MUL"
+ "REAL_INV_LT_1" > "HOLLight.hollight.REAL_INV_LT_1"
"REAL_INV_LE_1" > "HOLLight.hollight.REAL_INV_LE_1"
"REAL_INV_INV" > "HOLLight.hollight.REAL_INV_INV"
+ "REAL_INV_EQ_1" > "HOLLight.hollight.REAL_INV_EQ_1"
"REAL_INV_EQ_0" > "HOLLight.hollight.REAL_INV_EQ_0"
"REAL_INV_DIV" > "HOLLight.hollight.REAL_INV_DIV"
+ "REAL_INV_1_LT" > "HOLLight.hollight.REAL_INV_1_LT"
"REAL_INV_1_LE" > "HOLLight.hollight.REAL_INV_1_LE"
"REAL_INV_1" > "HOLLight.hollight.REAL_INV_1"
+ "REAL_INTEGRAL" > "HOLLight.hollight.REAL_INTEGRAL"
"REAL_HREAL_LEMMA2" > "HOLLight.hollight.REAL_HREAL_LEMMA2"
"REAL_HREAL_LEMMA1" > "HOLLight.hollight.REAL_HREAL_LEMMA1"
"REAL_EQ_SUB_RADD" > "HOLLight.hollight.REAL_EQ_SUB_RADD"
"REAL_EQ_SUB_LADD" > "HOLLight.hollight.REAL_EQ_SUB_LADD"
+ "REAL_EQ_SQUARE_ABS" > "HOLLight.hollight.REAL_EQ_SQUARE_ABS"
"REAL_EQ_RDIV_EQ" > "HOLLight.hollight.REAL_EQ_RDIV_EQ"
"REAL_EQ_RCANCEL_IMP" > "HOLLight.hollight.REAL_EQ_RCANCEL_IMP"
"REAL_EQ_NEG2" > "HOLLight.hollight.REAL_EQ_NEG2"
@@ -848,6 +1088,7 @@
"REAL_EQ_MUL_LCANCEL" > "HOLLight.hollight.REAL_EQ_MUL_LCANCEL"
"REAL_EQ_LDIV_EQ" > "HOLLight.hollight.REAL_EQ_LDIV_EQ"
"REAL_EQ_LCANCEL_IMP" > "HOLLight.hollight.REAL_EQ_LCANCEL_IMP"
+ "REAL_EQ_INV2" > "HOLLight.hollight.REAL_EQ_INV2"
"REAL_EQ_IMP_LE" > "HOLLight.hollight.REAL_EQ_IMP_LE"
"REAL_EQ_ADD_RCANCEL_0" > "HOLLight.hollight.REAL_EQ_ADD_RCANCEL_0"
"REAL_EQ_ADD_RCANCEL" > "HOLLight.hollight.REAL_EQ_ADD_RCANCEL"
@@ -865,6 +1106,8 @@
"REAL_DIFFSQ" > "HOLLight.hollight.REAL_DIFFSQ"
"REAL_COMPLETE_SOMEPOS" > "HOLLight.hollight.REAL_COMPLETE_SOMEPOS"
"REAL_COMPLETE" > "HOLLight.hollight.REAL_COMPLETE"
+ "REAL_BOUNDS_LT" > "HOLLight.hollight.REAL_BOUNDS_LT"
+ "REAL_BOUNDS_LE" > "HOLLight.hollight.REAL_BOUNDS_LE"
"REAL_ADD_SUB2" > "HOLLight.hollight.REAL_ADD_SUB2"
"REAL_ADD_SUB" > "HOLLight.hollight.REAL_ADD_SUB"
"REAL_ADD_RINV" > "HOLLight.hollight.REAL_ADD_RINV"
@@ -881,6 +1124,7 @@
"REAL_ABS_STILLNZ" > "HOLLight.hollight.REAL_ABS_STILLNZ"
"REAL_ABS_SIGN2" > "HOLLight.hollight.REAL_ABS_SIGN2"
"REAL_ABS_SIGN" > "HOLLight.hollight.REAL_ABS_SIGN"
+ "REAL_ABS_SGN" > "HOLLight.hollight.REAL_ABS_SGN"
"REAL_ABS_REFL" > "HOLLight.hollight.REAL_ABS_REFL"
"REAL_ABS_POW" > "HOLLight.hollight.REAL_ABS_POW"
"REAL_ABS_POS" > "HOLLight.hollight.REAL_ABS_POS"
@@ -906,44 +1150,50 @@
"RAT_LEMMA3" > "HOLLight.hollight.RAT_LEMMA3"
"RAT_LEMMA2" > "HOLLight.hollight.RAT_LEMMA2"
"RAT_LEMMA1" > "HOLLight.hollight.RAT_LEMMA1"
- "PSUBSET_def" > "HOLLight.hollight.PSUBSET_def"
"PSUBSET_UNIV" > "HOLLight.hollight.PSUBSET_UNIV"
- "PSUBSET_TRANS" > "HOLLight.hollight.PSUBSET_TRANS"
- "PSUBSET_SUBSET_TRANS" > "HOLLight.hollight.PSUBSET_SUBSET_TRANS"
+ "PSUBSET_TRANS" > "Orderings.order_less_trans"
+ "PSUBSET_SUBSET_TRANS" > "Orderings.order_less_le_trans"
"PSUBSET_MEMBER" > "HOLLight.hollight.PSUBSET_MEMBER"
- "PSUBSET_IRREFL" > "HOLLight.hollight.PSUBSET_IRREFL"
+ "PSUBSET_IRREFL" > "Orderings.order_less_irrefl"
"PSUBSET_INSERT_SUBSET" > "HOLLight.hollight.PSUBSET_INSERT_SUBSET"
+ "PSUBSET_ALT" > "HOLLight.hollight.PSUBSET_ALT"
"PRE_ELIM_THM" > "HOLLight.hollight.PRE_ELIM_THM"
- "PASTECART_FST_SND" > "HOLLight.hollight.PASTECART_FST_SND"
- "PASTECART_EQ" > "HOLLight.hollight.PASTECART_EQ"
+ "POWERSET_CLAUSES" > "HOLLight.hollight.POWERSET_CLAUSES"
"PASSOC_def" > "HOLLight.hollight.PASSOC_def"
- "PAIR_SURJECTIVE" > "Datatype.prod.nchotomy"
+ "PAIR_SURJECTIVE" > "Product_Type.prod.nchotomy"
"PAIR_EXISTS_THM" > "HOLLight.hollight.PAIR_EXISTS_THM"
- "PAIR_EQ" > "Datatype.prod.simps_1"
+ "PAIR_EQ" > "Product_Type.Pair_eq"
"PAIRWISE_def" > "HOLLight.hollight.PAIRWISE_def"
- "PAIR" > "HOL4Compat.PAIR"
- "OUTR_def" > "HOLLight.hollight.OUTR_def"
- "OUTL_def" > "HOLLight.hollight.OUTL_def"
+ "PAIRWISE_SING" > "HOLLight.hollight.PAIRWISE_SING"
+ "PAIRWISE_MONO" > "HOLLight.hollight.PAIRWISE_MONO"
+ "PAIRWISE_EMPTY" > "HOLLight.hollight.PAIRWISE_EMPTY"
+ "PAIR" > "HOLLightCompat.PAIR"
"OR_EXISTS_THM" > "HOL.ex_disj_distrib"
"OR_CLAUSES" > "HOLLight.hollight.OR_CLAUSES"
- "ONE" > "HOLLight.hollight.ONE"
- "ODD_def" > "HOLLight.hollight.ODD_def"
+ "ONE" > "Nat.One_nat_def"
+ "ODD_SUB" > "HOLLight.hollight.ODD_SUB"
"ODD_MULT" > "HOLLight.hollight.ODD_MULT"
"ODD_MOD" > "HOLLight.hollight.ODD_MOD"
"ODD_EXP" > "HOLLight.hollight.ODD_EXP"
- "ODD_EXISTS" > "HOLLight.hollight.ODD_EXISTS"
+ "ODD_EXISTS" > "Parity.odd_Suc_mult_two_ex"
"ODD_DOUBLE" > "HOLLight.hollight.ODD_DOUBLE"
- "ODD_ADD" > "HOLLight.hollight.ODD_ADD"
+ "ODD_ADD" > "Parity.odd_add"
+ "NUM_REP_def" > "HOLLight.hollight.NUM_REP_def"
+ "NUM_OF_INT_OF_NUM" > "HOLLight.hollight.NUM_OF_INT_OF_NUM"
+ "NUM_OF_INT" > "HOLLight.hollight.NUM_OF_INT"
"NUM_INTEGRAL_LEMMA" > "HOLLight.hollight.NUM_INTEGRAL_LEMMA"
"NUM_INTEGRAL" > "HOLLight.hollight.NUM_INTEGRAL"
+ "NUM_GCD" > "HOLLight.hollight.NUM_GCD"
"NUMSUM_def" > "HOLLight.hollight.NUMSUM_def"
"NUMSUM_INJ" > "HOLLight.hollight.NUMSUM_INJ"
"NUMSND_def" > "HOLLight.hollight.NUMSND_def"
- "NUMSEG_SING" > "HOLLight.hollight.NUMSEG_SING"
+ "NUMSEG_SING" > "SetInterval.order_class.atLeastAtMost_singleton"
"NUMSEG_RREC" > "HOLLight.hollight.NUMSEG_RREC"
- "NUMSEG_REC" > "HOLLight.hollight.NUMSEG_REC"
- "NUMSEG_OFFSET_IMAGE" > "HOLLight.hollight.NUMSEG_OFFSET_IMAGE"
+ "NUMSEG_REC" > "SetInterval.atLeastAtMostSuc_conv"
+ "NUMSEG_OFFSET_IMAGE" > "SetInterval.image_add_atLeastAtMost"
+ "NUMSEG_LT" > "HOLLight.hollight.NUMSEG_LT"
"NUMSEG_LREC" > "HOLLight.hollight.NUMSEG_LREC"
+ "NUMSEG_LE" > "HOLLight.hollight.NUMSEG_LE"
"NUMSEG_EMPTY" > "HOLLight.hollight.NUMSEG_EMPTY"
"NUMSEG_COMBINE_R" > "HOLLight.hollight.NUMSEG_COMBINE_R"
"NUMSEG_COMBINE_L" > "HOLLight.hollight.NUMSEG_COMBINE_L"
@@ -955,10 +1205,11 @@
"NUMPAIR_INJ" > "HOLLight.hollight.NUMPAIR_INJ"
"NUMLEFT_def" > "HOLLight.hollight.NUMLEFT_def"
"NUMFST_def" > "HOLLight.hollight.NUMFST_def"
- "NULL_def" > "HOLLight.hollight.NULL_def"
"NSUM_UNION_RZERO" > "HOLLight.hollight.NSUM_UNION_RZERO"
+ "NSUM_UNION_NONZERO" > "HOLLight.hollight.NSUM_UNION_NONZERO"
"NSUM_UNION_LZERO" > "HOLLight.hollight.NSUM_UNION_LZERO"
"NSUM_UNION_EQ" > "HOLLight.hollight.NSUM_UNION_EQ"
+ "NSUM_UNIONS_NONZERO" > "HOLLight.hollight.NSUM_UNIONS_NONZERO"
"NSUM_UNION" > "HOLLight.hollight.NSUM_UNION"
"NSUM_TRIV_NUMSEG" > "HOLLight.hollight.NSUM_TRIV_NUMSEG"
"NSUM_SWAP_NUMSEG" > "HOLLight.hollight.NSUM_SWAP_NUMSEG"
@@ -969,13 +1220,11 @@
"NSUM_SUBSET" > "HOLLight.hollight.NSUM_SUBSET"
"NSUM_SING_NUMSEG" > "HOLLight.hollight.NSUM_SING_NUMSEG"
"NSUM_SING" > "HOLLight.hollight.NSUM_SING"
+ "NSUM_RMUL" > "HOLLight.hollight.NSUM_RMUL"
"NSUM_RESTRICT_SET" > "HOLLight.hollight.NSUM_RESTRICT_SET"
"NSUM_RESTRICT" > "HOLLight.hollight.NSUM_RESTRICT"
- "NSUM_POS_LE_NUMSEG" > "HOLLight.hollight.NSUM_POS_LE_NUMSEG"
- "NSUM_POS_LE" > "HOLLight.hollight.NSUM_POS_LE"
- "NSUM_POS_EQ_0_NUMSEG" > "HOLLight.hollight.NSUM_POS_EQ_0_NUMSEG"
- "NSUM_POS_EQ_0" > "HOLLight.hollight.NSUM_POS_EQ_0"
"NSUM_POS_BOUND" > "HOLLight.hollight.NSUM_POS_BOUND"
+ "NSUM_PAIR" > "HOLLight.hollight.NSUM_PAIR"
"NSUM_OFFSET_0" > "HOLLight.hollight.NSUM_OFFSET_0"
"NSUM_OFFSET" > "HOLLight.hollight.NSUM_OFFSET"
"NSUM_NSUM_RESTRICT" > "HOLLight.hollight.NSUM_NSUM_RESTRICT"
@@ -984,26 +1233,35 @@
"NSUM_MULTICOUNT" > "HOLLight.hollight.NSUM_MULTICOUNT"
"NSUM_LT_ALL" > "HOLLight.hollight.NSUM_LT_ALL"
"NSUM_LT" > "HOLLight.hollight.NSUM_LT"
+ "NSUM_LMUL" > "HOLLight.hollight.NSUM_LMUL"
"NSUM_LE_NUMSEG" > "HOLLight.hollight.NSUM_LE_NUMSEG"
"NSUM_LE" > "HOLLight.hollight.NSUM_LE"
+ "NSUM_INJECTION" > "HOLLight.hollight.NSUM_INJECTION"
+ "NSUM_INCL_EXCL" > "HOLLight.hollight.NSUM_INCL_EXCL"
+ "NSUM_IMAGE_NONZERO" > "HOLLight.hollight.NSUM_IMAGE_NONZERO"
"NSUM_IMAGE_GEN" > "HOLLight.hollight.NSUM_IMAGE_GEN"
"NSUM_IMAGE" > "HOLLight.hollight.NSUM_IMAGE"
+ "NSUM_GROUP" > "HOLLight.hollight.NSUM_GROUP"
"NSUM_EQ_SUPERSET" > "HOLLight.hollight.NSUM_EQ_SUPERSET"
"NSUM_EQ_NUMSEG" > "HOLLight.hollight.NSUM_EQ_NUMSEG"
+ "NSUM_EQ_GENERAL_INVERSES" > "HOLLight.hollight.NSUM_EQ_GENERAL_INVERSES"
"NSUM_EQ_GENERAL" > "HOLLight.hollight.NSUM_EQ_GENERAL"
"NSUM_EQ_0_NUMSEG" > "HOLLight.hollight.NSUM_EQ_0_NUMSEG"
+ "NSUM_EQ_0_IFF_NUMSEG" > "HOLLight.hollight.NSUM_EQ_0_IFF_NUMSEG"
+ "NSUM_EQ_0_IFF" > "HOLLight.hollight.NSUM_EQ_0_IFF"
"NSUM_EQ_0" > "HOLLight.hollight.NSUM_EQ_0"
"NSUM_EQ" > "HOLLight.hollight.NSUM_EQ"
"NSUM_DIFF" > "HOLLight.hollight.NSUM_DIFF"
"NSUM_DELTA" > "HOLLight.hollight.NSUM_DELTA"
+ "NSUM_DELETE" > "HOLLight.hollight.NSUM_DELETE"
"NSUM_CONST_NUMSEG" > "HOLLight.hollight.NSUM_CONST_NUMSEG"
"NSUM_CONST" > "HOLLight.hollight.NSUM_CONST"
- "NSUM_CMUL_NUMSEG" > "HOLLight.hollight.NSUM_CMUL_NUMSEG"
- "NSUM_CMUL" > "HOLLight.hollight.NSUM_CMUL"
+ "NSUM_CLOSED" > "HOLLight.hollight.NSUM_CLOSED"
"NSUM_CLAUSES_RIGHT" > "HOLLight.hollight.NSUM_CLAUSES_RIGHT"
"NSUM_CLAUSES_NUMSEG" > "HOLLight.hollight.NSUM_CLAUSES_NUMSEG"
"NSUM_CLAUSES_LEFT" > "HOLLight.hollight.NSUM_CLAUSES_LEFT"
"NSUM_CLAUSES" > "HOLLight.hollight.NSUM_CLAUSES"
