(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
theory HOLLight imports "../HOLLightCompat" "../HOL4Syntax" begin
;setup_theory hollight
consts
"_FALSITY_" :: "bool" ("'_FALSITY'_")
defs
"_FALSITY__def": "_FALSITY_ == False"
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) &
(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) &
(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)"
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"
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"
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)"
by (import hollight IMP_CLAUSES)
lemma IMP_EQ_CLAUSE: "((x::'q_864::type) = 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))"
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)"
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))"
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)"
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))"
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))"
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))"
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"
by (import hollight SELECT_UNIQUE)
lemma EXCLUDED_MIDDLE: "ALL t::bool. t | ~ t"
by (import hollight EXCLUDED_MIDDLE)
constdefs
COND :: "bool => 'A => 'A => 'A"
"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"
by (import hollight COND_CLAUSES)
lemma COND_EXPAND: "ALL (b::bool) (t1::bool) t2::bool. COND b t1 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)"
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"
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"
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))"
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))"
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"
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"
by (import hollight o_ASSOC)
lemma I_O_ID: "ALL f::'A::type => 'B::type. id o f = f & f o id = f"
by (import hollight I_O_ID)
lemma EXISTS_ONE_REP: "EX x::bool. x"
by (import hollight EXISTS_ONE_REP)
lemma one_axiom: "ALL f::'A::type => unit. All (op = f)"
by (import hollight one_axiom)
lemma one_RECURSION: "ALL e::'A::type. EX x::unit => 'A::type. x () = e"
by (import hollight one_RECURSION)
lemma one_Axiom: "ALL e::'A::type. EX! fn::unit => 'A::type. fn () = e"
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)"
by (import hollight th_cond)
constdefs
LET_END :: "'A => 'A"
"LET_END == %t::'A::type. t"
lemma DEF_LET_END: "LET_END = (%t::'A::type. t)"
by (import hollight DEF_LET_END)
constdefs
GABS :: "('A => bool) => 'A"
"(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)
constdefs
GEQ :: "'A => 'A => bool"
"(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"
by (import hollight PAIR_EXISTS_THM)
constdefs
CURRY :: "('A * 'B => 'C) => 'A => 'B => 'C"
"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)
constdefs
UNCURRY :: "('A => 'B => 'C) => 'A * 'B => 'C"
"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))"
by (import hollight DEF_UNCURRY)
constdefs
PASSOC :: "(('A * 'B) * 'C => 'D) => 'A * 'B * 'C => 'D"
"PASSOC ==
%(u::('A::type * 'B::type) * 'C::type => 'D::type)
ua::'A::type * 'B::type * 'C::type.
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 ((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) &
(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)
lemma ADD_AC: "(m::nat) + (n::nat) = n + m &
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)"
by (import hollight EQ_ADD_LCANCEL_0)
lemma EQ_ADD_RCANCEL_0: "ALL (x::nat) xa::nat. (x + xa = xa) = (x = 0)"
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)"
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) &
(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)
lemma MULT_AC: "(m::nat) * (n::nat) = n * m &
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)
constdefs
EXP :: "nat => nat => nat"
"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)"
by (import hollight LT_ANTISYM)
lemma LET_ANTISYM: "ALL (m::nat) n::nat. ~ (<= m n & < n m)"
by (import hollight LET_ANTISYM)
lemma LTE_ANTISYM: "ALL (x::nat) xa::nat. ~ (< x xa & <= 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"
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"
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)"
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"
by (import hollight LT_ADD)
lemma LT_ADDR: "ALL (x::nat) xa::nat. < xa (x + xa) = < 0 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)"
by (import hollight LT_LMULT)
lemma LE_MULT_LCANCEL: "ALL (m::nat) (n::nat) p::nat. <= (m * n) (m * p) = (m = 0 | <= 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)"
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)"
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)"
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)"
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))"
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))"
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))"
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))"
by (import hollight num_MAX)
constdefs
EVEN :: "nat => bool"
"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)
constdefs
ODD :: "nat => bool"
"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"
by (import hollight NOT_EVEN)
lemma NOT_ODD: "ALL n::nat. (~ ODD n) = EVEN n"
by (import hollight NOT_ODD)
lemma EVEN_OR_ODD: "ALL n::nat. EVEN n | ODD n"
by (import hollight EVEN_OR_ODD)
lemma EVEN_AND_ODD: "ALL x::nat. ~ (EVEN x & 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)"
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)"
by (import hollight ODD_MULT)
lemma ODD_EXP: "ALL (m::nat) n::nat. ODD (EXP m n) = (ODD m | n = 0)"
by (import hollight ODD_EXP)
lemma EVEN_DOUBLE: "ALL n::nat. EVEN (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n)"
by (import hollight EVEN_DOUBLE)
lemma ODD_DOUBLE: "ALL x::nat. ODD (Suc (NUMERAL_BIT0 (NUMERAL_BIT1 0) * 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)))"
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)"
by (import hollight EVEN_ODD_DECOMPOSITION)
lemma SUB_0: "ALL x::nat. 0 - x = 0 & x - 0 = x"
by (import hollight SUB_0)
lemma SUB_PRESUC: "ALL (m::nat) n::nat. Pred (Suc m - n) = 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"
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)
constdefs
FACT :: "nat => nat"
"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)"
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)"
by (import hollight DIVMOD_EXIST_0)
constdefs
DIV :: "nat => nat => nat"
"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)
constdefs
MOD :: "nat => nat => nat"
"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"
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"
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"
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"
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"
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"
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"
by (import hollight DIV_LE)
lemma DIV_MUL_LE: "ALL (m::nat) n::nat. <= (n * DIV m 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"
by (import hollight MOD_MOD)
lemma MOD_MOD_REFL: "ALL (m::nat) n::nat. n ~= 0 --> MOD (MOD m n) n = MOD m 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"
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"
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)"
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)"
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)"
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"
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)"
by (import hollight MOD_EQ_0)
lemma EVEN_MOD: "ALL n::nat. EVEN n = (MOD n (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = 0)"
by (import hollight EVEN_MOD)
lemma ODD_MOD: "ALL n::nat. ODD n = (MOD n (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = NUMERAL_BIT1 0)"
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"
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"
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"
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"
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"
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)"
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"
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)"
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)"
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"
by (import hollight DIV_EQ_EXCLUSION)
lemma SUB_ELIM_THM: "(P::nat => bool) ((a::nat) - (b::nat)) =
(ALL x::nat. (b = a + x --> P 0) & (a = b + x --> P x))"
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))"
by (import hollight DIVMOD_ELIM_THM)
constdefs
eqeq :: "'q_9910 => 'q_9909 => ('q_9910 => 'q_9909 => bool) => bool"
"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_)
constdefs
mod_nat :: "nat => nat => nat => bool"
"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)
constdefs
minimal :: "(nat => bool) => nat"
"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))"
by (import hollight DEF_minimal)
lemma MINIMAL: "ALL P::nat => bool.
Ex P = (P (minimal P) & (ALL x::nat. < x (minimal P) --> ~ P x))"
by (import hollight MINIMAL)
constdefs
WF :: "('A => 'A => bool) => bool"
"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_354::'A::type => 'A::type => bool) =
(ALL P::'A::type => bool.
Ex P = (EX x::'A::type. P x & (ALL y::'A::type. u_354 y x --> ~ P y)))"
by (import hollight WF_EQ)
lemma WF_IND: "WF (u_354::'A::type => 'A::type => bool) =
(ALL P::'A::type => bool.
(ALL x::'A::type. (ALL y::'A::type. u_354 y x --> P y) --> P x) -->
All P)"
by (import hollight WF_IND)
lemma WF_DCHAIN: "WF (u_354::'A::type => 'A::type => bool) =
(~ (EX s::nat => 'A::type. ALL n::nat. u_354 (s (Suc n)) (s n)))"
by (import hollight WF_DCHAIN)
lemma WF_UREC: "WF (u_354::'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_354 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))"
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_354::'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_354"
by (import hollight WF_UREC_WF)
lemma WF_REC_INVARIANT: "WF (u_354::'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_354 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))"
by (import hollight WF_REC_INVARIANT)
lemma WF_REC: "WF (u_354::'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_354 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))"
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_354::'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_354"
by (import hollight WF_REC_WF)
lemma WF_EREC: "WF (u_354::'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_354 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))"
by (import hollight WF_EREC)
lemma WF_SUBSET: "(ALL (x::'A::type) y::'A::type.
(u_354::'A::type => 'A::type => bool) x y -->
(u_473::'A::type => 'A::type => bool) x y) &
WF u_473 -->
WF u_354"
by (import hollight WF_SUBSET)
lemma WF_MEASURE_GEN: "ALL m::'A::type => 'B::type.
WF (u_354::'B::type => 'B::type => bool) -->
WF (%(x::'A::type) x'::'A::type. u_354 (m 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)))))"
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)))))"
by (import hollight WF_LEX)
lemma WF_POINTWISE: "WF (u_354::'A::type => 'A::type => bool) &
WF (u_473::'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_354 x1 x2 & u_473 y1 y2)))))"
by (import hollight WF_POINTWISE)
lemma WF_num: "WF <"
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)"
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)"
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)"
by (import hollight MEASURE_LE)
lemma WF_REFL: "ALL x::'A::type. WF (u_354::'A::type => 'A::type => bool) --> ~ u_354 x 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)"
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_354::'A::type => 'A::type => bool) &
(ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type) x::'A::type.
(ALL z::'A::type. u_354 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_354 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_354 y (G f x) --> u_354 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))"
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))"
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))"
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)"
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) ((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))
((NUMERAL_BIT0::nat => nat) x)))
((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))
((NUMERAL_BIT1::nat => nat) x)))
((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)
(%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)
((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))
((NUMERAL_BIT1::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))
((NUMERAL_BIT1::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))
((NUMERAL_BIT0::nat => nat)
((Suc::nat => nat)
((op +::nat => nat => nat) 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))
((op +::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))
((op +::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))
((op +::nat => nat => nat)
((NUMERAL_BIT1::nat => nat) x)
((op +::nat => nat => nat)
((NUMERAL_BIT0::nat => nat) xa)
((NUMERAL_BIT0::nat => nat)
((NUMERAL_BIT0::nat => nat)
((op *::nat => nat => 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) &
(ALL (m::nat) n::nat.