+ "NSUM_CASES" > "HOLLight.hollight.NSUM_CASES"
"NSUM_BOUND_LT_GEN" > "HOLLight.hollight.NSUM_BOUND_LT_GEN"
"NSUM_BOUND_LT_ALL" > "HOLLight.hollight.NSUM_BOUND_LT_ALL"
"NSUM_BOUND_LT" > "HOLLight.hollight.NSUM_BOUND_LT"
@@ -1012,27 +1270,26 @@
"NSUM_BIJECTION" > "HOLLight.hollight.NSUM_BIJECTION"
"NSUM_ADD_SPLIT" > "HOLLight.hollight.NSUM_ADD_SPLIT"
"NSUM_ADD_NUMSEG" > "HOLLight.hollight.NSUM_ADD_NUMSEG"
+ "NSUM_ADD_GEN" > "HOLLight.hollight.NSUM_ADD_GEN"
"NSUM_ADD" > "HOLLight.hollight.NSUM_ADD"
"NSUM_0" > "HOLLight.hollight.NSUM_0"
- "NOT_UNIV_PSUBSET" > "HOLLight.hollight.NOT_UNIV_PSUBSET"
- "NOT_SUC" > "Nat.nat.simps_1"
- "NOT_PSUBSET_EMPTY" > "HOLLight.hollight.NOT_PSUBSET_EMPTY"
+ "NOT_UNIV_PSUBSET" > "Complete_Lattice.complete_lattice_class.not_top_less"
+ "NOT_SUC" > "Nat.Suc_not_Zero"
+ "NOT_PSUBSET_EMPTY" > "Complete_Lattice.complete_lattice_class.not_less_bot"
"NOT_ODD" > "HOLLight.hollight.NOT_ODD"
- "NOT_LT" > "HOLLight.hollight.NOT_LT"
- "NOT_LE" > "HOLLight.hollight.NOT_LE"
+ "NOT_LT" > "Orderings.linorder_class.not_less"
+ "NOT_LE" > "Orderings.linorder_class.not_le"
"NOT_IN_EMPTY" > "HOLLight.hollight.NOT_IN_EMPTY"
- "NOT_INSERT_EMPTY" > "HOLLight.hollight.NOT_INSERT_EMPTY"
- "NOT_FORALL_THM" > "Inductive.basic_monos_15"
- "NOT_EXISTS_THM" > "Inductive.basic_monos_16"
- "NOT_EX" > "HOLLight.hollight.NOT_EX"
+ "NOT_INSERT_EMPTY" > "Set.insert_not_empty"
+ "NOT_FORALL_THM" > "HOL.not_all"
+ "NOT_EXISTS_THM" > "HOL.not_ex"
+ "NOT_EX" > "HOLLightList.NOT_EX"
"NOT_EVEN" > "HOLLight.hollight.NOT_EVEN"
"NOT_EQUAL_SETS" > "HOLLight.hollight.NOT_EQUAL_SETS"
- "NOT_EMPTY_INSERT" > "HOLLight.hollight.NOT_EMPTY_INSERT"
- "NOT_CONS_NIL" > "HOLLight.hollight.NOT_CONS_NIL"
+ "NOT_EMPTY_INSERT" > "Set.empty_not_insert"
+ "NOT_CONS_NIL" > "List.list.distinct_2"
"NOT_CLAUSES_WEAK" > "HOLLight.hollight.NOT_CLAUSES_WEAK"
- "NOT_ALL" > "HOLLight.hollight.NOT_ALL"
- "NONE_def" > "HOLLight.hollight.NONE_def"
- "NIL_def" > "HOLLight.hollight.NIL_def"
+ "NOT_ALL" > "HOLLightList.NOT_ALL"
"NEUTRAL_REAL_MUL" > "HOLLight.hollight.NEUTRAL_REAL_MUL"
"NEUTRAL_REAL_ADD" > "HOLLight.hollight.NEUTRAL_REAL_ADD"
"NEUTRAL_MUL" > "HOLLight.hollight.NEUTRAL_MUL"
@@ -1105,229 +1362,252 @@
"NADD_ADD_ASSOC" > "HOLLight.hollight.NADD_ADD_ASSOC"
"NADD_ADDITIVE" > "HOLLight.hollight.NADD_ADDITIVE"
"NADD_ADD" > "HOLLight.hollight.NADD_ADD"
- "MULT_SYM" > "Int.zmult_ac_2"
+ "MULT_SYM" > "Fields.linordered_field_class.sign_simps_40"
"MULT_SUC" > "Nat.mult_Suc_right"
- "MULT_EXP" > "HOLLight.hollight.MULT_EXP"
- "MULT_EQ_1" > "HOLLight.hollight.MULT_EQ_1"
+ "MULT_EXP" > "Power.comm_monoid_mult_class.power_mult_distrib"
+ "MULT_EQ_1" > "Nat.nat_mult_eq_1_iff"
"MULT_EQ_0" > "Nat.mult_is_0"
+ "MULT_DIV_LE" > "HOLLight.hollight.MULT_DIV_LE"
"MULT_CLAUSES" > "HOLLight.hollight.MULT_CLAUSES"
- "MULT_ASSOC" > "Int.zmult_ac_1"
+ "MULT_ASSOC" > "Fields.linordered_field_class.sign_simps_41"
"MULT_AC" > "HOLLight.hollight.MULT_AC"
- "MULT_2" > "HOLLight.hollight.MULT_2"
- "MULT_0" > "Nat.mult_0_right"
+ "MULT_2" > "Int.semiring_mult_2"
+ "MULT_0" > "Divides.arithmetic_simps_41"
"MONO_FORALL" > "Inductive.basic_monos_6"
"MONO_EXISTS" > "Inductive.basic_monos_5"
"MONO_COND" > "HOLLight.hollight.MONO_COND"
- "MONO_ALL2" > "HOLLight.hollight.MONO_ALL2"
- "MONO_ALL" > "HOLLight.hollight.MONO_ALL"
+ "MONO_ALL2" > "List.list_all2_mono"
+ "MONO_ALL" > "HOLLightList.MONO_ALL"
"MONOIDAL_REAL_MUL" > "HOLLight.hollight.MONOIDAL_REAL_MUL"
"MONOIDAL_REAL_ADD" > "HOLLight.hollight.MONOIDAL_REAL_ADD"
"MONOIDAL_MUL" > "HOLLight.hollight.MONOIDAL_MUL"
"MONOIDAL_ADD" > "HOLLight.hollight.MONOIDAL_ADD"
- "MOD_def" > "HOLLight.hollight.MOD_def"
+ "MONOIDAL_AC" > "HOLLight.hollight.MONOIDAL_AC"
"MOD_UNIQ" > "HOLLight.hollight.MOD_UNIQ"
"MOD_MULT_RMOD" > "HOLLight.hollight.MOD_MULT_RMOD"
"MOD_MULT_MOD2" > "HOLLight.hollight.MOD_MULT_MOD2"
"MOD_MULT_LMOD" > "HOLLight.hollight.MOD_MULT_LMOD"
- "MOD_MULT_ADD" > "HOLLight.hollight.MOD_MULT_ADD"
+ "MOD_MULT_ADD" > "Divides.mod_mult_self3"
"MOD_MULT2" > "HOLLight.hollight.MOD_MULT2"
- "MOD_MULT" > "HOLLight.hollight.MOD_MULT"
"MOD_MOD_REFL" > "HOLLight.hollight.MOD_MOD_REFL"
"MOD_MOD" > "HOLLight.hollight.MOD_MOD"
- "MOD_LT" > "HOLLight.hollight.MOD_LT"
+ "MOD_LT" > "Divides.mod_less"
"MOD_LE" > "HOLLight.hollight.MOD_LE"
"MOD_EXP_MOD" > "HOLLight.hollight.MOD_EXP_MOD"
"MOD_EXISTS" > "HOLLight.hollight.MOD_EXISTS"
"MOD_EQ_0" > "HOLLight.hollight.MOD_EQ_0"
"MOD_EQ" > "HOLLight.hollight.MOD_EQ"
"MOD_ADD_MOD" > "HOLLight.hollight.MOD_ADD_MOD"
- "MOD_1" > "HOLLight.hollight.MOD_1"
- "MOD_0" > "HOLLight.hollight.MOD_0"
"MK_REC_INJ" > "HOLLight.hollight.MK_REC_INJ"
"MINIMAL" > "HOLLight.hollight.MINIMAL"
- "MEM_def" > "HOLLight.hollight.MEM_def"
- "MEM_MAP" > "HOLLight.hollight.MEM_MAP"
- "MEM_LIST_OF_SET" > "HOLLight.hollight.MEM_LIST_OF_SET"
- "MEM_FILTER" > "HOLLight.hollight.MEM_FILTER"
- "MEM_EL" > "HOLLight.hollight.MEM_EL"
- "MEM_ASSOC" > "HOLLight.hollight.MEM_ASSOC"
- "MEM_APPEND" > "HOLLight.hollight.MEM_APPEND"
- "MEMBER_NOT_EMPTY" > "HOLLight.hollight.MEMBER_NOT_EMPTY"
+ "MEM_MAP" > "HOLLightList.MEM_MAP"
+ "MEM_FILTER" > "HOLLightList.MEM_FILTER"
+ "MEM_EXISTS_EL" > "HOLLightList.MEM_EXISTS_EL"
+ "MEM_EL" > "List.nth_mem"
+ "MEM_APPEND" > "HOLLightList.MEM_APPEND"
+ "MEMBER_NOT_EMPTY" > "Set.ex_in_conv"
"MEASURE_LE" > "HOLLight.hollight.MEASURE_LE"
- "MAP_o" > "HOLLight.hollight.MAP_o"
- "MAP_def" > "HOLLight.hollight.MAP_def"
- "MAP_SND_ZIP" > "HOLLight.hollight.MAP_SND_ZIP"
- "MAP_FST_ZIP" > "HOLLight.hollight.MAP_FST_ZIP"
- "MAP_EQ_DEGEN" > "HOLLight.hollight.MAP_EQ_DEGEN"
- "MAP_EQ_ALL2" > "HOLLight.hollight.MAP_EQ_ALL2"
- "MAP_EQ" > "HOLLight.hollight.MAP_EQ"
- "MAP_APPEND" > "HOLLight.hollight.MAP_APPEND"
- "MAP2_def" > "HOLLight.hollight.MAP2_def"
- "MAP2" > "HOLLight.hollight.MAP2"
- "LT_TRANS" > "HOLLight.hollight.LT_TRANS"
- "LT_SUC_LE" > "HOLLight.hollight.LT_SUC_LE"
- "LT_SUC" > "HOLLight.hollight.LT_SUC"
- "LT_REFL" > "HOLLight.hollight.LT_REFL"
- "LT_NZ" > "HOLLight.hollight.LT_NZ"
+ "MATCH_SEQPATTERN" > "HOLLight.hollight.MATCH_SEQPATTERN"
+ "MAP_o" > "List.map.compositionality"
+ "MAP_SND_ZIP" > "List.map_snd_zip"
+ "MAP_ID" > "List.map_ident"
+ "MAP_I" > "List.map.id"
+ "MAP_FST_ZIP" > "List.map_fst_zip"
+ "MAP_EQ_NIL" > "List.map_is_Nil_conv"
+ "MAP_EQ_DEGEN" > "HOLLightList.MAP_EQ_DEGEN"
+ "MAP_EQ_ALL2" > "HOLLightList.MAP_EQ_ALL2"
+ "MAP_EQ" > "HOLLightList.MAP_EQ"
+ "MAP_APPEND" > "List.map_append"
+ "MAP2" > "HOLLightList.MAP2"
+ "LT_TRANS" > "Orderings.order_less_trans"
+ "LT_SUC_LE" > "Nat.le_simps_2"
+ "LT_SUC" > "Nat.Suc_less_eq"
+ "LT_REFL" > "Nat.less_not_refl"
+ "LT_NZ" > "Nat.neq0_conv"
"LT_MULT_RCANCEL" > "HOLLight.hollight.LT_MULT_RCANCEL"
"LT_MULT_LCANCEL" > "HOLLight.hollight.LT_MULT_LCANCEL"
"LT_MULT2" > "HOLLight.hollight.LT_MULT2"
- "LT_MULT" > "HOLLight.hollight.LT_MULT"
+ "LT_MULT" > "Nat.nat_0_less_mult_iff"
"LT_LMULT" > "HOLLight.hollight.LT_LMULT"
- "LT_LE" > "HOLLight.hollight.LT_LE"
- "LT_IMP_LE" > "HOLLight.hollight.LT_IMP_LE"
+ "LT_LE" > "Nat.nat_less_le"
+ "LT_IMP_LE" > "FunDef.termination_basic_simps_5"
"LT_EXP" > "HOLLight.hollight.LT_EXP"
"LT_EXISTS" > "HOLLight.hollight.LT_EXISTS"
"LT_CASES" > "HOLLight.hollight.LT_CASES"
"LT_ANTISYM" > "HOLLight.hollight.LT_ANTISYM"
- "LT_ADD_RCANCEL" > "HOLLight.hollight.LT_ADD_RCANCEL"
- "LT_ADD_LCANCEL" > "HOLLight.hollight.LT_ADD_LCANCEL"
+ "LT_ADD_RCANCEL" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_cancel_right"
+ "LT_ADD_LCANCEL" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_cancel_left"
"LT_ADDR" > "HOLLight.hollight.LT_ADDR"
- "LT_ADD2" > "HOLLight.hollight.LT_ADD2"
+ "LT_ADD2" > "Groups.add_mono_thms_linordered_field_5"
"LT_ADD" > "HOLLight.hollight.LT_ADD"
- "LT_0" > "HOLLight.hollight.LT_0"
- "LTE_TRANS" > "HOLLight.hollight.LTE_TRANS"
+ "LT_0" > "Nat.zero_less_Suc"
+ "LTE_TRANS" > "Orderings.order_less_le_trans"
"LTE_CASES" > "HOLLight.hollight.LTE_CASES"
"LTE_ANTISYM" > "HOLLight.hollight.LTE_ANTISYM"
- "LTE_ADD2" > "HOLLight.hollight.LTE_ADD2"
- "LIST_OF_SET_PROPERTIES" > "HOLLight.hollight.LIST_OF_SET_PROPERTIES"
- "LE_TRANS" > "HOLLight.hollight.LE_TRANS"
- "LE_SUC_LT" > "HOLLight.hollight.LE_SUC_LT"
- "LE_SUC" > "HOLLight.hollight.LE_SUC"
- "LE_SQUARE_REFL" > "HOLLight.hollight.LE_SQUARE_REFL"
- "LE_REFL" > "HOLLight.hollight.LE_REFL"
+ "LTE_ADD2" > "Groups.add_mono_thms_linordered_field_3"
+ "LE_TRANS" > "Nat.le_trans"
+ "LE_SUC_LT" > "Nat.Suc_le_eq"
+ "LE_SUC" > "Nat.Suc_le_mono"
+ "LE_SQUARE_REFL" > "Nat.le_square"
+ "LE_REFL" > "Nat.le_refl"
"LE_RDIV_EQ" > "HOLLight.hollight.LE_RDIV_EQ"
"LE_MULT_RCANCEL" > "HOLLight.hollight.LE_MULT_RCANCEL"
"LE_MULT_LCANCEL" > "HOLLight.hollight.LE_MULT_LCANCEL"
- "LE_MULT2" > "HOLLight.hollight.LE_MULT2"
- "LE_LT" > "HOLLight.hollight.LE_LT"
+ "LE_MULT2" > "Nat.mult_le_mono"
+ "LE_LT" > "Nat.le_eq_less_or_eq"
"LE_LDIV_EQ" > "HOLLight.hollight.LE_LDIV_EQ"
"LE_LDIV" > "HOLLight.hollight.LE_LDIV"
"LE_EXP" > "HOLLight.hollight.LE_EXP"