EXP (NUMERAL_BIT0 m) (NUMERAL_BIT0 n) =
EXP (NUMERAL_BIT0 m) n * EXP (NUMERAL_BIT0 m) n) &
(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)) &
(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))"
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)"
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)"
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)"
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)"
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))))))))))))"
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)))))))))))))"
by (import hollight ARITH_SUB)
lemma right_th: "(s::nat) * NUMERAL_BIT1 (x::nat) = s + NUMERAL_BIT0 (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)"
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)"
by (import hollight NUM_INTEGRAL_LEMMA)
lemma NUM_INTEGRAL: "(ALL x::nat. 0 * x = 0) &
(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)"
by (import hollight INJ_INVERSE2)
constdefs
NUMPAIR :: "nat => nat => nat"
"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))"
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"
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)"
by (import hollight NUMPAIR_INJ)
constdefs
NUMFST :: "nat => nat"
"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)"
by (import hollight DEF_NUMFST)
constdefs
NUMSND :: "nat => nat"
"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)"
by (import hollight DEF_NUMSND)
constdefs
NUMSUM :: "bool => nat => nat"
"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))"
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)"
by (import hollight NUMSUM_INJ)
constdefs
NUMLEFT :: "nat => bool"
"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)"
by (import hollight DEF_NUMLEFT)
constdefs
NUMRIGHT :: "nat => nat"
"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)"
by (import hollight DEF_NUMRIGHT)
constdefs
INJN :: "nat => nat => 'A => bool"
"INJN == %(u::nat) (n::nat) a::'A::type. n = u"
lemma DEF_INJN: "INJN = (%(u::nat) (n::nat) a::'A::type. 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)))"
by (import hollight INJN_INJ)
constdefs
INJA :: "'A => nat => 'A => bool"
"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)"
by (import hollight DEF_INJA)
lemma INJA_INJ: "ALL (a1::'A::type) a2::'A::type. (INJA a1 = INJA a2) = (a1 = a2)"
by (import hollight INJA_INJ)
constdefs
INJF :: "(nat => nat => 'A => bool) => nat => 'A => bool"
"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))"
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)"
by (import hollight INJF_INJ)
constdefs
INJP :: "(nat => 'A => bool) => (nat => 'A => bool) => nat => 'A => bool"
"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)"
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))"
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')"
by (import hollight INJP_INJ)
constdefs
ZCONSTR :: "nat => 'A => (nat => nat => 'A => bool) => nat => 'A => bool"
"ZCONSTR ==
%(u::nat) (ua::'A::type) ub::nat => nat => 'A::type => 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.
INJP (INJN (Suc u)) (INJP (INJA ua) (INJF ub)))"
by (import hollight DEF_ZCONSTR)
constdefs
ZBOT :: "nat => 'A => bool"
"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)"
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"
by (import hollight ZCONSTR_ZBOT)
constdefs
ZRECSPACE :: "(nat => 'A => bool) => bool"
"ZRECSPACE ==
%a::nat => 'A::type => bool.
ALL ZRECSPACE'::(nat => 'A::type => bool) => bool.
(ALL a::nat => 'A::type => bool.
a = ZBOT |
(EX (c::nat) (i::'A::type) r::nat => nat => 'A::type => 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 = ZBOT |
(EX (c::nat) (i::'A::type) r::nat => nat => 'A::type => 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"])
by (import hollight TYDEF_recspace)
syntax
"_dest_rec" :: _ ("'_dest'_rec")
syntax
"_mk_rec" :: _ ("'_mk'_rec")
lemmas "TYDEF_recspace_@intern" = typedef_hol2hollight
[where a="a :: 'A recspace" and r=r ,
OF type_definition_recspace]
constdefs
BOTTOM :: "'A recspace"
"(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))"
by (import hollight DEF_BOTTOM)
constdefs
CONSTR :: "nat => 'A => (nat => 'A recspace) => 'A recspace"
"(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))))"
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))))"
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"
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)"
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"
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))"
by (import hollight CONSTR_REC)
constdefs
FCONS :: "'A => (nat => 'A) => nat => 'A"
"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)"
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))"
by (import hollight DEF_FCONS)
lemma FCONS_UNDO: "ALL f::nat => 'A::type. f = FCONS (f 0) (f o Suc)"
by (import hollight FCONS_UNDO)
constdefs
FNIL :: "nat => 'A"
"FNIL == %u::nat. SOME x::'A::type. True"
lemma DEF_FNIL: "FNIL = (%u::nat. SOME x::'A::type. True)"
by (import hollight DEF_FNIL)
typedef (open) ('A, 'B) sum = "(Collect::(('A::type * 'B::type) recspace => bool)
=> ('A::type * 'B::type) recspace set)
(%a::('A::type * 'B::type) recspace.
(All::((('A::type * 'B::type) recspace => bool) => bool) => bool)
(%sum'::('A::type * 'B::type) recspace => bool.
(op -->::bool => bool => bool)
((All::(('A::type * 'B::type) recspace => bool) => bool)
(%a::('A::type * 'B::type) recspace.
(op -->::bool => bool => bool)
((op |::bool => bool => bool)
((Ex::('A::type => bool) => bool)
(%aa::'A::type.
(op =::('A::type * 'B::type) recspace
=> ('A::type * 'B::type) recspace => bool)
a ((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
((NUMERAL::nat => nat) (0::nat))
((Pair::'A::type
=> 'B::type => 'A::type * 'B::type)
aa ((Eps::('B::type => bool) => 'B::type)
(%v::'B::type. True::bool)))
(%n::nat.
BOTTOM::('A::type *
'B::type) recspace))))
((Ex::('B::type => bool) => bool)
(%aa::'B::type.
(op =::('A::type * 'B::type) recspace
=> ('A::type * 'B::type) recspace => bool)
a ((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
((Suc::nat => nat)
((NUMERAL::nat => nat) (0::nat)))
((Pair::'A::type
=> 'B::type => 'A::type * 'B::type)
((Eps::('A::type => bool) => 'A::type)
(%v::'A::type. True::bool))
aa)
(%n::nat.
BOTTOM::('A::type *
'B::type) recspace)))))
(sum' a)))
(sum' a)))" morphisms "_dest_sum" "_mk_sum"
apply (rule light_ex_imp_nonempty[where t="(CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
((NUMERAL::nat => nat) (0::nat))
((Pair::'A::type => 'B::type => 'A::type * 'B::type) (a::'A::type)
((Eps::('B::type => bool) => 'B::type) (%v::'B::type. True::bool)))
(%n::nat. BOTTOM::('A::type * 'B::type) recspace)"])
by (import hollight TYDEF_sum)
syntax
"_dest_sum" :: _ ("'_dest'_sum")
syntax
"_mk_sum" :: _ ("'_mk'_sum")
lemmas "TYDEF_sum_@intern" = typedef_hol2hollight
[where a="a :: ('A, 'B) sum" and r=r ,
OF type_definition_sum]
constdefs
INL :: "'A => ('A, 'B) sum"
"(op ==::('A::type => ('A::type, 'B::type) sum)
=> ('A::type => ('A::type, 'B::type) sum) => prop)
(INL::'A::type => ('A::type, 'B::type) sum)
(%a::'A::type.
(_mk_sum::('A::type * 'B::type) recspace => ('A::type, 'B::type) sum)
((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
(0::nat)
((Pair::'A::type => 'B::type => 'A::type * 'B::type) a
((Eps::('B::type => bool) => 'B::type)
(%v::'B::type. True::bool)))
(%n::nat. BOTTOM::('A::type * 'B::type) recspace)))"
lemma DEF_INL: "(op =::('A::type => ('A::type, 'B::type) sum)
=> ('A::type => ('A::type, 'B::type) sum) => bool)
(INL::'A::type => ('A::type, 'B::type) sum)
(%a::'A::type.
(_mk_sum::('A::type * 'B::type) recspace => ('A::type, 'B::type) sum)
((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
(0::nat)
((Pair::'A::type => 'B::type => 'A::type * 'B::type) a
((Eps::('B::type => bool) => 'B::type)
(%v::'B::type. True::bool)))
(%n::nat. BOTTOM::('A::type * 'B::type) recspace)))"
by (import hollight DEF_INL)
constdefs
INR :: "'B => ('A, 'B) sum"
"(op ==::('B::type => ('A::type, 'B::type) sum)
=> ('B::type => ('A::type, 'B::type) sum) => prop)
(INR::'B::type => ('A::type, 'B::type) sum)
(%a::'B::type.
(_mk_sum::('A::type * 'B::type) recspace => ('A::type, 'B::type) sum)
((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
((Suc::nat => nat) (0::nat))
((Pair::'A::type => 'B::type => 'A::type * 'B::type)
((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
a)
(%n::nat. BOTTOM::('A::type * 'B::type) recspace)))"
lemma DEF_INR: "(op =::('B::type => ('A::type, 'B::type) sum)
=> ('B::type => ('A::type, 'B::type) sum) => bool)
(INR::'B::type => ('A::type, 'B::type) sum)
(%a::'B::type.
(_mk_sum::('A::type * 'B::type) recspace => ('A::type, 'B::type) sum)
((CONSTR::nat
=> 'A::type * 'B::type
=> (nat => ('A::type * 'B::type) recspace)
=> ('A::type * 'B::type) recspace)
((Suc::nat => nat) (0::nat))
((Pair::'A::type => 'B::type => 'A::type * 'B::type)
((Eps::('A::type => bool) => 'A::type) (%v::'A::type. True::bool))
a)
(%n::nat. BOTTOM::('A::type * 'B::type) recspace)))"
by (import hollight DEF_INR)
consts
OUTL :: "('A, 'B) sum => 'A"
defs
OUTL_def: "hollight.OUTL ==
SOME OUTL::('A::type, 'B::type) sum => 'A::type.
ALL x::'A::type. OUTL (INL x) = x"
lemma DEF_OUTL: "hollight.OUTL =
(SOME OUTL::('A::type, 'B::type) sum => 'A::type.
ALL x::'A::type. OUTL (INL x) = x)"
by (import hollight DEF_OUTL)
consts
OUTR :: "('A, 'B) sum => 'B"
defs
OUTR_def: "hollight.OUTR ==
SOME OUTR::('A::type, 'B::type) sum => 'B::type.
ALL y::'B::type. OUTR (INR y) = y"
lemma DEF_OUTR: "hollight.OUTR =
(SOME OUTR::('A::type, 'B::type) sum => '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)
syntax
"_dest_option" :: _ ("'_dest'_option")
syntax
"_mk_option" :: _ ("'_mk'_option")
lemmas "TYDEF_option_@intern" = typedef_hol2hollight
[where a="a :: 'A hollight.option" and r=r ,
OF type_definition_option]
constdefs
NONE :: "'A hollight.option"
"(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)
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]
constdefs
NIL :: "'A hollight.list"
"(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)
constdefs
CONS :: "'A => 'A hollight.list => 'A hollight.list"
"(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)
constdefs
ISO :: "('A => 'B) => ('B => 'A) => bool"
"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_16585::type => 'q_16582::type)
(g::'q_16582::type => 'q_16585::type) -->
(ALL P::'q_16585::type => bool. All P = (ALL x::'q_16582::type. P (g x))) &
(ALL P::'q_16585::type => bool. Ex P = (EX x::'q_16582::type. P (g x))) &
(ALL (a::'q_16585::type) b::'q_16582::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
"_10288" :: "N_2" ("'_10288")
defs
"_10288_def": "(op ==::N_2 => N_2 => prop) (_10288::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__10288: "(op =::N_2 => N_2 => bool) (_10288::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__10288)
consts
"_10289" :: "N_2" ("'_10289")
defs
"_10289_def": "(op ==::N_2 => N_2 => prop) (_10289::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__10289: "(op =::N_2 => N_2 => bool) (_10289::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__10289)
constdefs
two_1 :: "N_2"
"two_1 == _10288"
lemma DEF_two_1: "two_1 = _10288"
by (import hollight DEF_two_1)
constdefs
two_2 :: "N_2"
"two_2 == _10289"
lemma DEF_two_2: "two_2 = _10289"
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
"_10312" :: "N_3" ("'_10312")
defs
"_10312_def": "(op ==::N_3 => N_3 => prop) (_10312::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__10312: "(op =::N_3 => N_3 => bool) (_10312::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__10312)
consts
"_10313" :: "N_3" ("'_10313")
defs
"_10313_def": "(op ==::N_3 => N_3 => prop) (_10313::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__10313: "(op =::N_3 => N_3 => bool) (_10313::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__10313)
consts
"_10314" :: "N_3" ("'_10314")
defs
"_10314_def": "(op ==::N_3 => N_3 => prop) (_10314::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__10314: "(op =::N_3 => N_3 => bool) (_10314::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__10314)
constdefs
three_1 :: "N_3"
"three_1 == _10312"
lemma DEF_three_1: "three_1 = _10312"
by (import hollight DEF_three_1)
constdefs
three_2 :: "N_3"
"three_2 == _10313"
lemma DEF_three_2: "three_2 = _10313"
by (import hollight DEF_three_2)
constdefs
three_3 :: "N_3"
"three_3 == _10314"
lemma DEF_three_3: "three_3 = _10314"
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)
constdefs
HD :: "'A hollight.list => 'A"
"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)
constdefs
TL :: "'A hollight.list => 'A hollight.list"
"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)
constdefs
APPEND :: "'A hollight.list => 'A hollight.list => 'A hollight.list"
"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)
constdefs
REVERSE :: "'A hollight.list => 'A hollight.list"
"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)
constdefs
LENGTH :: "'A hollight.list => nat"
"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)
constdefs
MAP :: "('A => 'B) => 'A hollight.list => 'B hollight.list"
"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)
constdefs
LAST :: "'A hollight.list => 'A"
"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)
constdefs
REPLICATE :: "nat => 'q_16809 => 'q_16809 hollight.list"
"REPLICATE ==
SOME REPLICATE::nat => 'q_16809::type => 'q_16809::type hollight.list.
(ALL x::'q_16809::type. REPLICATE 0 x = NIL) &
(ALL (n::nat) x::'q_16809::type.
REPLICATE (Suc n) x = CONS x (REPLICATE n x))"
lemma DEF_REPLICATE: "REPLICATE =
(SOME REPLICATE::nat => 'q_16809::type => 'q_16809::type hollight.list.
(ALL x::'q_16809::type. REPLICATE 0 x = NIL) &
(ALL (n::nat) x::'q_16809::type.
REPLICATE (Suc n) x = CONS x (REPLICATE n x)))"
by (import hollight DEF_REPLICATE)
constdefs
NULL :: "'q_16824 hollight.list => bool"
"NULL ==
SOME NULL::'q_16824::type hollight.list => bool.
NULL NIL = True &
(ALL (h::'q_16824::type) t::'q_16824::type hollight.list.
NULL (CONS h t) = False)"
lemma DEF_NULL: "NULL =
(SOME NULL::'q_16824::type hollight.list => bool.
NULL NIL = True &
(ALL (h::'q_16824::type) t::'q_16824::type hollight.list.
NULL (CONS h t) = False))"
by (import hollight DEF_NULL)
constdefs
ALL_list :: "('q_16844 => bool) => 'q_16844 hollight.list => bool"
"ALL_list ==
SOME u::('q_16844::type => bool) => 'q_16844::type hollight.list => bool.
(ALL P::'q_16844::type => bool. u P NIL = True) &
(ALL (h::'q_16844::type) (P::'q_16844::type => bool)
t::'q_16844::type hollight.list. u P (CONS h t) = (P h & u P t))"
lemma DEF_ALL: "ALL_list =
(SOME u::('q_16844::type => bool) => 'q_16844::type hollight.list => bool.
(ALL P::'q_16844::type => bool. u P NIL = True) &
(ALL (h::'q_16844::type) (P::'q_16844::type => bool)
t::'q_16844::type hollight.list. u P (CONS h t) = (P h & u P t)))"
by (import hollight DEF_ALL)
consts
EX :: "('q_16865 => bool) => 'q_16865 hollight.list => bool" ("EX")
defs
EX_def: "EX ==
SOME u::('q_16865::type => bool) => 'q_16865::type hollight.list => bool.
(ALL P::'q_16865::type => bool. u P NIL = False) &
(ALL (h::'q_16865::type) (P::'q_16865::type => bool)
t::'q_16865::type hollight.list. u P (CONS h t) = (P h | u P t))"
lemma DEF_EX: "EX =
(SOME u::('q_16865::type => bool) => 'q_16865::type hollight.list => bool.
(ALL P::'q_16865::type => bool. u P NIL = False) &
(ALL (h::'q_16865::type) (P::'q_16865::type => bool)
t::'q_16865::type hollight.list. u P (CONS h t) = (P h | u P t)))"
by (import hollight DEF_EX)
constdefs
ITLIST :: "('q_16888 => 'q_16887 => 'q_16887)
=> 'q_16888 hollight.list => 'q_16887 => 'q_16887"
"ITLIST ==
SOME ITLIST::('q_16888::type => 'q_16887::type => 'q_16887::type)
=> 'q_16888::type hollight.list
=> 'q_16887::type => 'q_16887::type.
(ALL (f::'q_16888::type => 'q_16887::type => 'q_16887::type)
b::'q_16887::type. ITLIST f NIL b = b) &
(ALL (h::'q_16888::type)
(f::'q_16888::type => 'q_16887::type => 'q_16887::type)
(t::'q_16888::type hollight.list) b::'q_16887::type.
ITLIST f (CONS h t) b = f h (ITLIST f t b))"
lemma DEF_ITLIST: "ITLIST =
(SOME ITLIST::('q_16888::type => 'q_16887::type => 'q_16887::type)
=> 'q_16888::type hollight.list
=> 'q_16887::type => 'q_16887::type.
(ALL (f::'q_16888::type => 'q_16887::type => 'q_16887::type)
b::'q_16887::type. ITLIST f NIL b = b) &
(ALL (h::'q_16888::type)
(f::'q_16888::type => 'q_16887::type => 'q_16887::type)
(t::'q_16888::type hollight.list) b::'q_16887::type.
ITLIST f (CONS h t) b = f h (ITLIST f t b)))"
by (import hollight DEF_ITLIST)
constdefs
MEM :: "'q_16913 => 'q_16913 hollight.list => bool"
"MEM ==
SOME MEM::'q_16913::type => 'q_16913::type hollight.list => bool.
(ALL x::'q_16913::type. MEM x NIL = False) &
(ALL (h::'q_16913::type) (x::'q_16913::type)
t::'q_16913::type hollight.list.
MEM x (CONS h t) = (x = h | MEM x t))"
lemma DEF_MEM: "MEM =
(SOME MEM::'q_16913::type => 'q_16913::type hollight.list => bool.
(ALL x::'q_16913::type. MEM x NIL = False) &
(ALL (h::'q_16913::type) (x::'q_16913::type)
t::'q_16913::type hollight.list.
MEM x (CONS h t) = (x = h | MEM x t)))"
by (import hollight DEF_MEM)
constdefs
ALL2 :: "('q_16946 => 'q_16953 => bool)
=> 'q_16946 hollight.list => 'q_16953 hollight.list => bool"
"ALL2 ==
SOME ALL2::('q_16946::type => 'q_16953::type => bool)
=> 'q_16946::type hollight.list
=> 'q_16953::type hollight.list => bool.
(ALL (P::'q_16946::type => 'q_16953::type => bool)
l2::'q_16953::type hollight.list. ALL2 P NIL l2 = (l2 = NIL)) &
(ALL (h1::'q_16946::type) (P::'q_16946::type => 'q_16953::type => bool)
(t1::'q_16946::type hollight.list) l2::'q_16953::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_16946::type => 'q_16953::type => bool)
=> 'q_16946::type hollight.list
=> 'q_16953::type hollight.list => bool.
(ALL (P::'q_16946::type => 'q_16953::type => bool)
l2::'q_16953::type hollight.list. ALL2 P NIL l2 = (l2 = NIL)) &
(ALL (h1::'q_16946::type) (P::'q_16946::type => 'q_16953::type => bool)
(t1::'q_16946::type hollight.list) l2::'q_16953::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_17008::type => 'q_17007::type => bool) NIL NIL = True &
ALL2 P (CONS (h1::'q_17008::type) (t1::'q_17008::type hollight.list)) NIL =
False &
ALL2 P NIL (CONS (h2::'q_17007::type) (t2::'q_17007::type hollight.list)) =
False &
ALL2 P (CONS h1 t1) (CONS h2 t2) = (P h1 h2 & ALL2 P t1 t2)"
by (import hollight ALL2)
constdefs
MAP2 :: "('q_17038 => 'q_17045 => 'q_17035)
=> 'q_17038 hollight.list
=> 'q_17045 hollight.list => 'q_17035 hollight.list"
"MAP2 ==
SOME MAP2::('q_17038::type => 'q_17045::type => 'q_17035::type)
=> 'q_17038::type hollight.list
=> 'q_17045::type hollight.list
=> 'q_17035::type hollight.list.
(ALL (f::'q_17038::type => 'q_17045::type => 'q_17035::type)
l::'q_17045::type hollight.list. MAP2 f NIL l = NIL) &
(ALL (h1::'q_17038::type)
(f::'q_17038::type => 'q_17045::type => 'q_17035::type)
(t1::'q_17038::type hollight.list) l::'q_17045::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_17038::type => 'q_17045::type => 'q_17035::type)
=> 'q_17038::type hollight.list
=> 'q_17045::type hollight.list
=> 'q_17035::type hollight.list.
(ALL (f::'q_17038::type => 'q_17045::type => 'q_17035::type)
l::'q_17045::type hollight.list. MAP2 f NIL l = NIL) &
(ALL (h1::'q_17038::type)
(f::'q_17038::type => 'q_17045::type => 'q_17035::type)
(t1::'q_17038::type hollight.list) l::'q_17045::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_17080::type => 'q_17079::type => 'q_17086::type) NIL NIL = NIL &
MAP2 f (CONS (h1::'q_17080::type) (t1::'q_17080::type hollight.list))
(CONS (h2::'q_17079::type) (t2::'q_17079::type hollight.list)) =
CONS (f h1 h2) (MAP2 f t1 t2)"
by (import hollight MAP2)
constdefs
EL :: "nat => 'q_17106 hollight.list => 'q_17106"
"EL ==
SOME EL::nat => 'q_17106::type hollight.list => 'q_17106::type.
(ALL l::'q_17106::type hollight.list. EL 0 l = HD l) &
(ALL (n::nat) l::'q_17106::type hollight.list.
EL (Suc n) l = EL n (TL l))"
lemma DEF_EL: "EL =
(SOME EL::nat => 'q_17106::type hollight.list => 'q_17106::type.
(ALL l::'q_17106::type hollight.list. EL 0 l = HD l) &
(ALL (n::nat) l::'q_17106::type hollight.list.