- "LE_EXISTS" > "HOLLight.hollight.LE_EXISTS"
- "LE_CASES" > "HOLLight.hollight.LE_CASES"
- "LE_ANTISYM" > "HOLLight.hollight.LE_ANTISYM"
- "LE_ADD_RCANCEL" > "HOLLight.hollight.LE_ADD_RCANCEL"
- "LE_ADD_LCANCEL" > "HOLLight.hollight.LE_ADD_LCANCEL"
- "LE_ADDR" > "HOLLight.hollight.LE_ADDR"
- "LE_ADD2" > "HOLLight.hollight.LE_ADD2"
- "LE_ADD" > "HOLLight.hollight.LE_ADD"
- "LE_0" > "HOLLight.hollight.LE_0"
- "LET_TRANS" > "HOLLight.hollight.LET_TRANS"
+ "LE_EXISTS" > "Nat.le_iff_add"
+ "LE_CASES" > "Nat.nat_le_linear"
+ "LE_C" > "HOLLight.hollight.LE_C"
+ "LE_ANTISYM" > "Orderings.order_class.eq_iff"
+ "LE_ADD_RCANCEL" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_right"
+ "LE_ADD_LCANCEL" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_left"
+ "LE_ADDR" > "Nat.le_add2"
+ "LE_ADD2" > "Groups.add_mono_thms_linordered_semiring_1"
+ "LE_ADD" > "Nat.le_add1"
+ "LE_1" > "HOLLight.hollight.LE_1"
+ "LE_0" > "Nat.le0"
+ "LET_TRANS" > "Orderings.order_le_less_trans"
"LET_END_def" > "HOLLight.hollight.LET_END_def"
- "LET_CASES" > "HOLLight.hollight.LET_CASES"
+ "LET_CASES" > "Orderings.linorder_class.le_less_linear"
"LET_ANTISYM" > "HOLLight.hollight.LET_ANTISYM"
- "LET_ADD2" > "HOLLight.hollight.LET_ADD2"
- "LENGTH_def" > "HOLLight.hollight.LENGTH_def"
- "LENGTH_REPLICATE" > "HOLLight.hollight.LENGTH_REPLICATE"
- "LENGTH_MAP2" > "HOLLight.hollight.LENGTH_MAP2"
- "LENGTH_MAP" > "HOLLight.hollight.LENGTH_MAP"
- "LENGTH_LIST_OF_SET" > "HOLLight.hollight.LENGTH_LIST_OF_SET"
- "LENGTH_EQ_NIL" > "HOLLight.hollight.LENGTH_EQ_NIL"
- "LENGTH_EQ_CONS" > "HOLLight.hollight.LENGTH_EQ_CONS"
- "LENGTH_APPEND" > "HOLLight.hollight.LENGTH_APPEND"
- "LEFT_SUB_DISTRIB" > "Nat.nat_distrib_4"
+ "LET_ADD2" > "Groups.add_mono_thms_linordered_field_4"
+ "LENGTH_TL" > "HOLLightList.LENGTH_TL"
+ "LENGTH_REPLICATE" > "List.length_replicate"
+ "LENGTH_MAP2" > "HOLLightList.LENGTH_MAP2"
+ "LENGTH_MAP" > "List.length_map"
+ "LENGTH_EQ_NIL" > "List.length_0_conv"
+ "LENGTH_EQ_CONS" > "List.length_Suc_conv"
+ "LENGTH_APPEND" > "List.length_append"
+ "LEFT_SUB_DISTRIB" > "Nat.diff_mult_distrib2"
"LEFT_OR_FORALL_THM" > "HOL.all_simps_3"
"LEFT_OR_EXISTS_THM" > "HOL.ex_simps_3"
- "LEFT_OR_DISTRIB" > "HOL.conj_disj_distribL"
- "LEFT_IMP_FORALL_THM" > "HOL.imp_all"
- "LEFT_IMP_EXISTS_THM" > "HOL.imp_ex"
+ "LEFT_OR_DISTRIB" > "Groebner_Basis.dnf_1"
+ "LEFT_IMP_FORALL_THM" > "HOL.ex_simps_5"
+ "LEFT_IMP_EXISTS_THM" > "HOL.all_simps_5"
"LEFT_FORALL_OR_THM" > "HOL.all_simps_3"
- "LEFT_FORALL_IMP_THM" > "HOL.imp_ex"
- "LEFT_EXISTS_IMP_THM" > "HOL.imp_all"
+ "LEFT_FORALL_IMP_THM" > "HOL.all_simps_5"
+ "LEFT_EXISTS_IMP_THM" > "HOL.ex_simps_5"
"LEFT_EXISTS_AND_THM" > "HOL.ex_simps_1"
"LEFT_AND_FORALL_THM" > "HOL.all_simps_1"
"LEFT_AND_EXISTS_THM" > "HOL.ex_simps_1"
- "LEFT_ADD_DISTRIB" > "Nat.nat_distrib_2"
- "LAST_def" > "HOLLight.hollight.LAST_def"
- "LAST_CLAUSES" > "HOLLight.hollight.LAST_CLAUSES"
+ "LEFT_ADD_DISTRIB" > "Fields.linordered_field_class.sign_simps_25"
+ "LAST_EL" > "List.last_conv_nth"
+ "LAST_CLAUSES" > "HOLLightList.LAST_CLAUSES"
+ "LAST_APPEND" > "List.last_append"
"LAMBDA_UNIQUE" > "HOLLight.hollight.LAMBDA_UNIQUE"
+ "LAMBDA_PAIR_THM" > "HOLLight.hollight.LAMBDA_PAIR_THM"
"LAMBDA_ETA" > "HOLLight.hollight.LAMBDA_ETA"
"LAMBDA_BETA" > "HOLLight.hollight.LAMBDA_BETA"
- "I_THM" > "Fun.id_apply"
+ "I_THM" > "HOL.refl"
"I_O_ID" > "HOLLight.hollight.I_O_ID"
"ITSET_def" > "HOLLight.hollight.ITSET_def"
"ITSET_EQ" > "HOLLight.hollight.ITSET_EQ"
- "ITLIST_def" > "HOLLight.hollight.ITLIST_def"
- "ITLIST_EXTRA" > "HOLLight.hollight.ITLIST_EXTRA"
- "ITLIST2_def" > "HOLLight.hollight.ITLIST2_def"
- "ITLIST2" > "HOLLight.hollight.ITLIST2"
+ "ITLIST_EXTRA" > "HOLLightList.ITLIST_EXTRA"
+ "ITLIST_APPEND" > "List.foldr_append"
+ "ITLIST2" > "HOLLightList.ITLIST2"
+ "ITERATE_UNION_NONZERO" > "HOLLight.hollight.ITERATE_UNION_NONZERO"
"ITERATE_UNION_GEN" > "HOLLight.hollight.ITERATE_UNION_GEN"
"ITERATE_UNION" > "HOLLight.hollight.ITERATE_UNION"
"ITERATE_SUPPORT" > "HOLLight.hollight.ITERATE_SUPPORT"
+ "ITERATE_SUPERSET" > "HOLLight.hollight.ITERATE_SUPERSET"
"ITERATE_SING" > "HOLLight.hollight.ITERATE_SING"
"ITERATE_RELATED" > "HOLLight.hollight.ITERATE_RELATED"
+ "ITERATE_PAIR" > "HOLLight.hollight.ITERATE_PAIR"
+ "ITERATE_OP_GEN" > "HOLLight.hollight.ITERATE_OP_GEN"
+ "ITERATE_OP" > "HOLLight.hollight.ITERATE_OP"
"ITERATE_ITERATE_PRODUCT" > "HOLLight.hollight.ITERATE_ITERATE_PRODUCT"
+ "ITERATE_INJECTION" > "HOLLight.hollight.ITERATE_INJECTION"
+ "ITERATE_INCL_EXCL" > "HOLLight.hollight.ITERATE_INCL_EXCL"
+ "ITERATE_IMAGE_NONZERO" > "HOLLight.hollight.ITERATE_IMAGE_NONZERO"
"ITERATE_IMAGE" > "HOLLight.hollight.ITERATE_IMAGE"
+ "ITERATE_EXPAND_CASES" > "HOLLight.hollight.ITERATE_EXPAND_CASES"
"ITERATE_EQ_NEUTRAL" > "HOLLight.hollight.ITERATE_EQ_NEUTRAL"
+ "ITERATE_EQ_GENERAL_INVERSES" > "HOLLight.hollight.ITERATE_EQ_GENERAL_INVERSES"
"ITERATE_EQ_GENERAL" > "HOLLight.hollight.ITERATE_EQ_GENERAL"
"ITERATE_EQ" > "HOLLight.hollight.ITERATE_EQ"
"ITERATE_DIFF_GEN" > "HOLLight.hollight.ITERATE_DIFF_GEN"
"ITERATE_DIFF" > "HOLLight.hollight.ITERATE_DIFF"
"ITERATE_DELTA" > "HOLLight.hollight.ITERATE_DELTA"
- "ITERATE_CLOSED_GEN" > "HOLLight.hollight.ITERATE_CLOSED_GEN"
+ "ITERATE_DELETE" > "HOLLight.hollight.ITERATE_DELETE"
"ITERATE_CLOSED" > "HOLLight.hollight.ITERATE_CLOSED"
+ "ITERATE_CLAUSES_NUMSEG" > "HOLLight.hollight.ITERATE_CLAUSES_NUMSEG"
"ITERATE_CLAUSES_GEN" > "HOLLight.hollight.ITERATE_CLAUSES_GEN"
"ITERATE_CLAUSES" > "HOLLight.hollight.ITERATE_CLAUSES"
+ "ITERATE_CASES" > "HOLLight.hollight.ITERATE_CASES"
"ITERATE_BIJECTION" > "HOLLight.hollight.ITERATE_BIJECTION"
"ISO_def" > "HOLLight.hollight.ISO_def"
"ISO_USAGE" > "HOLLight.hollight.ISO_USAGE"
"ISO_REFL" > "HOLLight.hollight.ISO_REFL"
"ISO_FUN" > "HOLLight.hollight.ISO_FUN"
- "IN_def" > "HOLLight.hollight.IN_def"
- "IN_UNIV" > "HOLLight.hollight.IN_UNIV"
+ "IN_UNIV" > "Set.UNIV_I"
"IN_UNIONS" > "HOLLight.hollight.IN_UNIONS"
- "IN_UNION" > "HOLLight.hollight.IN_UNION"
+ "IN_UNION" > "Complete_Lattice.mem_simps_3"
"IN_SUPPORT" > "HOLLight.hollight.IN_SUPPORT"
- "IN_SING" > "HOLLight.hollight.IN_SING"
- "IN_SET_OF_LIST" > "HOLLight.hollight.IN_SET_OF_LIST"
+ "IN_SING" > "Set.singleton_iff"
"IN_REST" > "HOLLight.hollight.IN_REST"
- "IN_NUMSEG" > "HOLLight.hollight.IN_NUMSEG"
+ "IN_NUMSEG_0" > "HOLLight.hollight.IN_NUMSEG_0"
+ "IN_NUMSEG" > "SetInterval.ord_class.atLeastAtMost_iff"
"IN_INTERS" > "HOLLight.hollight.IN_INTERS"
- "IN_INTER" > "HOLLight.hollight.IN_INTER"
- "IN_INSERT" > "HOLLight.hollight.IN_INSERT"
+ "IN_INTER" > "Complete_Lattice.mem_simps_4"
+ "IN_INSERT" > "Complete_Lattice.mem_simps_1"
"IN_IMAGE" > "HOLLight.hollight.IN_IMAGE"
"IN_ELIM_THM" > "HOLLight.hollight.IN_ELIM_THM"
"IN_ELIM_PAIR_THM" > "HOLLight.hollight.IN_ELIM_PAIR_THM"
"IN_DISJOINT" > "HOLLight.hollight.IN_DISJOINT"
- "IN_DIFF" > "HOLLight.hollight.IN_DIFF"
+ "IN_DIFF" > "Complete_Lattice.mem_simps_6"
"IN_DELETE_EQ" > "HOLLight.hollight.IN_DELETE_EQ"
"IN_DELETE" > "HOLLight.hollight.IN_DELETE"
+ "IN_CROSS" > "HOLLight.hollight.IN_CROSS"
+ "INT_WOP" > "HOLLight.hollight.INT_WOP"
"INT_POW" > "HOLLight.hollight.INT_POW"
+ "INT_OF_NUM_OF_INT" > "HOLLight.hollight.INT_OF_NUM_OF_INT"
"INT_LT_DISCRETE" > "HOLLight.hollight.INT_LT_DISCRETE"
"INT_LT" > "HOLLight.hollight.INT_LT"
+ "INT_INTEGRAL" > "HOLLight.hollight.INT_INTEGRAL"
"INT_IMAGE" > "HOLLight.hollight.INT_IMAGE"
"INT_GT_DISCRETE" > "HOLLight.hollight.INT_GT_DISCRETE"
"INT_GT" > "HOLLight.hollight.INT_GT"
"INT_GE" > "HOLLight.hollight.INT_GE"
+ "INT_GCD_EXISTS_POS" > "HOLLight.hollight.INT_GCD_EXISTS_POS"
+ "INT_GCD_EXISTS" > "HOLLight.hollight.INT_GCD_EXISTS"
"INT_FORALL_POS" > "HOLLight.hollight.INT_FORALL_POS"
+ "INT_FORALL_ABS" > "HOLLight.hollight.INT_FORALL_ABS"
+ "INT_EXISTS_POS" > "HOLLight.hollight.INT_EXISTS_POS"
+ "INT_EXISTS_ABS" > "HOLLight.hollight.INT_EXISTS_ABS"
+ "INT_DIVMOD_UNIQ" > "HOLLight.hollight.INT_DIVMOD_UNIQ"
+ "INT_DIVMOD_EXIST_0" > "HOLLight.hollight.INT_DIVMOD_EXIST_0"
+ "INT_DIVISION" > "HOLLight.hollight.INT_DIVISION"
"INT_ARCH" > "HOLLight.hollight.INT_ARCH"
"INT_ABS_MUL_1" > "HOLLight.hollight.INT_ABS_MUL_1"
"INT_ABS" > "HOLLight.hollight.INT_ABS"
- "INTER_def" > "HOLLight.hollight.INTER_def"
"INTER_UNIV" > "HOLLight.hollight.INTER_UNIV"
+ "INTER_UNIONS" > "HOLLight.hollight.INTER_UNIONS"
"INTER_SUBSET" > "HOLLight.hollight.INTER_SUBSET"
- "INTER_OVER_UNION" > "HOLLight.hollight.INTER_OVER_UNION"
- "INTER_IDEMPOT" > "HOLLight.hollight.INTER_IDEMPOT"
+ "INTER_OVER_UNION" > "Lattices.distrib_lattice_class.distrib_1"
+ "INTER_IDEMPOT" > "Big_Operators.lattice_class.Inf_fin.idem"
"INTER_EMPTY" > "HOLLight.hollight.INTER_EMPTY"
- "INTER_COMM" > "HOLLight.hollight.INTER_COMM"
- "INTER_ASSOC" > "HOLLight.hollight.INTER_ASSOC"
+ "INTER_COMM" > "Lattices.lattice_class.inf_sup_aci_1"
+ "INTER_ASSOC" > "Lattices.lattice_class.inf_sup_aci_2"
"INTER_ACI" > "HOLLight.hollight.INTER_ACI"