EL (Suc n) l = EL n (TL l)))"
by (import hollight DEF_EL)
constdefs
FILTER :: "('q_17131 => bool) => 'q_17131 hollight.list => 'q_17131 hollight.list"
"FILTER ==
SOME FILTER::('q_17131::type => bool)
=> 'q_17131::type hollight.list
=> 'q_17131::type hollight.list.
(ALL P::'q_17131::type => bool. FILTER P NIL = NIL) &
(ALL (h::'q_17131::type) (P::'q_17131::type => bool)
t::'q_17131::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_17131::type => bool)
=> 'q_17131::type hollight.list
=> 'q_17131::type hollight.list.
(ALL P::'q_17131::type => bool. FILTER P NIL = NIL) &
(ALL (h::'q_17131::type) (P::'q_17131::type => bool)
t::'q_17131::type hollight.list.
FILTER P (CONS h t) =
COND (P h) (CONS h (FILTER P t)) (FILTER P t)))"
by (import hollight DEF_FILTER)
constdefs
ASSOC :: "'q_17160 => ('q_17160 * 'q_17154) hollight.list => 'q_17154"
"ASSOC ==
SOME ASSOC::'q_17160::type
=> ('q_17160::type * 'q_17154::type) hollight.list
=> 'q_17154::type.
ALL (h::'q_17160::type * 'q_17154::type) (a::'q_17160::type)
t::('q_17160::type * 'q_17154::type) hollight.list.
ASSOC a (CONS h t) = COND (fst h = a) (snd h) (ASSOC a t)"
lemma DEF_ASSOC: "ASSOC =
(SOME ASSOC::'q_17160::type
=> ('q_17160::type * 'q_17154::type) hollight.list
=> 'q_17154::type.
ALL (h::'q_17160::type * 'q_17154::type) (a::'q_17160::type)
t::('q_17160::type * 'q_17154::type) hollight.list.
ASSOC a (CONS h t) = COND (fst h = a) (snd h) (ASSOC a t))"
by (import hollight DEF_ASSOC)
constdefs
ITLIST2 :: "('q_17184 => 'q_17192 => 'q_17182 => 'q_17182)
=> 'q_17184 hollight.list => 'q_17192 hollight.list => 'q_17182 => 'q_17182"
"ITLIST2 ==
SOME ITLIST2::('q_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
=> 'q_17184::type hollight.list
=> 'q_17192::type hollight.list
=> 'q_17182::type => 'q_17182::type.
(ALL (f::'q_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
(l2::'q_17192::type hollight.list) b::'q_17182::type.
ITLIST2 f NIL l2 b = b) &
(ALL (h1::'q_17184::type)
(f::'q_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
(t1::'q_17184::type hollight.list) (l2::'q_17192::type hollight.list)
b::'q_17182::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_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
=> 'q_17184::type hollight.list
=> 'q_17192::type hollight.list
=> 'q_17182::type => 'q_17182::type.
(ALL (f::'q_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
(l2::'q_17192::type hollight.list) b::'q_17182::type.
ITLIST2 f NIL l2 b = b) &
(ALL (h1::'q_17184::type)
(f::'q_17184::type
=> 'q_17192::type => 'q_17182::type => 'q_17182::type)
(t1::'q_17184::type hollight.list)
(l2::'q_17192::type hollight.list) b::'q_17182::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_17226::type => 'q_17225::type => 'q_17224::type => 'q_17224::type)
NIL NIL (b::'q_17224::type) =
b &
ITLIST2 f (CONS (h1::'q_17226::type) (t1::'q_17226::type hollight.list))
(CONS (h2::'q_17225::type) (t2::'q_17225::type hollight.list)) b =
f h1 h2 (ITLIST2 f t1 t2 b)"
by (import hollight ITLIST2)
consts
ZIP :: "'q_17256 hollight.list
=> 'q_17264 hollight.list => ('q_17256 * 'q_17264) hollight.list"
defs
ZIP_def: "hollight.ZIP ==
SOME ZIP::'q_17256::type hollight.list
=> 'q_17264::type hollight.list
=> ('q_17256::type * 'q_17264::type) hollight.list.
(ALL l2::'q_17264::type hollight.list. ZIP NIL l2 = NIL) &
(ALL (h1::'q_17256::type) (t1::'q_17256::type hollight.list)
l2::'q_17264::type hollight.list.
ZIP (CONS h1 t1) l2 = CONS (h1, HD l2) (ZIP t1 (TL l2)))"
lemma DEF_ZIP: "hollight.ZIP =
(SOME ZIP::'q_17256::type hollight.list
=> 'q_17264::type hollight.list
=> ('q_17256::type * 'q_17264::type) hollight.list.
(ALL l2::'q_17264::type hollight.list. ZIP NIL l2 = NIL) &
(ALL (h1::'q_17256::type) (t1::'q_17256::type hollight.list)
l2::'q_17264::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_17275::type * 'q_17276::type) hollight.list
=> ('q_17275::type * 'q_17276::type) hollight.list => bool)
((hollight.ZIP::'q_17275::type hollight.list
=> 'q_17276::type hollight.list
=> ('q_17275::type * 'q_17276::type) hollight.list)
(NIL::'q_17275::type hollight.list)
(NIL::'q_17276::type hollight.list))
(NIL::('q_17275::type * 'q_17276::type) hollight.list))
((op =::('q_17300::type * 'q_17301::type) hollight.list
=> ('q_17300::type * 'q_17301::type) hollight.list => bool)
((hollight.ZIP::'q_17300::type hollight.list
=> 'q_17301::type hollight.list
=> ('q_17300::type * 'q_17301::type) hollight.list)
((CONS::'q_17300::type
=> 'q_17300::type hollight.list
=> 'q_17300::type hollight.list)
(h1::'q_17300::type) (t1::'q_17300::type hollight.list))
((CONS::'q_17301::type
=> 'q_17301::type hollight.list
=> 'q_17301::type hollight.list)
(h2::'q_17301::type) (t2::'q_17301::type hollight.list)))
((CONS::'q_17300::type * 'q_17301::type
=> ('q_17300::type * 'q_17301::type) hollight.list
=> ('q_17300::type * 'q_17301::type) hollight.list)
((Pair::'q_17300::type
=> 'q_17301::type => 'q_17300::type * 'q_17301::type)
h1 h2)
((hollight.ZIP::'q_17300::type hollight.list
=> 'q_17301::type hollight.list
=> ('q_17300::type * 'q_17301::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_17608::type hollight.list) n::nat.
(LENGTH l = Suc n) =
(EX (h::'q_17608::type) t::'q_17608::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_17672::type => 'q_17683::type)
(g::'q_17672::type => 'q_17683::type) l::'q_17672::type hollight.list.
ALL_list (%x::'q_17672::type. f x = g x) l --> MAP f l = MAP g l"
by (import hollight MAP_EQ)
lemma ALL_IMP: "ALL (P::'q_17713::type => bool) (Q::'q_17713::type => bool)
l::'q_17713::type hollight.list.
(ALL x::'q_17713::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_17741::type => bool) l::'q_17741::type hollight.list.
(~ EX P l) = ALL_list (%x::'q_17741::type. ~ P x) l"
by (import hollight NOT_EX)
lemma NOT_ALL: "ALL (P::'q_17763::type => bool) l::'q_17763::type hollight.list.
(~ ALL_list P l) = EX (%x::'q_17763::type. ~ P x) l"
by (import hollight NOT_ALL)
lemma ALL_MAP: "ALL (P::'q_17785::type => bool) (f::'q_17784::type => 'q_17785::type)
l::'q_17784::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_17803::type. True))"
by (import hollight ALL_T)
lemma MAP_EQ_ALL2: "ALL (l::'q_17828::type hollight.list) m::'q_17828::type hollight.list.
ALL2
(%(x::'q_17828::type) y::'q_17828::type.
(f::'q_17828::type => 'q_17839::type) x = f y)
l m -->
MAP f l = MAP f m"
by (import hollight MAP_EQ_ALL2)
lemma ALL2_MAP: "ALL (P::'q_17870::type => 'q_17871::type => bool)
(f::'q_17871::type => 'q_17870::type) l::'q_17871::type hollight.list.
ALL2 P (MAP f l) l = ALL_list (%a::'q_17871::type. P (f a) a) l"
by (import hollight ALL2_MAP)
lemma MAP_EQ_DEGEN: "ALL (l::'q_17888::type hollight.list) f::'q_17888::type => 'q_17888::type.
ALL_list (%x::'q_17888::type. f x = x) l --> MAP f l = l"
by (import hollight MAP_EQ_DEGEN)
lemma ALL2_AND_RIGHT: "ALL (l::'q_17931::type hollight.list) (m::'q_17930::type hollight.list)
(P::'q_17931::type => bool) Q::'q_17931::type => 'q_17930::type => bool.
ALL2 (%(x::'q_17931::type) y::'q_17930::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_17968::type hollight.list.
ITLIST (f::'q_17968::type => 'q_17967::type => 'q_17967::type)
(APPEND l (CONS (a::'q_17968::type) NIL)) (b::'q_17967::type) =
ITLIST f l (f a b)"
by (import hollight ITLIST_EXTRA)
lemma ALL_MP: "ALL (P::'q_17994::type => bool) (Q::'q_17994::type => bool)
l::'q_17994::type hollight.list.
ALL_list (%x::'q_17994::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_18024::type hollight.list.
(ALL_list (P::'q_18024::type => bool) x &
ALL_list (Q::'q_18024::type => bool) x) =
ALL_list (%x::'q_18024::type. P x & Q x) x"
by (import hollight AND_ALL)
lemma EX_IMP: "ALL (P::'q_18054::type => bool) (Q::'q_18054::type => bool)
l::'q_18054::type hollight.list.
(ALL x::'q_18054::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_18081::type => bool) l::'q_18081::type hollight.list.
(ALL x::'q_18081::type. MEM x l --> P x) = ALL_list P l"
by (import hollight ALL_MEM)
lemma LENGTH_REPLICATE: "ALL (n::nat) x::'q_18099::type. LENGTH (REPLICATE n x) = n"
by (import hollight LENGTH_REPLICATE)
lemma EX_MAP: "ALL (P::'q_18123::type => bool) (f::'q_18122::type => 'q_18123::type)
l::'q_18122::type hollight.list. EX P (MAP f l) = EX (P o f) l"
by (import hollight EX_MAP)
lemma EXISTS_EX: "ALL (P::'q_18161::type => 'q_18160::type => bool)
l::'q_18160::type hollight.list.
(EX x::'q_18161::type. EX (P x) l) =
EX (%s::'q_18160::type. EX x::'q_18161::type. P x s) l"
by (import hollight EXISTS_EX)
lemma FORALL_ALL: "ALL (P::'q_18191::type => 'q_18190::type => bool)
l::'q_18190::type hollight.list.
(ALL x::'q_18191::type. ALL_list (P x) l) =
ALL_list (%s::'q_18190::type. ALL x::'q_18191::type. P x s) l"
by (import hollight FORALL_ALL)
lemma MEM_APPEND: "ALL (x::'q_18219::type) (l1::'q_18219::type hollight.list)
l2::'q_18219::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_18255::type => 'q_18252::type) (y::'q_18252::type)
l::'q_18255::type hollight.list.
MEM y (MAP f l) = (EX x::'q_18255::type. MEM x l & y = f x)"
by (import hollight MEM_MAP)
lemma FILTER_APPEND: "ALL (P::'q_18286::type => bool) (l1::'q_18286::type hollight.list)
l2::'q_18286::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_18313::type => bool) (f::'q_18320::type => 'q_18313::type)
l::'q_18320::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_18347::type => bool) (l::'q_18347::type hollight.list)
x::'q_18347::type. MEM x (FILTER P l) = (P x & MEM x l)"
by (import hollight MEM_FILTER)
lemma EX_MEM: "ALL (P::'q_18368::type => bool) l::'q_18368::type hollight.list.
(EX x::'q_18368::type. P x & MEM x l) = EX P l"
by (import hollight EX_MEM)
lemma MAP_FST_ZIP: "ALL (l1::'q_18388::type hollight.list) l2::'q_18390::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_18419::type hollight.list) l2::'q_18421::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_18465::type * 'q_18449::type) hollight.list) x::'q_18465::type.
MEM (x, ASSOC x l) l = MEM x (MAP fst l)"
by (import hollight MEM_ASSOC)
lemma ALL_APPEND: "ALL (P::'q_18486::type => bool) (l1::'q_18486::type hollight.list)
l2::'q_18486::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_18509::type hollight.list) n::nat.
< n (LENGTH l) --> MEM (EL n l) l"
by (import hollight MEM_EL)
lemma ALL2_MAP2: "ALL (l::'q_18552::type hollight.list) m::'q_18553::type hollight.list.
ALL2 (P::'q_18551::type => 'q_18550::type => bool)
(MAP (f::'q_18552::type => 'q_18551::type) l)
(MAP (g::'q_18553::type => 'q_18550::type) m) =
ALL2 (%(x::'q_18552::type) y::'q_18553::type. P (f x) (g y)) l m"
by (import hollight ALL2_MAP2)
lemma AND_ALL2: "ALL (P::'q_18599::type => 'q_18598::type => bool)
(Q::'q_18599::type => 'q_18598::type => bool)
(x::'q_18599::type hollight.list) xa::'q_18598::type hollight.list.
(ALL2 P x xa & ALL2 Q x xa) =
ALL2 (%(x::'q_18599::type) y::'q_18598::type. P x y & Q x y) x xa"
by (import hollight AND_ALL2)
lemma ALL2_ALL: "ALL (P::'q_18621::type => 'q_18621::type => bool)
l::'q_18621::type hollight.list.
ALL2 P l l = ALL_list (%x::'q_18621::type. P x x) l"
by (import hollight ALL2_ALL)
lemma APPEND_EQ_NIL: "ALL (x::'q_18650::type hollight.list) xa::'q_18650::type hollight.list.
(APPEND x xa = NIL) = (x = NIL & xa = NIL)"
by (import hollight APPEND_EQ_NIL)
lemma LENGTH_MAP2: "ALL (f::'q_18670::type => 'q_18672::type => 'q_18683::type)
(l::'q_18670::type hollight.list) m::'q_18672::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)
constdefs
dist :: "nat * nat => nat"
"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))"
by (import hollight DEF_dist)
lemma DIST_REFL: "ALL x::nat. dist (x, x) = 0"
by (import hollight DIST_REFL)
lemma DIST_LZERO: "ALL x::nat. dist (0, x) = x"
by (import hollight DIST_LZERO)
lemma DIST_RZERO: "ALL x::nat. dist (x, 0) = x"
by (import hollight DIST_RZERO)
lemma DIST_SYM: "ALL (x::nat) xa::nat. dist (x, xa) = 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)"
by (import hollight DIST_LADD)
lemma DIST_RADD: "ALL (x::nat) (xa::nat) xb::nat. dist (x + xa, xb + xa) = dist (x, xb)"
by (import hollight DIST_RADD)
lemma DIST_LADD_0: "ALL (x::nat) xa::nat. 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"
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)"
by (import hollight DIST_LMUL)
lemma DIST_RMUL: "ALL (x::nat) (xa::nat) xb::nat. dist (x, xa) * xb = 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)"
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))"
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))"
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"
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))"
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"
by (import hollight BOUNDS_LINEAR)
lemma BOUNDS_LINEAR_0: "ALL (A::nat) B::nat. (ALL n::nat. <= (A * n) B) = (A = 0)"
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))"
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)))"
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))"
by (import hollight BOUNDS_IGNORE)
constdefs
is_nadd :: "(nat => nat) => bool"
"is_nadd ==
%u::nat => nat.
EX B::nat.
ALL (m::nat) n::nat. <= (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)))"
by (import hollight DEF_is_nadd)
lemma is_nadd_0: "is_nadd (%n::nat. 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"])
by (import hollight TYDEF_nadd)
syntax
dest_nadd :: _
syntax
mk_nadd :: _
lemmas "TYDEF_nadd_@intern" = typedef_hol2hollight
[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))"
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)"
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)"
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"
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"
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)"
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))"
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)"
by (import hollight NADD_ALTMUL)
constdefs
nadd_eq :: "nadd => nadd => bool"
"nadd_eq ==
%(u::nadd) ua::nadd.
EX B::nat. ALL n::nat. <= (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)"
by (import hollight DEF_nadd_eq)
lemma NADD_EQ_REFL: "ALL x::nadd. 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"
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"
by (import hollight NADD_EQ_TRANS)
constdefs
nadd_of_num :: "nat => nadd"
"nadd_of_num == %u::nat. mk_nadd (op * u)"
lemma DEF_nadd_of_num: "nadd_of_num = (%u::nat. 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"
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)"
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)"
by (import hollight NADD_OF_NUM_EQ)
constdefs
nadd_le :: "nadd => nadd => bool"
"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))"
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'"
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'"
by (import hollight NADD_LE_WELLDEF)
lemma NADD_LE_REFL: "ALL x::nadd. 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"
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"
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))"
by (import hollight NADD_LE_TOTAL_LEMMA)
lemma NADD_LE_TOTAL: "ALL (x::nadd) y::nadd. 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)"
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"
by (import hollight NADD_OF_NUM_LE)
constdefs
nadd_add :: "nadd => nadd => nadd"
"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))"
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)"
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')"
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)"
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)"
by (import hollight NADD_ADD_ASSOC)
lemma NADD_ADD_LID: "ALL x::nadd. 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"
by (import hollight NADD_ADD_LCANCEL)
lemma NADD_LE_ADD: "ALL (x::nadd) y::nadd. 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))"
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))"
by (import hollight NADD_OF_NUM_ADD)
constdefs
nadd_mul :: "nadd => nadd => nadd"
"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)))"
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))"
by (import hollight NADD_MUL)
lemma NADD_MUL_SYM: "ALL (x::nadd) y::nadd. 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)"
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"
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))"
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')"
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')"
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))"
by (import hollight NADD_OF_NUM_MUL)
lemma NADD_LE_0: "All (nadd_le (nadd_of_num 0))"
by (import hollight NADD_LE_0)
lemma NADD_EQ_IMP_LE: "ALL (x::nadd) y::nadd. 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)"
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)"
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"
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"
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))"
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))"
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)"
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"
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'))"
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)"
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)"
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))"
by (import hollight NADD_LBOUND)
constdefs
nadd_rinv :: "nadd => nat => nat"
"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))"
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))"
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)"
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))"
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)))"
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)))"
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))))"
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))))"
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)))"
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))"
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)))"
by (import hollight NADD_MUL_LINV_LEMMA8)
constdefs
nadd_inv :: "nadd => nadd"
"nadd_inv ==
%u::nadd.
COND (nadd_eq u (nadd_of_num 0)) (nadd_of_num 0) (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)))"
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)"
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))"
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)"
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)"])
by (import hollight TYDEF_hreal)
syntax
dest_hreal :: _
syntax
mk_hreal :: _
lemmas "TYDEF_hreal_@intern" = typedef_hol2hollight
[where a="a :: hreal" and r=r ,
OF type_definition_hreal]
constdefs
hreal_of_num :: "nat => hreal"
"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)))"
by (import hollight DEF_hreal_of_num)
constdefs
hreal_add :: "hreal => hreal => hreal"
"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)"
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))"
by (import hollight DEF_hreal_add)
constdefs
hreal_mul :: "hreal => hreal => hreal"
"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)"
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))"
by (import hollight DEF_hreal_mul)
constdefs
hreal_le :: "hreal => hreal => bool"
"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"
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)"
by (import hollight DEF_hreal_le)
constdefs
hreal_inv :: "hreal => hreal"
"hreal_inv ==
%x::hreal.
mk_hreal
(%u::nadd. EX xa::nadd. 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))"
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)"
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)"
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)"
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"
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"
by (import hollight HREAL_LE_ADD_RCANCEL)
lemma HREAL_ADD_RID: "ALL x::hreal. 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)"
by (import hollight HREAL_ADD_RDISTRIB)
lemma HREAL_MUL_LZERO: "ALL m::hreal. 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"
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) &
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)"
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)"
by (import hollight HREAL_LE_MUL_RCANCEL_IMP)
constdefs
treal_of_num :: "nat => hreal * hreal"
"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))"
by (import hollight DEF_treal_of_num)
constdefs
treal_neg :: "hreal * hreal => hreal * hreal"
"treal_neg == %u::hreal * hreal. (snd u, fst u)"
lemma DEF_treal_neg: "treal_neg = (%u::hreal * hreal. (snd u, fst u))"
by (import hollight DEF_treal_neg)
constdefs
treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal"
"treal_add ==
%(u::hreal * hreal) ua::hreal * hreal.
(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)))"
by (import hollight DEF_treal_add)
constdefs
treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal"
"treal_mul ==
%(u::hreal * hreal) ua::hreal * hreal.
(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.
(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)
constdefs
treal_le :: "hreal * hreal => hreal * hreal => bool"
"treal_le ==
%(u::hreal * hreal) ua::hreal * hreal.
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)))"
by (import hollight DEF_treal_le)
constdefs
treal_inv :: "hreal * hreal => hreal * hreal"
"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)))"
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))))"
by (import hollight DEF_treal_inv)
constdefs
treal_eq :: "hreal * hreal => hreal * hreal => bool"
"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))"
by (import hollight DEF_treal_eq)
lemma TREAL_EQ_REFL: "ALL x::hreal * hreal. 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"
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"
by (import hollight TREAL_EQ_TRANS)
lemma TREAL_EQ_AP: "ALL (x::hreal * hreal) y::hreal * hreal. x = y --> treal_eq x y"
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)"
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"
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))"
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))"
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"
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"
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)"
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)"
by (import hollight TREAL_ADD_ASSOC)
lemma TREAL_ADD_LID: "ALL x::hreal * hreal. 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)"
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)"
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)"
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"
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))"
by (import hollight TREAL_ADD_LDISTRIB)
lemma TREAL_LE_REFL: "ALL x::hreal * hreal. 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"
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"
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"
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)"
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)"
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))"
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)"
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)"
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)"
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)"
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)"
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)"
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"
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"
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)"
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)"])
by (import hollight TYDEF_real)
syntax
dest_real :: _
syntax
mk_real :: _
lemmas "TYDEF_real_@intern" = typedef_hol2hollight
[where a="a :: hollight.real" and r=r ,
OF type_definition_real]
constdefs
real_of_num :: "nat => hollight.real"
"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)))"
by (import hollight DEF_real_of_num)
constdefs
real_neg :: "hollight.real => hollight.real"
"real_neg ==
%x1::hollight.real.
mk_real
(%u::hreal * hreal.