- "INTERS_def" > "HOLLight.hollight.INTERS_def"
- "INTERS_INSERT" > "HOLLight.hollight.INTERS_INSERT"
- "INTERS_2" > "HOLLight.hollight.INTERS_2"
- "INTERS_1" > "HOLLight.hollight.INTERS_1"
- "INTERS_0" > "HOLLight.hollight.INTERS_0"
- "INSERT_def" > "HOLLight.hollight.INSERT_def"
+ "INTERS_UNIONS" > "HOLLight.hollight.INTERS_UNIONS"
+ "INTERS_INSERT" > "Complete_Lattice.Inter_insert"
+ "INTERS_IMAGE" > "HOLLight.hollight.INTERS_IMAGE"
+ "INTERS_GSPEC" > "HOLLight.hollight.INTERS_GSPEC"
+ "INTERS_2" > "Complete_Lattice.Int_eq_Inter"
+ "INTERS_1" > "Complete_Lattice.complete_lattice_class.Inf_singleton"
+ "INTERS_0" > "Complete_Lattice.Inter_empty"
"INSERT_UNIV" > "HOLLight.hollight.INSERT_UNIV"
- "INSERT_UNION_EQ" > "HOLLight.hollight.INSERT_UNION_EQ"
+ "INSERT_UNION_EQ" > "Set.Un_insert_left"
"INSERT_UNION" > "HOLLight.hollight.INSERT_UNION"
- "INSERT_SUBSET" > "HOLLight.hollight.INSERT_SUBSET"
- "INSERT_INTER" > "HOLLight.hollight.INSERT_INTER"
- "INSERT_INSERT" > "HOLLight.hollight.INSERT_INSERT"
- "INSERT_DIFF" > "HOLLight.hollight.INSERT_DIFF"
- "INSERT_DELETE" > "HOLLight.hollight.INSERT_DELETE"
- "INSERT_COMM" > "HOLLight.hollight.INSERT_COMM"
+ "INSERT_SUBSET" > "Set.insert_subset"
+ "INSERT_INTER" > "Set.Int_insert_left"
+ "INSERT_INSERT" > "Set.insert_absorb2"
+ "INSERT_DIFF" > "Set.insert_Diff_if"
+ "INSERT_DELETE" > "Set.insert_Diff"
+ "INSERT_COMM" > "Set.insert_commute"
"INSERT_AC" > "HOLLight.hollight.INSERT_AC"
"INSERT" > "HOLLight.hollight.INSERT"
"INJ_def" > "HOLLight.hollight.INJ_def"
@@ -1339,26 +1619,36 @@
"INJF_def" > "HOLLight.hollight.INJF_def"
"INJF_INJ" > "HOLLight.hollight.INJF_INJ"
"INJECTIVE_ON_LEFT_INVERSE" > "HOLLight.hollight.INJECTIVE_ON_LEFT_INVERSE"
+ "INJECTIVE_ON_IMAGE" > "HOLLight.hollight.INJECTIVE_ON_IMAGE"
+ "INJECTIVE_MAP" > "HOLLightList.INJECTIVE_MAP"
"INJECTIVE_LEFT_INVERSE" > "HOLLight.hollight.INJECTIVE_LEFT_INVERSE"
+ "INJECTIVE_IMAGE" > "HOLLight.hollight.INJECTIVE_IMAGE"
"INJA_def" > "HOLLight.hollight.INJA_def"
"INJA_INJ" > "HOLLight.hollight.INJA_INJ"
- "INFINITE_def" > "HOLLight.hollight.INFINITE_def"
- "INFINITE_NONEMPTY" > "HOLLight.hollight.INFINITE_NONEMPTY"
+ "INFINITE_NONEMPTY" > "Infinite_Set.infinite_imp_nonempty"
"INFINITE_IMAGE_INJ" > "HOLLight.hollight.INFINITE_IMAGE_INJ"
- "INFINITE_DIFF_FINITE" > "HOLLight.hollight.INFINITE_DIFF_FINITE"
+ "INFINITE_DIFF_FINITE" > "Infinite_Set.Diff_infinite_finite"
+ "IND_SUC_def" > "HOLLight.hollight.IND_SUC_def"
+ "IND_SUC_0_EXISTS" > "HOLLight.hollight.IND_SUC_0_EXISTS"
+ "IND_0_def" > "HOLLight.hollight.IND_0_def"
"IMP_EQ_CLAUSE" > "HOLLight.hollight.IMP_EQ_CLAUSE"
+ "IMP_CONJ_ALT" > "HOLLight.hollight.IMP_CONJ_ALT"
"IMP_CONJ" > "HOL.imp_conjL"
"IMP_CLAUSES" > "HOLLight.hollight.IMP_CLAUSES"
- "IMAGE_o" > "HOLLight.hollight.IMAGE_o"
- "IMAGE_def" > "HOLLight.hollight.IMAGE_def"
- "IMAGE_UNION" > "HOLLight.hollight.IMAGE_UNION"
- "IMAGE_SUBSET" > "HOLLight.hollight.IMAGE_SUBSET"
+ "IMAGE_o" > "Fun.image_compose"
+ "IMAGE_UNIONS" > "HOLLight.hollight.IMAGE_UNIONS"
+ "IMAGE_UNION" > "Set.image_Un"
+ "IMAGE_SUBSET" > "Set.image_mono"
+ "IMAGE_INTER_INJ" > "HOLLight.hollight.IMAGE_INTER_INJ"
+ "IMAGE_INJECTIVE_IMAGE_OF_SUBSET" > "HOLLight.hollight.IMAGE_INJECTIVE_IMAGE_OF_SUBSET"
"IMAGE_IMP_INJECTIVE_GEN" > "HOLLight.hollight.IMAGE_IMP_INJECTIVE_GEN"
"IMAGE_IMP_INJECTIVE" > "HOLLight.hollight.IMAGE_IMP_INJECTIVE"
- "IMAGE_EQ_EMPTY" > "HOLLight.hollight.IMAGE_EQ_EMPTY"
+ "IMAGE_ID" > "Fun.image_ident"
+ "IMAGE_I" > "Fun.image_id"
+ "IMAGE_EQ_EMPTY" > "Set.image_is_empty"
"IMAGE_DIFF_INJ" > "HOLLight.hollight.IMAGE_DIFF_INJ"
"IMAGE_DELETE_INJ" > "HOLLight.hollight.IMAGE_DELETE_INJ"
- "IMAGE_CONST" > "HOLLight.hollight.IMAGE_CONST"
+ "IMAGE_CONST" > "Set.image_constant_conv"
"IMAGE_CLAUSES" > "HOLLight.hollight.IMAGE_CLAUSES"
"HREAL_MUL_RZERO" > "HOLLight.hollight.HREAL_MUL_RZERO"
"HREAL_MUL_LZERO" > "HOLLight.hollight.HREAL_MUL_LZERO"
@@ -1372,7 +1662,7 @@
"HREAL_ADD_RID" > "HOLLight.hollight.HREAL_ADD_RID"
"HREAL_ADD_RDISTRIB" > "HOLLight.hollight.HREAL_ADD_RDISTRIB"
"HREAL_ADD_AC" > "HOLLight.hollight.HREAL_ADD_AC"
- "HD_def" > "HOLLight.hollight.HD_def"
+ "HD_APPEND" > "List.hd_append"
"HAS_SIZE_def" > "HOLLight.hollight.HAS_SIZE_def"
"HAS_SIZE_UNIONS" > "HOLLight.hollight.HAS_SIZE_UNIONS"
"HAS_SIZE_UNION" > "HOLLight.hollight.HAS_SIZE_UNION"
@@ -1385,31 +1675,40 @@
"HAS_SIZE_NUMSEG_1" > "HOLLight.hollight.HAS_SIZE_NUMSEG_1"
"HAS_SIZE_NUMSEG" > "HOLLight.hollight.HAS_SIZE_NUMSEG"
"HAS_SIZE_INDEX" > "HOLLight.hollight.HAS_SIZE_INDEX"
+ "HAS_SIZE_IMAGE_INJ_EQ" > "HOLLight.hollight.HAS_SIZE_IMAGE_INJ_EQ"
"HAS_SIZE_IMAGE_INJ" > "HOLLight.hollight.HAS_SIZE_IMAGE_INJ"
"HAS_SIZE_FUNSPACE" > "HOLLight.hollight.HAS_SIZE_FUNSPACE"
"HAS_SIZE_FINITE_IMAGE" > "HOLLight.hollight.HAS_SIZE_FINITE_IMAGE"
+ "HAS_SIZE_DIFF" > "HOLLight.hollight.HAS_SIZE_DIFF"
+ "HAS_SIZE_CROSS" > "HOLLight.hollight.HAS_SIZE_CROSS"
"HAS_SIZE_CLAUSES" > "HOLLight.hollight.HAS_SIZE_CLAUSES"
"HAS_SIZE_CARD" > "HOLLight.hollight.HAS_SIZE_CARD"
+ "HAS_SIZE_1" > "HOLLight.hollight.HAS_SIZE_1"
"HAS_SIZE_0" > "HOLLight.hollight.HAS_SIZE_0"
- "GSPEC_def" > "HOLLight.hollight.GSPEC_def"
- "GEQ_def" > "HOLLight.hollight.GEQ_def"
- "GABS_def" > "HOLLight.hollight.GABS_def"
+ "GE_C" > "HOLLight.hollight.GE_C"
+ "FUN_IN_IMAGE" > "Set.imageI"
"FUN_EQ_THM" > "Fun.fun_eq_iff"
"FUNCTION_FACTORS_RIGHT" > "HOLLight.hollight.FUNCTION_FACTORS_RIGHT"
"FUNCTION_FACTORS_LEFT" > "HOLLight.hollight.FUNCTION_FACTORS_LEFT"
- "FSTCART_PASTECART" > "HOLLight.hollight.FSTCART_PASTECART"
"FST" > "Product_Type.fst_conv"
+ "FORALL_UNWIND_THM2" > "HOL.simp_thms_41"
+ "FORALL_UNWIND_THM1" > "HOL.simp_thms_42"
+ "FORALL_UNCURRY" > "HOLLight.hollight.FORALL_UNCURRY"
+ "FORALL_TRIPLED_THM" > "HOLLight.hollight.FORALL_TRIPLED_THM"
+ "FORALL_SUBSET_IMAGE" > "HOLLight.hollight.FORALL_SUBSET_IMAGE"
"FORALL_SIMP" > "HOL.simp_thms_35"
- "FORALL_PASTECART" > "HOLLight.hollight.FORALL_PASTECART"
"FORALL_PAIR_THM" > "Product_Type.split_paired_All"
- "FORALL_NOT_THM" > "Inductive.basic_monos_16"
+ "FORALL_PAIRED_THM" > "HOLLight.hollight.FORALL_PAIRED_THM"
+ "FORALL_NOT_THM" > "HOL.not_ex"
"FORALL_IN_UNIONS" > "HOLLight.hollight.FORALL_IN_UNIONS"
+ "FORALL_IN_INSERT" > "HOLLight.hollight.FORALL_IN_INSERT"
"FORALL_IN_IMAGE" > "HOLLight.hollight.FORALL_IN_IMAGE"
+ "FORALL_IN_GSPEC" > "HOLLight.hollight.FORALL_IN_GSPEC"
"FORALL_IN_CLAUSES" > "HOLLight.hollight.FORALL_IN_CLAUSES"
"FORALL_FINITE_INDEX" > "HOLLight.hollight.FORALL_FINITE_INDEX"
"FORALL_BOOL_THM" > "Set.all_bool_eq"
"FORALL_AND_THM" > "HOL.all_conj_distrib"
- "FORALL_ALL" > "HOLLight.hollight.FORALL_ALL"
+ "FORALL_ALL" > "HOLLightList.FORALL_ALL"
"FNIL_def" > "HOLLight.hollight.FNIL_def"
"FINREC_def" > "HOLLight.hollight.FINREC_def"
"FINREC_UNIQUE_LEMMA" > "HOLLight.hollight.FINREC_UNIQUE_LEMMA"
@@ -1418,139 +1717,153 @@
"FINREC_FUN" > "HOLLight.hollight.FINREC_FUN"
"FINREC_EXISTS_LEMMA" > "HOLLight.hollight.FINREC_EXISTS_LEMMA"
"FINREC_1_LEMMA" > "HOLLight.hollight.FINREC_1_LEMMA"
- "FINITE_def" > "HOLLight.hollight.FINITE_def"
- "FINITE_UNION_IMP" > "HOLLight.hollight.FINITE_UNION_IMP"
+ "FINITE_UNION_IMP" > "Finite_Set.finite_UnI"
"FINITE_UNIONS" > "HOLLight.hollight.FINITE_UNIONS"
- "FINITE_UNION" > "HOLLight.hollight.FINITE_UNION"
+ "FINITE_UNION" > "Finite_Set.finite_Un"
"FINITE_SUPPORT_DELTA" > "HOLLight.hollight.FINITE_SUPPORT_DELTA"
"FINITE_SUPPORT" > "HOLLight.hollight.FINITE_SUPPORT"
+ "FINITE_SUM_IMAGE" > "HOLLight.hollight.FINITE_SUM_IMAGE"
"FINITE_SUBSET_IMAGE_IMP" > "HOLLight.hollight.FINITE_SUBSET_IMAGE_IMP"
"FINITE_SUBSET_IMAGE" > "HOLLight.hollight.FINITE_SUBSET_IMAGE"
- "FINITE_SUBSETS" > "HOLLight.hollight.FINITE_SUBSETS"
- "FINITE_SUBSET" > "HOLLight.hollight.FINITE_SUBSET"
- "FINITE_SET_OF_LIST" > "HOLLight.hollight.FINITE_SET_OF_LIST"
+ "FINITE_SUBSET" > "Finite_Set.finite_subset"
+ "FINITE_SING" > "HOLLight.hollight.FINITE_SING"
"FINITE_RESTRICT" > "HOLLight.hollight.FINITE_RESTRICT"
"FINITE_RECURSION_DELETE" > "HOLLight.hollight.FINITE_RECURSION_DELETE"
"FINITE_RECURSION" > "HOLLight.hollight.FINITE_RECURSION"
+ "FINITE_REAL_INTERVAL" > "HOLLight.hollight.FINITE_REAL_INTERVAL"
"FINITE_PRODUCT_DEPENDENT" > "HOLLight.hollight.FINITE_PRODUCT_DEPENDENT"
"FINITE_PRODUCT" > "HOLLight.hollight.FINITE_PRODUCT"
"FINITE_POWERSET" > "HOLLight.hollight.FINITE_POWERSET"
"FINITE_NUMSEG_LT" > "HOLLight.hollight.FINITE_NUMSEG_LT"
"FINITE_NUMSEG_LE" > "HOLLight.hollight.FINITE_NUMSEG_LE"
- "FINITE_NUMSEG" > "HOLLight.hollight.FINITE_NUMSEG"
- "FINITE_INTER" > "HOLLight.hollight.FINITE_INTER"
- "FINITE_INSERT" > "HOLLight.hollight.FINITE_INSERT"
- "FINITE_INDUCT_STRONG" > "HOLLight.hollight.FINITE_INDUCT_STRONG"
- "FINITE_INDEX_WORKS_FINITE" > "HOLLight.hollight.FINITE_INDEX_WORKS_FINITE"
+ "FINITE_NUMSEG" > "SetInterval.finite_atLeastAtMost"
+ "FINITE_INTSEG" > "HOLLight.hollight.FINITE_INTSEG"
+ "FINITE_INTER" > "Finite_Set.finite_Int"
+ "FINITE_INSERT" > "Finite_Set.finite_insert"
+ "FINITE_INDUCT_STRONG" > "Finite_Set.finite_induct"
+ "FINITE_INDUCT_DELETE" > "HOLLight.hollight.FINITE_INDUCT_DELETE"
"FINITE_INDEX_WORKS" > "HOLLight.hollight.FINITE_INDEX_WORKS"
"FINITE_INDEX_NUMSEG" > "HOLLight.hollight.FINITE_INDEX_NUMSEG"
"FINITE_INDEX_NUMBERS" > "HOLLight.hollight.FINITE_INDEX_NUMBERS"
+ "FINITE_INDEX_INRANGE" > "HOLLight.hollight.FINITE_INDEX_INRANGE"
"FINITE_INDEX_INJ" > "HOLLight.hollight.FINITE_INDEX_INJ"
"FINITE_IMAGE_INJ_GENERAL" > "HOLLight.hollight.FINITE_IMAGE_INJ_GENERAL"
+ "FINITE_IMAGE_INJ_EQ" > "HOLLight.hollight.FINITE_IMAGE_INJ_EQ"
"FINITE_IMAGE_INJ" > "HOLLight.hollight.FINITE_IMAGE_INJ"
"FINITE_IMAGE_IMAGE" > "HOLLight.hollight.FINITE_IMAGE_IMAGE"
"FINITE_IMAGE_EXPAND" > "HOLLight.hollight.FINITE_IMAGE_EXPAND"
- "FINITE_IMAGE" > "HOLLight.hollight.FINITE_IMAGE"
- "FINITE_HAS_SIZE_LEMMA" > "HOLLight.hollight.FINITE_HAS_SIZE_LEMMA"
+ "FINITE_IMAGE" > "Finite_Set.finite_imageI"
+ "FINITE_HAS_SIZE" > "HOLLight.hollight.FINITE_HAS_SIZE"
"FINITE_FUNSPACE" > "HOLLight.hollight.FINITE_FUNSPACE"
+ "FINITE_FINITE_UNIONS" > "HOLLight.hollight.FINITE_FINITE_UNIONS"
+ "FINITE_FINITE_PREIMAGE_GENERAL" > "HOLLight.hollight.FINITE_FINITE_PREIMAGE_GENERAL"
+ "FINITE_FINITE_PREIMAGE" > "HOLLight.hollight.FINITE_FINITE_PREIMAGE"
"FINITE_FINITE_IMAGE" > "HOLLight.hollight.FINITE_FINITE_IMAGE"
- "FINITE_DIFF" > "HOLLight.hollight.FINITE_DIFF"
+ "FINITE_EMPTY" > "Finite_Set.finite.emptyI"
+ "FINITE_DIFF" > "Finite_Set.finite_Diff"
"FINITE_DELETE_IMP" > "HOLLight.hollight.FINITE_DELETE_IMP"
"FINITE_DELETE" > "HOLLight.hollight.FINITE_DELETE"
- "FILTER_def" > "HOLLight.hollight.FILTER_def"
- "FILTER_MAP" > "HOLLight.hollight.FILTER_MAP"
- "FILTER_APPEND" > "HOLLight.hollight.FILTER_APPEND"
+ "FINITE_CROSS" > "HOLLight.hollight.FINITE_CROSS"
+ "FINITE_CART" > "HOLLight.hollight.FINITE_CART"
+ "FILTER_MAP" > "List.filter_map"
+ "FILTER_APPEND" > "List.filter_append"
"FCONS_def" > "HOLLight.hollight.FCONS_def"
"FCONS_UNDO" > "HOLLight.hollight.FCONS_UNDO"
- "FACT_def" > "HOLLight.hollight.FACT_def"
- "FACT_MONO" > "HOLLight.hollight.FACT_MONO"
- "FACT_LT" > "HOLLight.hollight.FACT_LT"
- "FACT_LE" > "HOLLight.hollight.FACT_LE"
- "EX_def" > "HOLLight.hollight.EX_def"
- "EX_MEM" > "HOLLight.hollight.EX_MEM"
- "EX_MAP" > "HOLLight.hollight.EX_MAP"
- "EX_IMP" > "HOLLight.hollight.EX_IMP"
- "EXTENSION" > "HOLLight.hollight.EXTENSION"
- "EXP_def" > "HOLLight.hollight.EXP_def"
- "EXP_ONE" > "HOLLight.hollight.EXP_ONE"
- "EXP_MULT" > "HOLLight.hollight.EXP_MULT"
+ "FACT_NZ" > "Fact.fact_nonzero_nat"
+ "FACT_MONO" > "Fact.fact_mono_nat"
+ "FACT_LT" > "Fact.fact_gt_zero_nat"
+ "FACT_LE" > "Fact.fact_ge_one_nat"
+ "EX_MEM" > "HOLLightList.EX_MEM"
+ "EX_IMP" > "HOLLightList.EX_IMP"
+ "EXTENSION" > "Set.set_eq_iff"
+ "EXP_ZERO" > "Power.power_0_left"
+ "EXP_ONE" > "Power.monoid_mult_class.power_one"
+ "EXP_MULT" > "Power.monoid_mult_class.power_mult"
+ "EXP_MONO_LT_IMP" > "HOLLight.hollight.EXP_MONO_LT_IMP"
+ "EXP_MONO_LT" > "HOLLight.hollight.EXP_MONO_LT"
+ "EXP_MONO_LE_IMP" > "HOLLight.hollight.EXP_MONO_LE_IMP"
+ "EXP_MONO_LE" > "HOLLight.hollight.EXP_MONO_LE"
+ "EXP_MONO_EQ" > "HOLLight.hollight.EXP_MONO_EQ"
"EXP_LT_0" > "HOLLight.hollight.EXP_LT_0"
- "EXP_EQ_0" > "HOLLight.hollight.EXP_EQ_0"
- "EXP_ADD" > "HOLLight.hollight.EXP_ADD"
- "EXP_2" > "HOLLight.hollight.EXP_2"
- "EXP_1" > "HOLLight.hollight.EXP_1"
- "EXISTS_UNIQUE_THM" > "HOL4Compat.EXISTS_UNIQUE_DEF"
+ "EXP_EQ_1" > "HOLLight.hollight.EXP_EQ_1"
+ "EXP_EQ_0" > "Power.power_eq_0_iff"
+ "EXP_ADD" > "Power.monoid_mult_class.power_add"
+ "EXP_2" > "Nat_Numeral.monoid_mult_class.power2_eq_square"
+ "EXP_1" > "Power.monoid_mult_class.power_one_right"
+ "EXISTS_UNIQUE_THM" > "HOLLightCompat.EXISTS_UNIQUE_THM"
"EXISTS_UNIQUE_REFL" > "HOL.ex1_eq_1"
"EXISTS_UNIQUE_ALT" > "HOLLight.hollight.EXISTS_UNIQUE_ALT"
"EXISTS_UNIQUE" > "HOL.Ex1_def"
+ "EXISTS_UNCURRY" > "HOLLight.hollight.EXISTS_UNCURRY"
+ "EXISTS_TRIPLED_THM" > "HOLLight.hollight.EXISTS_TRIPLED_THM"
"EXISTS_THM" > "HOL4Setup.EXISTS_DEF"
+ "EXISTS_SUBSET_IMAGE" > "HOLLight.hollight.EXISTS_SUBSET_IMAGE"
"EXISTS_SIMP" > "HOL.simp_thms_36"
"EXISTS_REFL" > "HOL.simp_thms_37"
- "EXISTS_PASTECART" > "HOLLight.hollight.EXISTS_PASTECART"
"EXISTS_PAIR_THM" > "Product_Type.split_paired_Ex"
+ "EXISTS_PAIRED_THM" > "HOLLight.hollight.EXISTS_PAIRED_THM"
"EXISTS_OR_THM" > "HOL.ex_disj_distrib"
"EXISTS_ONE_REP" > "HOLLight.hollight.EXISTS_ONE_REP"
- "EXISTS_NOT_THM" > "Inductive.basic_monos_15"
+ "EXISTS_NOT_THM" > "HOL.not_all"
+ "EXISTS_IN_UNIONS" > "HOLLight.hollight.EXISTS_IN_UNIONS"
+ "EXISTS_IN_INSERT" > "HOLLight.hollight.EXISTS_IN_INSERT"
"EXISTS_IN_IMAGE" > "HOLLight.hollight.EXISTS_IN_IMAGE"
+ "EXISTS_IN_GSPEC" > "HOLLight.hollight.EXISTS_IN_GSPEC"
"EXISTS_IN_CLAUSES" > "HOLLight.hollight.EXISTS_IN_CLAUSES"
- "EXISTS_EX" > "HOLLight.hollight.EXISTS_EX"
+ "EXISTS_FINITE_SUBSET_IMAGE" > "HOLLight.hollight.EXISTS_FINITE_SUBSET_IMAGE"
+ "EXISTS_EX" > "HOLLightList.EXISTS_EX"
"EXISTS_BOOL_THM" > "Set.ex_bool_eq"
"EXCLUDED_MIDDLE" > "HOLLight.hollight.EXCLUDED_MIDDLE"
- "EVEN_def" > "HOLLight.hollight.EVEN_def"
+ "EVEN_SUB" > "HOLLight.hollight.EVEN_SUB"
"EVEN_OR_ODD" > "HOLLight.hollight.EVEN_OR_ODD"
"EVEN_ODD_DECOMPOSITION" > "HOLLight.hollight.EVEN_ODD_DECOMPOSITION"
- "EVEN_MULT" > "HOLLight.hollight.EVEN_MULT"
+ "EVEN_MULT" > "Parity.even_product_nat"
"EVEN_MOD" > "HOLLight.hollight.EVEN_MOD"
"EVEN_EXP" > "HOLLight.hollight.EVEN_EXP"
"EVEN_EXISTS_LEMMA" > "HOLLight.hollight.EVEN_EXISTS_LEMMA"
- "EVEN_EXISTS" > "HOLLight.hollight.EVEN_EXISTS"
+ "EVEN_EXISTS" > "Parity.even_mult_two_ex"
"EVEN_DOUBLE" > "HOLLight.hollight.EVEN_DOUBLE"
"EVEN_AND_ODD" > "HOLLight.hollight.EVEN_AND_ODD"
- "EVEN_ADD" > "HOLLight.hollight.EVEN_ADD"
+ "EVEN_ADD" > "Parity.even_add"
"EQ_UNIV" > "HOLLight.hollight.EQ_UNIV"
- "EQ_TRANS" > "Set.basic_trans_rules_31"
- "EQ_SYM_EQ" > "HOL.eq_sym_conv"
- "EQ_SYM" > "HOL.meta_eq_to_obj_eq"
- "EQ_SUC" > "Nat.nat.simps_3"
- "EQ_REFL_T" > "HOL.simp_thms_6"
- "EQ_REFL" > "Presburger.fm_modd_pinf"
+ "EQ_TRANS" > "HOL.trans"
+ "EQ_SYM_EQ" > "HOL.eq_ac_1"
+ "EQ_SYM" > "HOL.eq_reflection"
+ "EQ_REFL" > "HOL.refl"
"EQ_MULT_RCANCEL" > "Nat.mult_cancel2"
- "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
- "EQ_IMP_LE" > "HOLLight.hollight.EQ_IMP_LE"
- "EQ_EXT" > "HOL.meta_eq_to_obj_eq"
+ "EQ_MULT_LCANCEL" > "Numeral_Simprocs.nat_mult_eq_cancel_disj"
+ "EQ_IMP_LE" > "Nat.eq_imp_le"
+ "EQ_EXT" > "HOL.eq_reflection"
+ "EQ_EXP" > "HOLLight.hollight.EQ_EXP"
"EQ_CLAUSES" > "HOLLight.hollight.EQ_CLAUSES"
"EQ_ADD_RCANCEL_0" > "HOLLight.hollight.EQ_ADD_RCANCEL_0"
- "EQ_ADD_RCANCEL" > "Nat.nat_add_right_cancel"
+ "EQ_ADD_RCANCEL" > "Groups.cancel_semigroup_add_class.add_right_cancel"
"EQ_ADD_LCANCEL_0" > "HOLLight.hollight.EQ_ADD_LCANCEL_0"
- "EQ_ADD_LCANCEL" > "Nat.nat_add_left_cancel"
- "EMPTY_def" > "HOLLight.hollight.EMPTY_def"
+ "EQ_ADD_LCANCEL" > "Groups.cancel_semigroup_add_class.add_left_cancel"
"EMPTY_UNIONS" > "HOLLight.hollight.EMPTY_UNIONS"
- "EMPTY_UNION" > "HOLLight.hollight.EMPTY_UNION"
- "EMPTY_SUBSET" > "HOLLight.hollight.EMPTY_SUBSET"
+ "EMPTY_UNION" > "Lattices.bounded_lattice_bot_class.sup_eq_bot_iff"
+ "EMPTY_SUBSET" > "Orderings.bot_class.bot_least"
"EMPTY_NOT_UNIV" > "HOLLight.hollight.EMPTY_NOT_UNIV"
"EMPTY_GSPEC" > "HOLLight.hollight.EMPTY_GSPEC"
- "EMPTY_DIFF" > "HOLLight.hollight.EMPTY_DIFF"
+ "EMPTY_DIFF" > "Set.empty_Diff"
"EMPTY_DELETE" > "HOLLight.hollight.EMPTY_DELETE"
- "EL_def" > "HOLLight.hollight.EL_def"
- "DIV_def" > "HOLLight.hollight.DIV_def"
+ "EL_CONS" > "List.nth_Cons'"
+ "EL_APPEND" > "List.nth_append"
"DIV_UNIQ" > "HOLLight.hollight.DIV_UNIQ"
- "DIV_REFL" > "HOLLight.hollight.DIV_REFL"
+ "DIV_REFL" > "Divides.semiring_div_class.div_self"
"DIV_MUL_LE" > "HOLLight.hollight.DIV_MUL_LE"
"DIV_MULT2" > "HOLLight.hollight.DIV_MULT2"
- "DIV_MULT" > "HOLLight.hollight.DIV_MULT"
"DIV_MONO_LT" > "HOLLight.hollight.DIV_MONO_LT"
"DIV_MONO2" > "HOLLight.hollight.DIV_MONO2"
"DIV_MONO" > "HOLLight.hollight.DIV_MONO"
"DIV_MOD" > "HOLLight.hollight.DIV_MOD"
- "DIV_LT" > "HOLLight.hollight.DIV_LT"
+ "DIV_LT" > "Divides.div_less"
"DIV_LE_EXCLUSION" > "HOLLight.hollight.DIV_LE_EXCLUSION"
"DIV_LE" > "HOLLight.hollight.DIV_LE"
"DIV_EQ_EXCLUSION" > "HOLLight.hollight.DIV_EQ_EXCLUSION"
"DIV_EQ_0" > "HOLLight.hollight.DIV_EQ_0"
"DIV_DIV" > "HOLLight.hollight.DIV_DIV"
"DIV_ADD_MOD" > "HOLLight.hollight.DIV_ADD_MOD"
- "DIV_1" > "HOLLight.hollight.DIV_1"
- "DIV_0" > "HOLLight.hollight.DIV_0"
"DIVMOD_UNIQ_LEMMA" > "HOLLight.hollight.DIVMOD_UNIQ_LEMMA"
"DIVMOD_UNIQ" > "HOLLight.hollight.DIVMOD_UNIQ"
"DIVMOD_EXIST_0" > "HOLLight.hollight.DIVMOD_EXIST_0"
@@ -1559,7 +1872,6 @@
"DIVISION" > "HOLLight.hollight.DIVISION"
"DIST_TRIANGLE_LE" > "HOLLight.hollight.DIST_TRIANGLE_LE"
"DIST_TRIANGLES_LE" > "HOLLight.hollight.DIST_TRIANGLES_LE"
- "DIST_TRIANGLE" > "HOLLight.hollight.DIST_TRIANGLE"
"DIST_SYM" > "HOLLight.hollight.DIST_SYM"
"DIST_RZERO" > "HOLLight.hollight.DIST_RZERO"