EX x1a::hreal * hreal.
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))"
by (import hollight DEF_real_neg)
constdefs
real_add :: "hollight.real => hollight.real => hollight.real"
"real_add ==
%(x1::hollight.real) y1::hollight.real.
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)"
lemma DEF_real_add: "real_add =
(%(x1::hollight.real) y1::hollight.real.
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))"
by (import hollight DEF_real_add)
constdefs
real_mul :: "hollight.real => hollight.real => hollight.real"
"real_mul ==
%(x1::hollight.real) y1::hollight.real.
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)"
lemma DEF_real_mul: "real_mul =
(%(x1::hollight.real) y1::hollight.real.
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))"
by (import hollight DEF_real_mul)
constdefs
real_le :: "hollight.real => hollight.real => bool"
"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"
lemma DEF_real_le: "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)"
by (import hollight DEF_real_le)
constdefs
real_inv :: "hollight.real => hollight.real"
"real_inv ==
%x::hollight.real.
mk_real
(%u::hreal * hreal.
EX xa::hreal * hreal. 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))"
by (import hollight DEF_real_inv)
constdefs
real_sub :: "hollight.real => hollight.real => hollight.real"
"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))"
by (import hollight DEF_real_sub)
constdefs
real_lt :: "hollight.real => hollight.real => bool"
"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)"
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)"
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)"
by (import hollight DEF_real_gt)
constdefs
real_abs :: "hollight.real => hollight.real"
"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))"
by (import hollight DEF_real_abs)
constdefs
real_pow :: "hollight.real => nat => hollight.real"
"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))"
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)))"
by (import hollight DEF_real_pow)
constdefs
real_div :: "hollight.real => hollight.real => hollight.real"
"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))"
by (import hollight DEF_real_div)
constdefs
real_max :: "hollight.real => hollight.real => hollight.real"
"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)"
by (import hollight DEF_real_max)
constdefs
real_min :: "hollight.real => hollight.real => hollight.real"
"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)"
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))"
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))"
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'))"
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'))"
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) &
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"
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)"
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)"
by (import hollight REAL_EQ_ADD_RCANCEL)
lemma REAL_NEG_NEG: "ALL x::hollight.real. 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)"
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)"
by (import hollight REAL_MUL_LNEG)
lemma REAL_ADD_RID: "ALL x::hollight.real. real_add x (real_of_num 0) = x"
by (import hollight REAL_ADD_RID)
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)"
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"
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)"
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)"
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))"
by (import hollight REAL_POW_NEG)
lemma REAL_ABS_NUM: "ALL x::nat. 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"
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"
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"
by (import hollight REAL_LET_TOTAL)
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"
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"
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)"
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)"
by (import hollight REAL_LTE_ANTISYM)
lemma REAL_LT_REFL: "ALL x::hollight.real. ~ real_lt x x"
by (import hollight REAL_LT_REFL)
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)"
by (import hollight REAL_LET_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)"
by (import hollight REAL_ENTIRE)
lemma REAL_POW_2: "ALL x::hollight.real.
real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = 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.
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))"
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))"
by (import hollight REAL_POLY_NEG_CLAUSES)
lemma REAL_OF_NUM_LT: "ALL (x::nat) xa::nat. 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"
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"
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)"
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)"
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) &
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)"
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)"
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)"
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)"
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)"
by (import hollight REAL_EQ_ADD_RCANCEL_0)
lemma REAL_NOT_EQ: "ALL (x::hollight.real) y::hollight.real.
(x ~= y) = (real_lt x y | real_lt y x)"
by (import hollight REAL_NOT_EQ)
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)"
by (import hollight REAL_LET_ANTISYM)
lemma REAL_LT_TOTAL: "ALL (x::hollight.real) y::hollight.real. x = y | real_lt x y | real_lt y x"
by (import hollight REAL_LT_TOTAL)
lemma REAL_LE_01: "real_le (real_of_num 0) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight REAL_LE_01)
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)"
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)"
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)"
by (import hollight REAL_LT_RNEG)
lemma REAL_NEG_EQ_0: "ALL x::hollight.real. (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"
by (import hollight REAL_ADD_SUB)
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"
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"
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"
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"
by (import hollight REAL_LT_ADDL)
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)"
by (import hollight REAL_ADD2_SUB2)
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)"
by (import hollight REAL_LET_ADD2)
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)"
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)"
by (import hollight REAL_EQ_SUB_RADD)
lemma REAL_ADD_SUB2: "ALL (x::hollight.real) y::hollight.real.
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"
by (import hollight REAL_EQ_IMP_LE)
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)"
by (import hollight REAL_DIFFSQ)
lemma REAL_EQ_NEG2: "ALL (x::hollight.real) y::hollight.real. (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"
by (import hollight REAL_LT_NEG2)
lemma REAL_ABS_ZERO: "ALL x::hollight.real. (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)"
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))"
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"
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"
by (import hollight REAL_ABS_TRIANGLE_LT)
lemma REAL_ABS_POS: "ALL x::hollight.real. 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)"
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)"
by (import hollight REAL_ABS_NZ)
lemma REAL_ABS_ABS: "ALL x::hollight.real. 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)"
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"
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"
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)"
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"
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)"
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"
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"
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)"
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)"
by (import hollight REAL_ABS_CIRCLE)
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))"
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"
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)"
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))"
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))"
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)"
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"
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"
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"
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)"
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)"
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)"
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)"
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)"
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)"
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)"
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)"
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"
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"
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) &
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) &
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)"
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)"
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)"
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"
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)"
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)"
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)"
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)"
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)"
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)"
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"
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"
by (import hollight REAL_MUL_RINV_UNIQ)
lemma REAL_INV_INV: "ALL x::hollight.real. 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)"
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)"
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"
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)"
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"
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)"
by (import hollight REAL_LE_INV)
lemma REAL_INV_1: "real_inv (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight REAL_INV_1)
lemma REAL_DIV_1: "ALL x::hollight.real. real_div x (real_of_num (NUMERAL_BIT1 0)) = 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)"
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"
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"
by (import hollight REAL_DIV_LMUL)
lemma REAL_ABS_INV: "ALL x::hollight.real. 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)"
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)"
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"
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)"
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)"
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)"
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)"
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"
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)"
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"
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"
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"
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"
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"
by (import hollight REAL_LE_RCANCEL_IMP)
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"
by (import hollight REAL_LE_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"
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)"
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"
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)"
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)"
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)"
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"
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"
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"
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)"
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)"
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)"
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)"
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))"
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)"
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)"
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)"
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)"
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)"
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)"
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)"
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)"
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)"
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"
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"
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)"
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)"
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)))"
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)))"
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)"
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)"
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))"
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)))"
by (import hollight REAL_LE_SQUARE_ABS)
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))"
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))"
by (import hollight REAL_WLOG_LT)
constdefs
mod_real :: "hollight.real => hollight.real => hollight.real => bool"
"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)
constdefs
DECIMAL :: "nat => nat => hollight.real"
"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))"
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))"
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))"
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))"
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)"
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)"
by (import hollight RAT_LEMMA5)
constdefs
is_int :: "hollight.real => bool"
"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)"])
by (import hollight TYDEF_int)
syntax
dest_int :: _
syntax
mk_int :: _
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)"
by (import hollight dest_int_rep)
constdefs
int_le :: "hollight.int => hollight.int => bool"
"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))"
by (import hollight DEF_int_le)
constdefs
int_lt :: "hollight.int => hollight.int => bool"
"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))"
by (import hollight DEF_int_lt)
constdefs
int_ge :: "hollight.int => hollight.int => bool"
"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))"
by (import hollight DEF_int_ge)
constdefs
int_gt :: "hollight.int => hollight.int => bool"
"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))"
by (import hollight DEF_int_gt)
constdefs
int_of_num :: "nat => hollight.int"
"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))"
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"
by (import hollight int_of_num_th)
constdefs
int_neg :: "hollight.int => hollight.int"
"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)))"
by (import hollight DEF_int_neg)
lemma int_neg_th: "ALL x::hollight.int. dest_int (int_neg x) = real_neg (dest_int x)"
by (import hollight int_neg_th)
constdefs
int_add :: "hollight.int => hollight.int => hollight.int"
"int_add ==
%(u::hollight.int) ua::hollight.int.
mk_int (real_add (dest_int u) (dest_int ua))"
lemma DEF_int_add: "int_add =
(%(u::hollight.int) ua::hollight.int.
mk_int (real_add (dest_int u) (dest_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)"
by (import hollight int_add_th)
constdefs
int_sub :: "hollight.int => hollight.int => hollight.int"
"int_sub ==
%(u::hollight.int) ua::hollight.int.
mk_int (real_sub (dest_int u) (dest_int ua))"
lemma DEF_int_sub: "int_sub =
(%(u::hollight.int) ua::hollight.int.
mk_int (real_sub (dest_int u) (dest_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)"
by (import hollight int_sub_th)
constdefs
int_mul :: "hollight.int => hollight.int => hollight.int"
"int_mul ==
%(u::hollight.int) ua::hollight.int.
mk_int (real_mul (dest_int u) (dest_int ua))"
lemma DEF_int_mul: "int_mul =
(%(u::hollight.int) ua::hollight.int.
mk_int (real_mul (dest_int u) (dest_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)"
by (import hollight int_mul_th)
constdefs
int_abs :: "hollight.int => hollight.int"
"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)))"
by (import hollight DEF_int_abs)
lemma int_abs_th: "ALL x::hollight.int. dest_int (int_abs x) = real_abs (dest_int x)"
by (import hollight int_abs_th)
constdefs
int_max :: "hollight.int => hollight.int => hollight.int"
"int_max ==
%(u::hollight.int) ua::hollight.int.
mk_int (real_max (dest_int u) (dest_int ua))"
lemma DEF_int_max: "int_max =
(%(u::hollight.int) ua::hollight.int.
mk_int (real_max (dest_int u) (dest_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)"
by (import hollight int_max_th)
constdefs
int_min :: "hollight.int => hollight.int => hollight.int"
"int_min ==
%(u::hollight.int) ua::hollight.int.
mk_int (real_min (dest_int u) (dest_int ua))"
lemma DEF_int_min: "int_min =
(%(u::hollight.int) ua::hollight.int.
mk_int (real_min (dest_int u) (dest_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)"
by (import hollight int_min_th)
constdefs
int_pow :: "hollight.int => nat => hollight.int"
"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))"
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"
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))"
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"
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)))"
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)"
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))"
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))"
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)"
by (import hollight INT_ABS)
lemma INT_GE: "ALL (x::hollight.int) xa::hollight.int. 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"
by (import hollight INT_GT)
lemma INT_LT: "ALL (x::hollight.int) xa::hollight.int. 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))"
by (import hollight INT_ARCH)
constdefs
mod_int :: "hollight.int => hollight.int => hollight.int => bool"
"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)
constdefs
IN :: "'A => ('A => bool) => bool"
"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)
constdefs
GSPEC :: "('A => bool) => 'A => bool"
"GSPEC == %u::'A::type => bool. u"
lemma DEF_GSPEC: "GSPEC = (%u::'A::type => bool. u)"
by (import hollight DEF_GSPEC)
constdefs
SETSPEC :: "'q_36941 => bool => 'q_36941 => bool"
"SETSPEC == %(u::'q_36941::type) (ua::bool) ub::'q_36941::type. ua & u = ub"
lemma DEF_SETSPEC: "SETSPEC = (%(u::'q_36941::type) (ua::bool) ub::'q_36941::type. ua & u = ub)"
by (import hollight DEF_SETSPEC)
lemma IN_ELIM_THM: "(ALL (P::(bool => 'q_36974::type => bool) => bool) x::'q_36974::type.