"DIST_RMUL" > "HOLLight.hollight.DIST_RMUL"
@@ -1573,13 +1885,9 @@
"DIST_LADD" > "HOLLight.hollight.DIST_LADD"
"DIST_EQ_0" > "HOLLight.hollight.DIST_EQ_0"
"DIST_ELIM_THM" > "HOLLight.hollight.DIST_ELIM_THM"
- "DIST_ADDBOUND" > "HOLLight.hollight.DIST_ADDBOUND"
- "DIST_ADD2_REV" > "HOLLight.hollight.DIST_ADD2_REV"
- "DIST_ADD2" > "HOLLight.hollight.DIST_ADD2"
- "DISJ_SYM" > "HOL.disj_comms_1"
- "DISJ_ASSOC" > "Recdef.tfl_disj_assoc"
+ "DISJ_SYM" > "Groebner_Basis.dnf_4"
+ "DISJ_ASSOC" > "HOL.disj_ac_3"
"DISJ_ACI" > "HOLLight.hollight.DISJ_ACI"
- "DISJOINT_def" > "HOLLight.hollight.DISJOINT_def"
"DISJOINT_UNION" > "HOLLight.hollight.DISJOINT_UNION"
"DISJOINT_SYM" > "HOLLight.hollight.DISJOINT_SYM"
"DISJOINT_NUMSEG" > "HOLLight.hollight.DISJOINT_NUMSEG"
@@ -1587,28 +1895,26 @@
"DISJOINT_EMPTY_REFL" > "HOLLight.hollight.DISJOINT_EMPTY_REFL"
"DISJOINT_EMPTY" > "HOLLight.hollight.DISJOINT_EMPTY"
"DISJOINT_DELETE_SYM" > "HOLLight.hollight.DISJOINT_DELETE_SYM"
+ "DIMINDEX_UNIV" > "HOLLight.hollight.DIMINDEX_UNIV"
+ "DIMINDEX_UNIQUE" > "HOLLight.hollight.DIMINDEX_UNIQUE"
"DIMINDEX_NONZERO" > "HOLLight.hollight.DIMINDEX_NONZERO"
- "DIMINDEX_HAS_SIZE_FINITE_SUM" > "HOLLight.hollight.DIMINDEX_HAS_SIZE_FINITE_SUM"
"DIMINDEX_GE_1" > "HOLLight.hollight.DIMINDEX_GE_1"
- "DIMINDEX_FINITE_SUM" > "HOLLight.hollight.DIMINDEX_FINITE_SUM"
"DIMINDEX_FINITE_IMAGE" > "HOLLight.hollight.DIMINDEX_FINITE_IMAGE"
- "DIFF_def" > "HOLLight.hollight.DIFF_def"
- "DIFF_UNIV" > "HOLLight.hollight.DIFF_UNIV"
- "DIFF_INSERT" > "HOLLight.hollight.DIFF_INSERT"
- "DIFF_EQ_EMPTY" > "HOLLight.hollight.DIFF_EQ_EMPTY"
- "DIFF_EMPTY" > "HOLLight.hollight.DIFF_EMPTY"
- "DIFF_DIFF" > "HOLLight.hollight.DIFF_DIFF"
+ "DIFF_UNIV" > "Set.Diff_UNIV"
+ "DIFF_INTERS" > "HOLLight.hollight.DIFF_INTERS"
+ "DIFF_INSERT" > "Set.Diff_insert2"
+ "DIFF_EQ_EMPTY" > "Set.Diff_cancel"
+ "DIFF_EMPTY" > "Set.Diff_empty"
+ "DIFF_DIFF" > "Set.Diff_idemp"
"DEST_REC_INJ" > "HOLLight.hollight.recspace._dest_rec_inject"
- "DELETE_def" > "HOLLight.hollight.DELETE_def"
"DELETE_SUBSET" > "HOLLight.hollight.DELETE_SUBSET"
"DELETE_NON_ELEMENT" > "HOLLight.hollight.DELETE_NON_ELEMENT"
"DELETE_INTER" > "HOLLight.hollight.DELETE_INTER"
"DELETE_INSERT" > "HOLLight.hollight.DELETE_INSERT"
"DELETE_DELETE" > "HOLLight.hollight.DELETE_DELETE"
"DELETE_COMM" > "HOLLight.hollight.DELETE_COMM"
- "DEF_~" > "HOL.simp_thms_19"
- "DEF_two_2" > "HOLLight.hollight.DEF_two_2"
- "DEF_two_1" > "HOLLight.hollight.DEF_two_1"
+ "DEF_~" > "Groebner_Basis.bool_simps_19"
+ "DEF_vector" > "HOLLight.hollight.DEF_vector"
"DEF_treal_of_num" > "HOLLight.hollight.DEF_treal_of_num"
"DEF_treal_neg" > "HOLLight.hollight.DEF_treal_neg"
"DEF_treal_mul" > "HOLLight.hollight.DEF_treal_mul"
@@ -1616,20 +1922,20 @@
"DEF_treal_inv" > "HOLLight.hollight.DEF_treal_inv"
"DEF_treal_eq" > "HOLLight.hollight.DEF_treal_eq"
"DEF_treal_add" > "HOLLight.hollight.DEF_treal_add"
- "DEF_three_3" > "HOLLight.hollight.DEF_three_3"
- "DEF_three_2" > "HOLLight.hollight.DEF_three_2"
- "DEF_three_1" > "HOLLight.hollight.DEF_three_1"
"DEF_tailadmissible" > "HOLLight.hollight.DEF_tailadmissible"
"DEF_support" > "HOLLight.hollight.DEF_support"
"DEF_superadmissible" > "HOLLight.hollight.DEF_superadmissible"
"DEF_sum" > "HOLLight.hollight.DEF_sum"
"DEF_sndcart" > "HOLLight.hollight.DEF_sndcart"
- "DEF_set_of_list" > "HOLLight.hollight.DEF_set_of_list"
+ "DEF_set_of_list" > "HOLLightList.DEF_set_of_list"
+ "DEF_rem" > "HOLLight.hollight.DEF_rem"
"DEF_real_sub" > "HOLLight.hollight.DEF_real_sub"
+ "DEF_real_sgn" > "HOLLight.hollight.DEF_real_sgn"
"DEF_real_pow" > "HOLLight.hollight.DEF_real_pow"
"DEF_real_of_num" > "HOLLight.hollight.DEF_real_of_num"
"DEF_real_neg" > "HOLLight.hollight.DEF_real_neg"
"DEF_real_mul" > "HOLLight.hollight.DEF_real_mul"
+ "DEF_real_mod" > "HOLLight.hollight.DEF_real_mod"
"DEF_real_min" > "HOLLight.hollight.DEF_real_min"
"DEF_real_max" > "HOLLight.hollight.DEF_real_max"
"DEF_real_lt" > "HOLLight.hollight.DEF_real_lt"
@@ -1642,7 +1948,12 @@
"DEF_real_abs" > "HOLLight.hollight.DEF_real_abs"
"DEF_pastecart" > "HOLLight.hollight.DEF_pastecart"
"DEF_pairwise" > "HOLLight.hollight.DEF_pairwise"
- "DEF_o" > "Fun.o_apply"
+ "DEF_o" > "Fun.comp_def"
+ "DEF_num_of_int" > "HOLLight.hollight.DEF_num_of_int"
+ "DEF_num_mod" > "HOLLight.hollight.DEF_num_mod"
+ "DEF_num_gcd" > "HOLLight.hollight.DEF_num_gcd"
+ "DEF_num_divides" > "HOLLight.hollight.DEF_num_divides"
+ "DEF_num_coprime" > "HOLLight.hollight.DEF_num_coprime"
"DEF_nsum" > "HOLLight.hollight.DEF_nsum"
"DEF_neutral" > "HOLLight.hollight.DEF_neutral"
"DEF_nadd_rinv" > "HOLLight.hollight.DEF_nadd_rinv"
@@ -1653,28 +1964,27 @@
"DEF_nadd_eq" > "HOLLight.hollight.DEF_nadd_eq"
"DEF_nadd_add" > "HOLLight.hollight.DEF_nadd_add"
"DEF_monoidal" > "HOLLight.hollight.DEF_monoidal"
- "DEF_mod_real" > "HOLLight.hollight.DEF_mod_real"
- "DEF_mod_nat" > "HOLLight.hollight.DEF_mod_nat"
- "DEF_mod_int" > "HOLLight.hollight.DEF_mod_int"
- "DEF_mk_pair" > "Product_Type.Pair_Rep_def"
"DEF_minimal" > "HOLLight.hollight.DEF_minimal"
- "DEF_measure" > "HOLLight.hollight.DEF_measure"
- "DEF_list_of_set" > "HOLLight.hollight.DEF_list_of_set"
"DEF_lambda" > "HOLLight.hollight.DEF_lambda"
"DEF_iterate" > "HOLLight.hollight.DEF_iterate"
"DEF_is_nadd" > "HOLLight.hollight.DEF_is_nadd"
- "DEF_is_int" > "HOLLight.hollight.DEF_is_int"
+ "DEF_integer" > "HOLLight.hollight.DEF_integer"
"DEF_int_sub" > "HOLLight.hollight.DEF_int_sub"
+ "DEF_int_sgn" > "HOLLight.hollight.DEF_int_sgn"
"DEF_int_pow" > "HOLLight.hollight.DEF_int_pow"
"DEF_int_of_num" > "HOLLight.hollight.DEF_int_of_num"
"DEF_int_neg" > "HOLLight.hollight.DEF_int_neg"
"DEF_int_mul" > "HOLLight.hollight.DEF_int_mul"
+ "DEF_int_mod" > "HOLLight.hollight.DEF_int_mod"
"DEF_int_min" > "HOLLight.hollight.DEF_int_min"
"DEF_int_max" > "HOLLight.hollight.DEF_int_max"
"DEF_int_lt" > "HOLLight.hollight.DEF_int_lt"
"DEF_int_le" > "HOLLight.hollight.DEF_int_le"
"DEF_int_gt" > "HOLLight.hollight.DEF_int_gt"
"DEF_int_ge" > "HOLLight.hollight.DEF_int_ge"
+ "DEF_int_gcd" > "HOLLight.hollight.DEF_int_gcd"
+ "DEF_int_divides" > "HOLLight.hollight.DEF_int_divides"
+ "DEF_int_coprime" > "HOLLight.hollight.DEF_int_coprime"
"DEF_int_add" > "HOLLight.hollight.DEF_int_add"
"DEF_int_abs" > "HOLLight.hollight.DEF_int_abs"
"DEF_hreal_of_num" > "HOLLight.hollight.DEF_hreal_of_num"
@@ -1683,178 +1993,178 @@
"DEF_hreal_inv" > "HOLLight.hollight.DEF_hreal_inv"
"DEF_hreal_add" > "HOLLight.hollight.DEF_hreal_add"
"DEF_fstcart" > "HOLLight.hollight.DEF_fstcart"
- "DEF_finite_index" > "HOLLight.hollight.DEF_finite_index"
+ "DEF_div" > "HOLLight.hollight.DEF_div"
"DEF_dist" > "HOLLight.hollight.DEF_dist"
"DEF_dimindex" > "HOLLight.hollight.DEF_dimindex"
"DEF_admissible" > "HOLLight.hollight.DEF_admissible"
- "DEF__star_" > "HOLLightCompat.mult_altdef"
- "DEF__slash__backslash_" > "HOLLightCompat.light_and_def"
- "DEF__questionmark__exclamationmark_" > "HOL4Compat.EXISTS_UNIQUE_DEF"
+ "DEF__star_" > "HOLLightCompat.DEF__star_"
+ "DEF__slash__backslash_" > "HOLLightCompat.DEF__slash__backslash_"
+ "DEF__questionmark__exclamationmark_" > "HOLLightCompat.EXISTS_UNIQUE_THM"
"DEF__questionmark_" > "HOL.Ex_def"
- "DEF__lessthan__equal_" > "HOLLight.hollight.DEF__lessthan__equal_"
- "DEF__lessthan_" > "HOLLight.hollight.DEF__lessthan_"
- "DEF__greaterthan__equal_" > "HOLLight.hollight.DEF__greaterthan__equal_"
- "DEF__greaterthan_" > "HOLLight.hollight.DEF__greaterthan_"
+ "DEF__lessthan__equal__c" > "HOLLight.hollight.DEF__lessthan__equal__c"
+ "DEF__lessthan__equal_" > "HOLLightCompat.DEF__lessthan__equal_"
+ "DEF__lessthan__c" > "HOLLight.hollight.DEF__lessthan__c"
+ "DEF__lessthan_" > "HOLLightCompat.DEF__lessthan_"
+ "DEF__greaterthan__equal__c" > "HOLLight.hollight.DEF__greaterthan__equal__c"
+ "DEF__greaterthan__equal_" > "HOLLightCompat.DEF__greaterthan__equal_"
+ "DEF__greaterthan__c" > "HOLLight.hollight.DEF__greaterthan__c"
+ "DEF__greaterthan_" > "HOLLightCompat.DEF__greaterthan_"
"DEF__exclamationmark_" > "HOL.All_def"
- "DEF__equal__equal__greaterthan_" > "HOLLightCompat.light_imp_def"
+ "DEF__equal__equal__greaterthan_" > "HOLLightCompat.DEF__equal__equal__greaterthan_"
"DEF__equal__equal_" > "HOLLight.hollight.DEF__equal__equal_"
- "DEF__dot__dot_" > "HOLLight.hollight.DEF__dot__dot_"
+ "DEF__equal__c" > "HOLLight.hollight.DEF__equal__c"
+ "DEF__dot__dot_" > "HOLLightCompat.DEF__dot__dot_"
"DEF__backslash__slash_" > "HOL.or_def"
+ "DEF__UNGUARDED_PATTERN" > "HOLLight.hollight.DEF__UNGUARDED_PATTERN"
+ "DEF__SEQPATTERN" > "HOLLight.hollight.DEF__SEQPATTERN"
+ "DEF__MATCH" > "HOLLight.hollight.DEF__MATCH"
+ "DEF__GUARDED_PATTERN" > "HOLLight.hollight.DEF__GUARDED_PATTERN"
+ "DEF__FUNCTION" > "HOLLight.hollight.DEF__FUNCTION"
"DEF__FALSITY_" > "HOLLight.hollight.DEF__FALSITY_"
- "DEF__10328" > "HOLLight.hollight.DEF__10328"
- "DEF__10327" > "HOLLight.hollight.DEF__10327"
- "DEF__10326" > "HOLLight.hollight.DEF__10326"
- "DEF__10303" > "HOLLight.hollight.DEF__10303"
- "DEF__10302" > "HOLLight.hollight.DEF__10302"
+ "DEF__11937" > "HOLLight.hollight.DEF__11937"
"DEF_ZRECSPACE" > "HOLLight.hollight.DEF_ZRECSPACE"
- "DEF_ZIP" > "HOLLight.hollight.DEF_ZIP"
"DEF_ZCONSTR" > "HOLLight.hollight.DEF_ZCONSTR"
"DEF_ZBOT" > "HOLLight.hollight.DEF_ZBOT"
- "DEF_WF" > "HOLLight.hollight.DEF_WF"
- "DEF_UNIV" > "HOLLight.hollight.DEF_UNIV"
- "DEF_UNIONS" > "HOLLight.hollight.DEF_UNIONS"
- "DEF_UNION" > "HOLLight.hollight.DEF_UNION"
+ "DEF_WF" > "HOLLightCompat.DEF_WF"
+ "DEF_UNIV" > "HOLLightCompat.DEF_UNIV"
+ "DEF_UNIONS" > "HOLLightCompat.DEF_UNIONS"
+ "DEF_UNION" > "HOLLightCompat.DEF_UNION"
"DEF_UNCURRY" > "HOLLight.hollight.DEF_UNCURRY"
- "DEF_TL" > "HOLLight.hollight.DEF_TL"
"DEF_T" > "HOL.True_def"
"DEF_SURJ" > "HOLLight.hollight.DEF_SURJ"
- "DEF_SUBSET" > "HOLLight.hollight.DEF_SUBSET"
- "DEF_SOME" > "HOLLight.hollight.DEF_SOME"
- "DEF_SND" > "HOLLightCompat.snd_altdef"
+ "DEF_SUBSET" > "HOLLightCompat.DEF_SUBSET"
+ "DEF_SND" > "HOLLightCompat.DEF_SND"
"DEF_SING" > "HOLLight.hollight.DEF_SING"
- "DEF_SETSPEC" > "HOLLight.hollight.DEF_SETSPEC"
- "DEF_REVERSE" > "HOLLight.hollight.DEF_REVERSE"
+ "DEF_SETSPEC" > "HOLLightCompat.SETSPEC_def"
+ "DEF_REVERSE" > "HOLLightList.DEF_REVERSE"
"DEF_REST" > "HOLLight.hollight.DEF_REST"
- "DEF_REPLICATE" > "HOLLight.hollight.DEF_REPLICATE"
- "DEF_PSUBSET" > "HOLLight.hollight.DEF_PSUBSET"
- "DEF_PRE" > "HOLLightCompat.Pred_altdef"
+ "DEF_REPLICATE" > "HOLLightList.DEF_REPLICATE"
+ "DEF_PSUBSET" > "HOLLightCompat.DEF_PSUBSET"
+ "DEF_PRE" > "HOLLightCompat.DEF_PRE"
"DEF_PASSOC" > "HOLLight.hollight.DEF_PASSOC"
"DEF_PAIRWISE" > "HOLLight.hollight.DEF_PAIRWISE"
- "DEF_OUTR" > "HOLLight.hollight.DEF_OUTR"
- "DEF_OUTL" > "HOLLight.hollight.DEF_OUTL"
"DEF_ONTO" > "Fun.surj_def"
- "DEF_ONE_ONE" > "HOL4Setup.ONE_ONE_DEF"
- "DEF_ODD" > "HOLLight.hollight.DEF_ODD"
+ "DEF_ONE_ONE" > "HOLLightCompat.DEF_ONE_ONE"
+ "DEF_ODD" > "HOLLightCompat.DEF_ODD"
+ "DEF_NUM_REP" > "HOLLight.hollight.DEF_NUM_REP"
"DEF_NUMSUM" > "HOLLight.hollight.DEF_NUMSUM"
"DEF_NUMSND" > "HOLLight.hollight.DEF_NUMSND"
"DEF_NUMRIGHT" > "HOLLight.hollight.DEF_NUMRIGHT"
"DEF_NUMPAIR" > "HOLLight.hollight.DEF_NUMPAIR"
"DEF_NUMLEFT" > "HOLLight.hollight.DEF_NUMLEFT"
"DEF_NUMFST" > "HOLLight.hollight.DEF_NUMFST"
- "DEF_NUMERAL" > "HOLLightCompat.NUMERAL_rew"
- "DEF_NULL" > "HOLLight.hollight.DEF_NULL"
- "DEF_NONE" > "HOLLight.hollight.DEF_NONE"
- "DEF_NIL" > "HOLLight.hollight.DEF_NIL"
- "DEF_MOD" > "HOLLight.hollight.DEF_MOD"
- "DEF_MEM" > "HOLLight.hollight.DEF_MEM"
- "DEF_MAP2" > "HOLLight.hollight.DEF_MAP2"
- "DEF_MAP" > "HOLLight.hollight.DEF_MAP"
+ "DEF_NUMERAL" > "HOLLightCompat.NUMERAL_def"
+ "DEF_NULL" > "HOLLightList.DEF_NULL"
+ "DEF_MOD" > "HOLLightCompat.DEF_MOD"
+ "DEF_MIN" > "Orderings.ord_class.min_def"
+ "DEF_MEM" > "HOLLightList.DEF_MEM"
+ "DEF_MEASURE" > "HOLLightCompat.MEASURE_def"
+ "DEF_MAX" > "Orderings.ord_class.max_def"
+ "DEF_MAP" > "HOLLightList.DEF_MAP"
"DEF_LET_END" > "HOLLight.hollight.DEF_LET_END"
- "DEF_LET" > "HOL4Compat.LET_def"
- "DEF_LENGTH" > "HOLLight.hollight.DEF_LENGTH"
- "DEF_LAST" > "HOLLight.hollight.DEF_LAST"
+ "DEF_LET" > "HOLLightCompat.LET_def"
+ "DEF_LENGTH" > "HOLLightList.DEF_LENGTH"
+ "DEF_LAST" > "HOLLightList.DEF_LAST"
"DEF_ITSET" > "HOLLight.hollight.DEF_ITSET"
- "DEF_ITLIST2" > "HOLLight.hollight.DEF_ITLIST2"
- "DEF_ITLIST" > "HOLLight.hollight.DEF_ITLIST"
+ "DEF_ITLIST" > "HOLLightList.DEF_ITLIST"
"DEF_ISO" > "HOLLight.hollight.DEF_ISO"
- "DEF_INTERS" > "HOLLight.hollight.DEF_INTERS"
- "DEF_INTER" > "HOLLight.hollight.DEF_INTER"
- "DEF_INSERT" > "HOLLight.hollight.DEF_INSERT"
+ "DEF_INTERS" > "HOLLightCompat.DEF_INTERS"
+ "DEF_INTER" > "HOLLightCompat.DEF_INTER"
+ "DEF_INSERT" > "HOLLightCompat.DEF_INSERT"
"DEF_INJP" > "HOLLight.hollight.DEF_INJP"
"DEF_INJN" > "HOLLight.hollight.DEF_INJN"
"DEF_INJF" > "HOLLight.hollight.DEF_INJF"
"DEF_INJA" > "HOLLight.hollight.DEF_INJA"
"DEF_INJ" > "HOLLight.hollight.DEF_INJ"
- "DEF_INFINITE" > "HOLLight.hollight.DEF_INFINITE"
- "DEF_IN" > "HOLLight.hollight.DEF_IN"
- "DEF_IMAGE" > "HOLLight.hollight.DEF_IMAGE"
+ "DEF_INFINITE" > "HOLLightCompat.DEF_INFINITE"
+ "DEF_IND_SUC" > "HOLLight.hollight.DEF_IND_SUC"
+ "DEF_IND_0" > "HOLLight.hollight.DEF_IND_0"
+ "DEF_IN" > "Set.mem_def"
+ "DEF_IMAGE" > "HOLLightCompat.DEF_IMAGE"
"DEF_I" > "Fun.id_apply"
- "DEF_HD" > "HOLLight.hollight.DEF_HD"
"DEF_HAS_SIZE" > "HOLLight.hollight.DEF_HAS_SIZE"
- "DEF_GSPEC" > "HOLLight.hollight.DEF_GSPEC"
- "DEF_GEQ" > "HOLLight.hollight.DEF_GEQ"
- "DEF_GABS" > "HOLLight.hollight.DEF_GABS"
- "DEF_FST" > "HOLLightCompat.fst_altdef"
+ "DEF_GSPEC" > "HOLLightCompat.DEF_GSPEC"
+ "DEF_GEQ" > "HOLLightCompat.DEF_GEQ"
+ "DEF_GABS" > "HOLLightCompat.DEF_GABS"
+ "DEF_FST" > "HOLLightCompat.DEF_FST"
"DEF_FNIL" > "HOLLight.hollight.DEF_FNIL"
"DEF_FINREC" > "HOLLight.hollight.DEF_FINREC"
- "DEF_FINITE" > "HOLLight.hollight.DEF_FINITE"
- "DEF_FILTER" > "HOLLight.hollight.DEF_FILTER"
+ "DEF_FINITE" > "HOLLightCompat.DEF_FINITE"
+ "DEF_FILTER" > "HOLLightList.DEF_FILTER"
"DEF_FCONS" > "HOLLight.hollight.DEF_FCONS"
- "DEF_FACT" > "HOLLight.hollight.DEF_FACT"
+ "DEF_FACT" > "HOLLightCompat.DEF_FACT"
"DEF_F" > "HOL.False_def"
- "DEF_EXP" > "HOLLight.hollight.DEF_EXP"
- "DEF_EX" > "HOLLight.hollight.DEF_EX"
- "DEF_EVEN" > "HOLLight.hollight.DEF_EVEN"
- "DEF_EMPTY" > "HOLLight.hollight.DEF_EMPTY"
- "DEF_EL" > "HOLLight.hollight.DEF_EL"
- "DEF_DIV" > "HOLLight.hollight.DEF_DIV"
- "DEF_DISJOINT" > "HOLLight.hollight.DEF_DISJOINT"
- "DEF_DIFF" > "HOLLight.hollight.DEF_DIFF"
- "DEF_DELETE" > "HOLLight.hollight.DEF_DELETE"
+ "DEF_EXP" > "HOLLightCompat.DEF_EXP"
+ "DEF_EX" > "HOLLightList.DEF_EX"
+ "DEF_EVEN" > "HOLLightCompat.DEF_EVEN"
+ "DEF_EMPTY" > "HOLLightCompat.DEF_EMPTY"
+ "DEF_EL" > "HOLLightList.DEF_EL"
+ "DEF_DIV" > "HOLLightCompat.DEF_DIV"
+ "DEF_DISJOINT" > "HOLLightCompat.DEF_DISJOINT"
+ "DEF_DIFF" > "HOLLightCompat.DEF_DIFF"
+ "DEF_DELETE" > "HOLLightCompat.DEF_DELETE"
"DEF_DECIMAL" > "HOLLight.hollight.DEF_DECIMAL"
- "DEF_CURRY" > "HOLLight.hollight.DEF_CURRY"
+ "DEF_CURRY" > "Product_Type.curry_conv"
+ "DEF_CROSS" > "HOLLight.hollight.DEF_CROSS"
"DEF_COUNTABLE" > "HOLLight.hollight.DEF_COUNTABLE"
"DEF_CONSTR" > "HOLLight.hollight.DEF_CONSTR"
- "DEF_CONS" > "HOLLight.hollight.DEF_CONS"
- "DEF_COND" > "HOLLight.hollight.DEF_COND"
- "DEF_CHOICE" > "HOLLight.hollight.DEF_CHOICE"
+ "DEF_COND" > "HOLLightCompat.COND_DEF"
+ "DEF_CHOICE" > "HOLLightCompat.DEF_CHOICE"
"DEF_CASEWISE" > "HOLLight.hollight.DEF_CASEWISE"
- "DEF_CARD_LT" > "HOLLight.hollight.DEF_CARD_LT"
- "DEF_CARD_LE" > "HOLLight.hollight.DEF_CARD_LE"
- "DEF_CARD_GT" > "HOLLight.hollight.DEF_CARD_GT"
- "DEF_CARD_GE" > "HOLLight.hollight.DEF_CARD_GE"
- "DEF_CARD_EQ" > "HOLLight.hollight.DEF_CARD_EQ"
"DEF_CARD" > "HOLLight.hollight.DEF_CARD"
+ "DEF_BUTLAST" > "HOLLightList.DEF_BUTLAST"
"DEF_BOTTOM" > "HOLLight.hollight.DEF_BOTTOM"
- "DEF_BIT1" > "HOLLightCompat.NUMERAL_BIT1_altdef"
- "DEF_BIT0" > "HOLLightCompat.NUMERAL_BIT0_def"
+ "DEF_BIT1" > "HOLLightCompat.BIT1_DEF"
+ "DEF_BIT0" > "HOLLightCompat.BIT0_DEF"
"DEF_BIJ" > "HOLLight.hollight.DEF_BIJ"
- "DEF_ASSOC" > "HOLLight.hollight.DEF_ASSOC"
- "DEF_APPEND" > "HOLLight.hollight.DEF_APPEND"
- "DEF_ALL2" > "HOLLight.hollight.DEF_ALL2"
- "DEF_ALL" > "HOLLight.hollight.DEF_ALL"
- "DEF_-" > "HOLLightCompat.sub_altdef"
- "DEF_," > "Product_Type.Pair_def"
- "DEF_+" > "HOLLightCompat.add_altdef"
+ "DEF_ASCII" > "HOLLight.hollight.DEF_ASCII"
+ "DEF_APPEND" > "HOLLightList.DEF_APPEND"
+ "DEF_ALL2" > "HOLLightList.DEF_ALL2"
+ "DEF_ALL" > "HOLLightList.DEF_ALL"
+ "DEF_-" > "HOLLightCompat.DEF_MINUS"
+ "DEF_+" > "HOLLightCompat.DEF_PLUS"
"DEF_$" > "HOLLight.hollight.DEF_$"
"DECOMPOSITION" > "HOLLight.hollight.DECOMPOSITION"
"DECIMAL_def" > "HOLLight.hollight.DECIMAL_def"
- "CURRY_def" > "HOLLight.hollight.CURRY_def"
+ "CROSS_def" > "HOLLight.hollight.CROSS_def"
+ "CROSS_EQ_EMPTY" > "HOLLight.hollight.CROSS_EQ_EMPTY"
"COUNTABLE_def" > "HOLLight.hollight.COUNTABLE_def"
- "CONS_def" > "HOLLight.hollight.CONS_def"
- "CONS_11" > "HOLLight.hollight.CONS_11"
+ "CONS_HD_TL" > "List.hd_Cons_tl"
+ "CONS_11" > "List.list.inject"
"CONSTR_def" > "HOLLight.hollight.CONSTR_def"
"CONSTR_REC" > "HOLLight.hollight.CONSTR_REC"
"CONSTR_INJ" > "HOLLight.hollight.CONSTR_INJ"
"CONSTR_IND" > "HOLLight.hollight.CONSTR_IND"
"CONSTR_BOT" > "HOLLight.hollight.CONSTR_BOT"
- "CONJ_SYM" > "HOL.conj_comms_1"
- "CONJ_ASSOC" > "HOL.conj_assoc"
+ "CONJ_SYM" > "Groebner_Basis.dnf_3"
+ "CONJ_ASSOC" > "HOL.conj_ac_3"
"CONJ_ACI" > "HOLLight.hollight.CONJ_ACI"
- "COND_def" > "HOLLight.hollight.COND_def"
"COND_RATOR" > "HOLLight.hollight.COND_RATOR"
- "COND_RAND" > "HOLLight.hollight.COND_RAND"
- "COND_ID" > "HOLLight.hollight.COND_ID"
+ "COND_RAND" > "HOL.if_distrib"
+ "COND_ID" > "HOL.if_cancel"
"COND_EXPAND" > "HOLLight.hollight.COND_EXPAND"
"COND_EQ_CLAUSE" > "HOLLight.hollight.COND_EQ_CLAUSE"