IN x (GSPEC (%v::'q_36974::type. P (SETSPEC v))) =
P (%(p::bool) t::'q_36974::type. p & x = t)) &
(ALL (p::'q_37005::type => bool) x::'q_37005::type.
IN x
(GSPEC (%v::'q_37005::type. EX y::'q_37005::type. SETSPEC v (p y) y)) =
p x) &
(ALL (P::(bool => 'q_37033::type => bool) => bool) x::'q_37033::type.
GSPEC (%v::'q_37033::type. P (SETSPEC v)) x =
P (%(p::bool) t::'q_37033::type. p & x = t)) &
(ALL (p::'q_37062::type => bool) x::'q_37062::type.
GSPEC (%v::'q_37062::type. EX y::'q_37062::type. SETSPEC v (p y) y) x =
p x) &
(ALL (p::'q_37079::type => bool) x::'q_37079::type. IN x p = p x)"
by (import hollight IN_ELIM_THM)
constdefs
EMPTY :: "'A => bool"
"EMPTY == %x::'A::type. False"
lemma DEF_EMPTY: "EMPTY = (%x::'A::type. False)"
by (import hollight DEF_EMPTY)
constdefs
INSERT :: "'A => ('A => bool) => 'A => bool"
"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)
consts
UNIV :: "'A => bool"
defs
UNIV_def: "hollight.UNIV == %x::'A::type. True"
lemma DEF_UNIV: "hollight.UNIV = (%x::'A::type. True)"
by (import hollight DEF_UNIV)
consts
UNION :: "('A => bool) => ('A => bool) => 'A => 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)
constdefs
UNIONS :: "(('A => bool) => bool) => 'A => bool"
"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)
consts
INTER :: "('A => bool) => ('A => bool) => 'A => bool"
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)
constdefs
INTERS :: "(('A => bool) => bool) => 'A => bool"
"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)
constdefs
DIFF :: "('A => bool) => ('A => bool) => 'A => bool"
"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)"
by (import hollight INSERT)
constdefs
DELETE :: "('A => bool) => 'A => 'A => bool"
"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)
constdefs
SUBSET :: "('A => bool) => ('A => bool) => bool"
"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)
constdefs
PSUBSET :: "('A => bool) => ('A => bool) => bool"
"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)
constdefs
DISJOINT :: "('A => bool) => ('A => bool) => bool"
"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)
constdefs
SING :: "('A => bool) => bool"
"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)"
by (import hollight DEF_SING)
constdefs
FINITE :: "('A => bool) => bool"
"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)
constdefs
INFINITE :: "('A => bool) => bool"
"INFINITE == %u::'A::type => bool. ~ FINITE u"
lemma DEF_INFINITE: "INFINITE = (%u::'A::type => bool. ~ FINITE u)"
by (import hollight DEF_INFINITE)
constdefs
IMAGE :: "('A => 'B) => ('A => bool) => 'B => bool"
"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)
constdefs
INJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool"
"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)"
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))"
by (import hollight DEF_INJ)
constdefs
SURJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool"
"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))"
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)))"
by (import hollight DEF_SURJ)
constdefs
BIJ :: "('A => 'B) => ('A => bool) => ('B => bool) => bool"
"BIJ ==
%(u::'A::type => 'B::type) (ua::'A::type => bool) ub::'B::type => 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)"
by (import hollight DEF_BIJ)
constdefs
CHOICE :: "('A => bool) => 'A"
"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)
constdefs
REST :: "('A => bool) => 'A => bool"
"REST == %u::'A::type => bool. DELETE u (CHOICE u)"
lemma DEF_REST: "REST = (%u::'A::type => bool. DELETE u (CHOICE u))"
by (import hollight DEF_REST)
constdefs
CARD_GE :: "('q_37578 => bool) => ('q_37575 => bool) => bool"
"CARD_GE ==
%(u::'q_37578::type => bool) ua::'q_37575::type => bool.
EX f::'q_37578::type => 'q_37575::type.
ALL y::'q_37575::type.
IN y ua --> (EX x::'q_37578::type. IN x u & y = f x)"
lemma DEF_CARD_GE: "CARD_GE =
(%(u::'q_37578::type => bool) ua::'q_37575::type => bool.
EX f::'q_37578::type => 'q_37575::type.
ALL y::'q_37575::type.
IN y ua --> (EX x::'q_37578::type. IN x u & y = f x))"
by (import hollight DEF_CARD_GE)
constdefs
CARD_LE :: "('q_37587 => bool) => ('q_37586 => bool) => bool"
"CARD_LE ==
%(u::'q_37587::type => bool) ua::'q_37586::type => bool. CARD_GE ua u"
lemma DEF_CARD_LE: "CARD_LE =
(%(u::'q_37587::type => bool) ua::'q_37586::type => bool. CARD_GE ua u)"
by (import hollight DEF_CARD_LE)
constdefs
CARD_EQ :: "('q_37597 => bool) => ('q_37598 => bool) => bool"
"CARD_EQ ==
%(u::'q_37597::type => bool) ua::'q_37598::type => bool.
CARD_LE u ua & CARD_LE ua u"
lemma DEF_CARD_EQ: "CARD_EQ =
(%(u::'q_37597::type => bool) ua::'q_37598::type => bool.
CARD_LE u ua & CARD_LE ua u)"
by (import hollight DEF_CARD_EQ)
constdefs
CARD_GT :: "('q_37612 => bool) => ('q_37613 => bool) => bool"
"CARD_GT ==
%(u::'q_37612::type => bool) ua::'q_37613::type => bool.
CARD_GE u ua & ~ CARD_GE ua u"
lemma DEF_CARD_GT: "CARD_GT =
(%(u::'q_37612::type => bool) ua::'q_37613::type => bool.
CARD_GE u ua & ~ CARD_GE ua u)"
by (import hollight DEF_CARD_GT)
constdefs
CARD_LT :: "('q_37628 => bool) => ('q_37629 => bool) => bool"
"CARD_LT ==
%(u::'q_37628::type => bool) ua::'q_37629::type => bool.
CARD_LE u ua & ~ CARD_LE ua u"
lemma DEF_CARD_LT: "CARD_LT =
(%(u::'q_37628::type => bool) ua::'q_37629::type => bool.
CARD_LE u ua & ~ CARD_LE ua u)"
by (import hollight DEF_CARD_LT)
constdefs
COUNTABLE :: "('q_37642 => bool) => bool"
"(op ==::(('q_37642::type => bool) => bool)
=> (('q_37642::type => bool) => bool) => prop)
(COUNTABLE::('q_37642::type => bool) => bool)
((CARD_GE::(nat => bool) => ('q_37642::type => bool) => bool)
(hollight.UNIV::nat => bool))"
lemma DEF_COUNTABLE: "(op =::(('q_37642::type => bool) => bool)
=> (('q_37642::type => bool) => bool) => bool)
(COUNTABLE::('q_37642::type => bool) => bool)
((CARD_GE::(nat => bool) => ('q_37642::type => bool) => bool)
(hollight.UNIV::nat => bool))"
by (import hollight DEF_COUNTABLE)
lemma NOT_IN_EMPTY: "ALL x::'A::type. ~ IN x EMPTY"
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)"
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)"
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)"
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)"
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)"
by (import hollight IN_REST)
lemma CHOICE_DEF: "ALL x::'A::type => bool. x ~= EMPTY --> IN (CHOICE 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))"
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))"
by (import hollight EMPTY_NOT_UNIV)
lemma EQ_UNIV: "(ALL x::'A::type. IN x (s::'A::type => bool)) = (s = hollight.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)"
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)"
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))"
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)"
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)"
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_38479::type => bool) (xa::'q_38479::type => bool)
xb::'q_38479::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)"
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)"
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)"
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))"
by (import hollight IN_DISJOINT)
lemma DISJOINT_SYM: "ALL (x::'A::type => bool) xa::'A::type => bool.
DISJOINT x xa = DISJOINT xa x"
by (import hollight DISJOINT_SYM)
lemma DISJOINT_EMPTY: "ALL x::'A::type => bool. DISJOINT EMPTY x & DISJOINT x EMPTY"
by (import hollight DISJOINT_EMPTY)
lemma DISJOINT_EMPTY_REFL: "ALL x::'A::type => bool. (x = EMPTY) = DISJOINT x 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)"
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_38897::type => bool) xa::'q_38897::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)"
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)"
by (import hollight SET_CASES)
lemma ABSORPTION: "ALL (x::'A::type) xa::'A::type => bool. IN x xa = (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"
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))"
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)"
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_39353::type)
(INSERT (y::'q_39353::type) (s::'q_39353::type => 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_39420::type => bool) (q::'q_39420::type => bool) =
hollight.INTER q p &
hollight.INTER (hollight.INTER p q) (r::'q_39420::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"
by (import hollight INTER_ACI)
lemma UNION_ACI: "hollight.UNION (p::'q_39486::type => bool) (q::'q_39486::type => bool) =
hollight.UNION q p &
hollight.UNION (hollight.UNION p q) (r::'q_39486::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"
by (import hollight UNION_ACI)
lemma DELETE_NON_ELEMENT: "ALL (x::'A::type) xa::'A::type => bool. (~ IN x xa) = (DELETE 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))"
by (import hollight IN_DELETE_EQ)
lemma EMPTY_DELETE: "ALL x::'A::type. DELETE EMPTY x = EMPTY"
by (import hollight EMPTY_DELETE)
lemma DELETE_DELETE: "ALL (x::'A::type) xa::'A::type => bool. DELETE (DELETE xa x) x = DELETE 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"
by (import hollight DELETE_COMM)
lemma DELETE_SUBSET: "ALL (x::'A::type) xa::'A::type => bool. SUBSET (DELETE xa x) 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)"
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"
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)"
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))"
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))"
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"
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"
by (import hollight DISJOINT_DELETE_SYM)
lemma UNIONS_0: "(op =::('q_39893::type => bool) => ('q_39893::type => bool) => bool)
((UNIONS::(('q_39893::type => bool) => bool) => 'q_39893::type => bool)
(EMPTY::('q_39893::type => bool) => bool))
(EMPTY::'q_39893::type => bool)"
by (import hollight UNIONS_0)
lemma UNIONS_1: "UNIONS (INSERT (s::'q_39899::type => bool) EMPTY) = s"
by (import hollight UNIONS_1)
lemma UNIONS_2: "UNIONS
(INSERT (s::'q_39919::type => bool)
(INSERT (t::'q_39919::type => bool) EMPTY)) =
hollight.UNION s t"
by (import hollight UNIONS_2)
lemma UNIONS_INSERT: "UNIONS
(INSERT (s::'q_39933::type => bool)
(u::('q_39933::type => bool) => bool)) =
hollight.UNION s (UNIONS u)"
by (import hollight UNIONS_INSERT)
lemma FORALL_IN_UNIONS: "ALL (x::'q_39975::type => bool) xa::('q_39975::type => bool) => bool.