- "COND_ELIM_THM" > "HOLLight.hollight.COND_ELIM_THM"
+ "COND_ELIM_THM" > "HOL.if_splits_1"
"COND_CLAUSES" > "HOLLight.hollight.COND_CLAUSES"
"COND_ABS" > "HOLLight.hollight.COND_ABS"
- "COMPONENT" > "HOLLight.hollight.COMPONENT"
+ "COMPONENT" > "Set.insertI1"
+ "CHOOSE_SUBSET_STRONG" > "HOLLight.hollight.CHOOSE_SUBSET_STRONG"
"CHOOSE_SUBSET" > "HOLLight.hollight.CHOOSE_SUBSET"
- "CHOICE_def" > "HOLLight.hollight.CHOICE_def"
"CHOICE_DEF" > "HOLLight.hollight.CHOICE_DEF"
"CASEWISE_def" > "HOLLight.hollight.CASEWISE_def"
- "CASEWISE_WORKS" > "HOLLight.hollight.CASEWISE_WORKS"
- "CASEWISE_CASES" > "HOLLight.hollight.CASEWISE_CASES"
- "CASEWISE" > "HOLLight.hollight.CASEWISE"
+ "CART_EQ_FULL" > "HOLLight.hollight.CART_EQ_FULL"
"CART_EQ" > "HOLLight.hollight.CART_EQ"
"CARD_def" > "HOLLight.hollight.CARD_def"
+ "CARD_UNION_OVERLAP_EQ" > "HOLLight.hollight.CARD_UNION_OVERLAP_EQ"
+ "CARD_UNION_OVERLAP" > "HOLLight.hollight.CARD_UNION_OVERLAP"
"CARD_UNION_LE" > "HOLLight.hollight.CARD_UNION_LE"
+ "CARD_UNION_GEN" > "HOLLight.hollight.CARD_UNION_GEN"
"CARD_UNION_EQ" > "HOLLight.hollight.CARD_UNION_EQ"
"CARD_UNIONS_LE" > "HOLLight.hollight.CARD_UNIONS_LE"
+ "CARD_UNIONS" > "HOLLight.hollight.CARD_UNIONS"
"CARD_UNION" > "HOLLight.hollight.CARD_UNION"
"CARD_SUBSET_LE" > "HOLLight.hollight.CARD_SUBSET_LE"
+ "CARD_SUBSET_IMAGE" > "HOLLight.hollight.CARD_SUBSET_IMAGE"
"CARD_SUBSET_EQ" > "HOLLight.hollight.CARD_SUBSET_EQ"
"CARD_SUBSET" > "HOLLight.hollight.CARD_SUBSET"
"CARD_PSUBSET" > "HOLLight.hollight.CARD_PSUBSET"
@@ -1865,22 +2175,20 @@
"CARD_NUMSEG_LE" > "HOLLight.hollight.CARD_NUMSEG_LE"
"CARD_NUMSEG_1" > "HOLLight.hollight.CARD_NUMSEG_1"
"CARD_NUMSEG" > "HOLLight.hollight.CARD_NUMSEG"
- "CARD_LT_def" > "HOLLight.hollight.CARD_LT_def"
- "CARD_LE_def" > "HOLLight.hollight.CARD_LE_def"
"CARD_LE_INJ" > "HOLLight.hollight.CARD_LE_INJ"
"CARD_IMAGE_LE" > "HOLLight.hollight.CARD_IMAGE_LE"
+ "CARD_IMAGE_INJ_EQ" > "HOLLight.hollight.CARD_IMAGE_INJ_EQ"
"CARD_IMAGE_INJ" > "HOLLight.hollight.CARD_IMAGE_INJ"
- "CARD_GT_def" > "HOLLight.hollight.CARD_GT_def"
- "CARD_GE_def" > "HOLLight.hollight.CARD_GE_def"
- "CARD_GE_TRANS" > "HOLLight.hollight.CARD_GE_TRANS"
- "CARD_GE_REFL" > "HOLLight.hollight.CARD_GE_REFL"
"CARD_FUNSPACE" > "HOLLight.hollight.CARD_FUNSPACE"
"CARD_FINITE_IMAGE" > "HOLLight.hollight.CARD_FINITE_IMAGE"
- "CARD_EQ_def" > "HOLLight.hollight.CARD_EQ_def"
"CARD_EQ_SUM" > "HOLLight.hollight.CARD_EQ_SUM"
"CARD_EQ_NSUM" > "HOLLight.hollight.CARD_EQ_NSUM"
+ "CARD_EQ_BIJECTIONS" > "HOLLight.hollight.CARD_EQ_BIJECTIONS"
+ "CARD_EQ_BIJECTION" > "HOLLight.hollight.CARD_EQ_BIJECTION"
"CARD_EQ_0" > "HOLLight.hollight.CARD_EQ_0"
+ "CARD_DIFF" > "HOLLight.hollight.CARD_DIFF"
"CARD_DELETE" > "HOLLight.hollight.CARD_DELETE"
+ "CARD_CROSS" > "HOLLight.hollight.CARD_CROSS"
"CARD_CLAUSES" > "HOLLight.hollight.CARD_CLAUSES"
"BOUNDS_NOTZERO" > "HOLLight.hollight.BOUNDS_NOTZERO"
"BOUNDS_LINEAR_0" > "HOLLight.hollight.BOUNDS_LINEAR_0"
@@ -1888,12 +2196,19 @@
"BOUNDS_IGNORE" > "HOLLight.hollight.BOUNDS_IGNORE"
"BOUNDS_DIVIDED" > "HOLLight.hollight.BOUNDS_DIVIDED"
"BOTTOM_def" > "HOLLight.hollight.BOTTOM_def"
- "BOOL_CASES_AX" > "Datatype.bool.nchotomy"
- "BIT1_THM" > "HOLLightCompat.NUMERAL_BIT1_altdef"
- "BIT0_THM" > "HOLLightCompat.NUMERAL_BIT0_def"
+ "BOOL_CASES_AX" > "HOL.True_or_False"
+ "BIT1_THM" > "HOLLight.hollight.BIT1_THM"
+ "BIT1" > "HOLLight.hollight.BIT1"
+ "BIT0_THM" > "Int.semiring_mult_2"
+ "BIT0" > "Int.semiring_mult_2"
"BIJ_def" > "HOLLight.hollight.BIJ_def"
- "BETA_THM" > "Presburger.fm_modd_pinf"
- "ASSOC_def" > "HOLLight.hollight.ASSOC_def"
+ "BIJECTIVE_ON_LEFT_RIGHT_INVERSE" > "HOLLight.hollight.BIJECTIVE_ON_LEFT_RIGHT_INVERSE"
+ "BIJECTIVE_LEFT_RIGHT_INVERSE" > "HOLLight.hollight.BIJECTIVE_LEFT_RIGHT_INVERSE"
+ "BIJECTIONS_HAS_SIZE_EQ" > "HOLLight.hollight.BIJECTIONS_HAS_SIZE_EQ"
+ "BIJECTIONS_HAS_SIZE" > "HOLLight.hollight.BIJECTIONS_HAS_SIZE"
+ "BIJECTIONS_CARD_EQ" > "HOLLight.hollight.BIJECTIONS_CARD_EQ"
+ "BETA_THM" > "HOL.eta_contract_eq"
+ "ASCII_def" > "HOLLight.hollight.ASCII_def"
"ARITH_ZERO" > "HOLLight.hollight.ARITH_ZERO"
"ARITH_SUC" > "HOLLight.hollight.ARITH_SUC"
"ARITH_SUB" > "HOLLight.hollight.ARITH_SUB"
@@ -1906,37 +2221,42 @@
"ARITH_EVEN" > "HOLLight.hollight.ARITH_EVEN"
"ARITH_EQ" > "HOLLight.hollight.ARITH_EQ"
"ARITH_ADD" > "HOLLight.hollight.ARITH_ADD"
- "APPEND_def" > "HOLLight.hollight.APPEND_def"
- "APPEND_NIL" > "HOLLight.hollight.APPEND_NIL"
- "APPEND_EQ_NIL" > "HOLLight.hollight.APPEND_EQ_NIL"
- "APPEND_ASSOC" > "HOLLight.hollight.APPEND_ASSOC"
+ "APPEND_NIL" > "List.append_Nil2"
+ "APPEND_EQ_NIL" > "List.append_is_Nil_conv"
+ "APPEND_BUTLAST_LAST" > "List.append_butlast_last_id"
+ "APPEND_ASSOC" > "List.append_assoc"
"AND_FORALL_THM" > "HOL.all_conj_distrib"
"AND_CLAUSES" > "HOLLight.hollight.AND_CLAUSES"
- "AND_ALL2" > "HOLLight.hollight.AND_ALL2"
- "AND_ALL" > "HOLLight.hollight.AND_ALL"
- "ALL_list_def" > "HOLLight.hollight.ALL_list_def"
- "ALL_T" > "HOLLight.hollight.ALL_T"
- "ALL_MP" > "HOLLight.hollight.ALL_MP"
- "ALL_MEM" > "HOLLight.hollight.ALL_MEM"
- "ALL_MAP" > "HOLLight.hollight.ALL_MAP"
- "ALL_IMP" > "HOLLight.hollight.ALL_IMP"
- "ALL_APPEND" > "HOLLight.hollight.ALL_APPEND"
- "ALL2_def" > "HOLLight.hollight.ALL2_def"
- "ALL2_MAP2" > "HOLLight.hollight.ALL2_MAP2"
- "ALL2_MAP" > "HOLLight.hollight.ALL2_MAP"
- "ALL2_AND_RIGHT" > "HOLLight.hollight.ALL2_AND_RIGHT"
- "ALL2_ALL" > "HOLLight.hollight.ALL2_ALL"
- "ALL2" > "HOLLight.hollight.ALL2"
+ "AND_ALL2" > "HOLLightList.AND_ALL2"
+ "AND_ALL" > "HOLLightList.AND_ALL"
+ "ALL_T" > "HOLLightList.ALL_T"
+ "ALL_MP" > "HOLLightList.ALL_MP"
+ "ALL_MEM" > "HOLLightList.ALL_MEM"
+ "ALL_IMP" > "HOLLightList.ALL_IMP"
+ "ALL_EL" > "List.list_all_length"
+ "ALL_APPEND" > "List.list_all_append"
+ "ALL2_MAP2" > "HOLLightList.ALL2_MAP2"
+ "ALL2_MAP" > "HOLLightList.ALL2_MAP"
+ "ALL2_AND_RIGHT" > "HOLLightList.ALL2_AND_RIGHT"
+ "ALL2_ALL" > "HOLLightList.ALL2_ALL"
+ "ALL2" > "HOLLightList.ALL2"
+ "ADMISSIBLE_UNGUARDED_PATTERN" > "HOLLight.hollight.ADMISSIBLE_UNGUARDED_PATTERN"
"ADMISSIBLE_SUM" > "HOLLight.hollight.ADMISSIBLE_SUM"
+ "ADMISSIBLE_SEQPATTERN" > "HOLLight.hollight.ADMISSIBLE_SEQPATTERN"
+ "ADMISSIBLE_RAND" > "HOLLight.hollight.ADMISSIBLE_RAND"
"ADMISSIBLE_NSUM" > "HOLLight.hollight.ADMISSIBLE_NSUM"
"ADMISSIBLE_NEST" > "HOLLight.hollight.ADMISSIBLE_NEST"
+ "ADMISSIBLE_MATCH_SEQPATTERN" > "HOLLight.hollight.ADMISSIBLE_MATCH_SEQPATTERN"
+ "ADMISSIBLE_MATCH" > "HOLLight.hollight.ADMISSIBLE_MATCH"
+ "ADMISSIBLE_MAP" > "HOLLight.hollight.ADMISSIBLE_MAP"
"ADMISSIBLE_LAMBDA" > "HOLLight.hollight.ADMISSIBLE_LAMBDA"
"ADMISSIBLE_IMP_SUPERADMISSIBLE" > "HOLLight.hollight.ADMISSIBLE_IMP_SUPERADMISSIBLE"
+ "ADMISSIBLE_GUARDED_PATTERN" > "HOLLight.hollight.ADMISSIBLE_GUARDED_PATTERN"
"ADMISSIBLE_CONST" > "HOLLight.hollight.ADMISSIBLE_CONST"
"ADMISSIBLE_COND" > "HOLLight.hollight.ADMISSIBLE_COND"
"ADMISSIBLE_COMB" > "HOLLight.hollight.ADMISSIBLE_COMB"
"ADMISSIBLE_BASE" > "HOLLight.hollight.ADMISSIBLE_BASE"
- "ADD_SYM" > "Finite_Set.AC_add.f.AC_2"
+ "ADD_SYM" > "Fields.linordered_field_class.sign_simps_43"
"ADD_SUC" > "Nat.add_Suc_right"
"ADD_SUBR2" > "Nat.diff_add_0"
"ADD_SUBR" > "HOLLight.hollight.ADD_SUBR"
@@ -1944,31 +2264,63 @@
"ADD_SUB" > "Nat.diff_add_inverse2"
"ADD_EQ_0" > "Nat.add_is_0"
"ADD_CLAUSES" > "HOLLight.hollight.ADD_CLAUSES"
- "ADD_ASSOC" > "Finite_Set.AC_add.f.AC_1"
+ "ADD_ASSOC" > "Fields.linordered_field_class.sign_simps_44"
"ADD_AC" > "HOLLight.hollight.ADD_AC"
- "ADD_0" > "Presburger.add_right_zero"
- "ADD1" > "HOLLight.hollight.ADD1"
- "ABS_SIMP" > "Presburger.fm_modd_pinf"
+ "ADD_0" > "Divides.arithmetic_simps_39"
+ "ADD1" > "Nat_Numeral.Suc_eq_plus1"
+ "ABS_SIMP" > "HOL.refl"
"ABSORPTION" > "HOLLight.hollight.ABSORPTION"
- ">_def" > "HOLLight.hollight.>_def"
- ">=_def" > "HOLLight.hollight.>=_def"
- "<_def" > "HOLLight.hollight.<_def"
- "<=_def" > "HOLLight.hollight.<=_def"
+ ">_c_def" > "HOLLight.hollight.>_c_def"
+ ">=_c_def" > "HOLLight.hollight.>=_c_def"
+ "=_c_def" > "HOLLight.hollight.=_c_def"
+ "<_c_def" > "HOLLight.hollight.<_c_def"
+ "<=_c_def" > "HOLLight.hollight.<=_c_def"
"$_def" > "HOLLight.hollight.$_def"
ignore_thms
"TYDEF_sum"
"TYDEF_prod"
+ "TYDEF_option"
"TYDEF_num"
+ "TYDEF_list"
"TYDEF_1"
- "IND_SUC_0_EXISTS"
+ "SNDCART_PASTECART"
+ "SET_OF_LIST_OF_SET"
+ "REP_ABS_PAIR"
+ "PASTECART_FST_SND"
+ "PASTECART_EQ"
+ "MEM_LIST_OF_SET"
+ "MEM_ASSOC"
+ "LIST_OF_SET_PROPERTIES"
+ "LENGTH_LIST_OF_SET"
+ "FSTCART_PASTECART"
+ "FORALL_PASTECART"
+ "FINITE_SET_OF_LIST"
+ "EX_MAP"
+ "EXISTS_PASTECART"
+ "EL_TL"
+ "DIMINDEX_HAS_SIZE_FINITE_SUM"
+ "DIMINDEX_FINITE_SUM"
"DEF_one"
+ "DEF_mk_pair"
+ "DEF_list_of_set"
"DEF__0"
+ "DEF_ZIP"
+ "DEF_TL"
"DEF_SUC"
- "DEF_NUM_REP"
+ "DEF_SOME"
+ "DEF_OUTR"
+ "DEF_OUTL"
+ "DEF_NONE"
+ "DEF_NIL"
+ "DEF_MAP2"
+ "DEF_ITLIST2"
"DEF_INR"
"DEF_INL"
- "DEF_IND_SUC"
- "DEF_IND_0"
+ "DEF_HD"
+ "DEF_CONS"
+ "DEF_ASSOC"
+ "DEF_,"
+ "ALL_MAP"
end