(ALL xb::'q_39975::type. IN xb (UNIONS xa) --> x xb) =
(ALL (t::'q_39975::type => bool) xb::'q_39975::type.
IN t xa & IN xb t --> x xb)"
by (import hollight FORALL_IN_UNIONS)
lemma EMPTY_UNIONS: "ALL x::('q_40001::type => bool) => bool.
(UNIONS x = EMPTY) =
(ALL xa::'q_40001::type => bool. IN xa x --> xa = EMPTY)"
by (import hollight EMPTY_UNIONS)
lemma IMAGE_CLAUSES: "IMAGE (f::'q_40027::type => 'q_40031::type) EMPTY = EMPTY &
IMAGE f (INSERT (x::'q_40027::type) (s::'q_40027::type => bool)) =
INSERT (f x) (IMAGE f s)"
by (import hollight IMAGE_CLAUSES)
lemma IMAGE_UNION: "ALL (x::'q_40054::type => 'q_40065::type) (xa::'q_40054::type => bool)
xb::'q_40054::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_40098::type => 'q_40094::type)
(xa::'q_40089::type => 'q_40098::type) xb::'q_40089::type => bool.
IMAGE (x o xa) xb = IMAGE x (IMAGE xa xb)"
by (import hollight IMAGE_o)
lemma IMAGE_SUBSET: "ALL (x::'q_40116::type => 'q_40127::type) (xa::'q_40116::type => bool)
xb::'q_40116::type => bool.
SUBSET xa xb --> SUBSET (IMAGE x xa) (IMAGE x xb)"
by (import hollight IMAGE_SUBSET)
lemma IMAGE_DIFF_INJ: "(ALL (x::'q_40158::type) y::'q_40158::type.
(f::'q_40158::type => 'q_40169::type) x = f y --> x = y) -->
IMAGE f (DIFF (s::'q_40158::type => bool) (t::'q_40158::type => bool)) =
DIFF (IMAGE f s) (IMAGE f t)"
by (import hollight IMAGE_DIFF_INJ)
lemma IMAGE_DELETE_INJ: "(ALL x::'q_40204::type.
(f::'q_40204::type => 'q_40203::type) x = f (a::'q_40204::type) -->
x = a) -->
IMAGE f (DELETE (s::'q_40204::type => bool) a) = DELETE (IMAGE f s) (f a)"
by (import hollight IMAGE_DELETE_INJ)
lemma IMAGE_EQ_EMPTY: "ALL (x::'q_40227::type => 'q_40223::type) xa::'q_40227::type => bool.
(IMAGE x xa = EMPTY) = (xa = EMPTY)"
by (import hollight IMAGE_EQ_EMPTY)
lemma FORALL_IN_IMAGE: "ALL (x::'q_40263::type => 'q_40262::type) xa::'q_40263::type => bool.
(ALL xb::'q_40262::type.
IN xb (IMAGE x xa) --> (P::'q_40262::type => bool) xb) =
(ALL xb::'q_40263::type. IN xb xa --> P (x xb))"
by (import hollight FORALL_IN_IMAGE)
lemma EXISTS_IN_IMAGE: "ALL (x::'q_40299::type => 'q_40298::type) xa::'q_40299::type => bool.
(EX xb::'q_40298::type.
IN xb (IMAGE x xa) & (P::'q_40298::type => bool) xb) =
(EX xb::'q_40299::type. IN 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_40385::type => bool) c::'q_40390::type.
IMAGE (%x::'q_40385::type. c) s = COND (s = EMPTY) EMPTY (INSERT c EMPTY)"
by (import hollight IMAGE_CONST)
lemma SIMPLE_IMAGE: "ALL (x::'q_40418::type => 'q_40422::type) xa::'q_40418::type => bool.
GSPEC
(%u::'q_40422::type.
EX xb::'q_40418::type. SETSPEC u (IN xb xa) (x xb)) =
IMAGE x xa"
by (import hollight SIMPLE_IMAGE)
lemma EMPTY_GSPEC: "GSPEC (%u::'q_40439::type. Ex (SETSPEC u False)) = EMPTY"
by (import hollight EMPTY_GSPEC)
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)"
by (import hollight FINITE_DELETE_IMP)
lemma FINITE_DELETE: "ALL (s::'A::type => bool) x::'A::type. FINITE (DELETE s x) = FINITE s"
by (import hollight FINITE_DELETE)
lemma FINITE_UNIONS: "ALL s::('q_40774::type => bool) => bool.
FINITE s -->
FINITE (UNIONS s) = (ALL t::'q_40774::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))"
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))"
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))"
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))"
by (import hollight INFINITE_IMAGE_INJ)
lemma INFINITE_NONEMPTY: "ALL s::'q_41257::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)"
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'))"
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_41555::type => bool) t::'q_41555::type => bool.
FINITE s --> FINITE (DIFF s t)"
by (import hollight FINITE_DIFF)
constdefs
FINREC :: "('q_41615 => 'q_41614 => 'q_41614)
=> 'q_41614 => ('q_41615 => bool) => 'q_41614 => nat => bool"
"FINREC ==
SOME FINREC::('q_41615::type => 'q_41614::type => 'q_41614::type)
=> 'q_41614::type
=> ('q_41615::type => bool)
=> 'q_41614::type => nat => bool.
(ALL (f::'q_41615::type => 'q_41614::type => 'q_41614::type)
(s::'q_41615::type => bool) (a::'q_41614::type) b::'q_41614::type.
FINREC f b s a 0 = (s = EMPTY & a = b)) &
(ALL (b::'q_41614::type) (s::'q_41615::type => bool) (n::nat)
(a::'q_41614::type)
f::'q_41615::type => 'q_41614::type => 'q_41614::type.
FINREC f b s a (Suc n) =
(EX (x::'q_41615::type) c::'q_41614::type.
IN x s & FINREC f b (DELETE s x) c n & a = f x c))"
lemma DEF_FINREC: "FINREC =
(SOME FINREC::('q_41615::type => 'q_41614::type => 'q_41614::type)
=> 'q_41614::type
=> ('q_41615::type => bool)
=> 'q_41614::type => nat => bool.
(ALL (f::'q_41615::type => 'q_41614::type => 'q_41614::type)
(s::'q_41615::type => bool) (a::'q_41614::type) b::'q_41614::type.
FINREC f b s a 0 = (s = EMPTY & a = b)) &
(ALL (b::'q_41614::type) (s::'q_41615::type => bool) (n::nat)
(a::'q_41614::type)
f::'q_41615::type => 'q_41614::type => 'q_41614::type.
FINREC f b s a (Suc n) =
(EX (x::'q_41615::type) c::'q_41614::type.
IN x s & FINREC f b (DELETE s x) c n & a = f x c)))"
by (import hollight DEF_FINREC)
lemma FINREC_1_LEMMA: "ALL (x::'q_41660::type => 'q_41659::type => 'q_41659::type)
(xa::'q_41659::type) (xb::'q_41660::type => bool) xc::'q_41659::type.
FINREC x xa xb xc (Suc 0) =
(EX xd::'q_41660::type. xb = INSERT xd EMPTY & 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)))"
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)"
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))"
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))"
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))))"
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))))"
by (import hollight SET_RECURSION_LEMMA)
constdefs
ITSET :: "('q_42316 => 'q_42315 => 'q_42315)
=> ('q_42316 => bool) => 'q_42315 => 'q_42315"
"ITSET ==
%(u::'q_42316::type => 'q_42315::type => 'q_42315::type)
(ua::'q_42316::type => bool) ub::'q_42315::type.
(SOME g::('q_42316::type => bool) => 'q_42315::type.
g EMPTY = ub &
(ALL (x::'q_42316::type) s::'q_42316::type => bool.
FINITE s --> g (INSERT x s) = COND (IN x s) (g s) (u x (g s))))
ua"
lemma DEF_ITSET: "ITSET =
(%(u::'q_42316::type => 'q_42315::type => 'q_42315::type)
(ua::'q_42316::type => bool) ub::'q_42315::type.
(SOME g::('q_42316::type => bool) => 'q_42315::type.
g EMPTY = ub &
(ALL (x::'q_42316::type) s::'q_42316::type => bool.
FINITE s --> g (INSERT x s) = COND (IN x s) (g s) (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)))"
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))"
by (import hollight FINITE_RECURSION_DELETE)
lemma ITSET_EQ: "ALL (x::'q_42621::type => bool)
(xa::'q_42621::type => 'q_42622::type => 'q_42622::type)
(xb::'q_42621::type => 'q_42622::type => 'q_42622::type)
xc::'q_42622::type.
FINITE x &
(ALL xc::'q_42621::type. IN xc x --> xa xc = xb xc) &
(ALL (x::'q_42621::type) (y::'q_42621::type) s::'q_42622::type.
x ~= y --> xa x (xa y s) = xa y (xa x s)) &
(ALL (x::'q_42621::type) (y::'q_42621::type) s::'q_42622::type.
x ~= y --> xb x (xb y s) = xb y (xb x s)) -->
ITSET xa x xc = ITSET xb x xc"
by (import hollight ITSET_EQ)
lemma SUBSET_RESTRICT: "ALL (x::'q_42655::type => bool) xa::'q_42655::type => bool.
SUBSET
(GSPEC
(%u::'q_42655::type.
EX xb::'q_42655::type. SETSPEC u (IN xb x & xa xb) xb))
x"
by (import hollight SUBSET_RESTRICT)
lemma FINITE_RESTRICT: "ALL (s::'A::type => bool) p::'q_42673::type.
FINITE s -->
FINITE
(GSPEC
(%u::'A::type.
EX x::'A::type. SETSPEC u (IN x s & (P::'A::type => bool) x) x))"
by (import hollight FINITE_RESTRICT)
constdefs
CARD :: "('q_42709 => bool) => nat"
"CARD == %u::'q_42709::type => bool. ITSET (%x::'q_42709::type. Suc) u 0"
lemma DEF_CARD: "CARD = (%u::'q_42709::type => bool. ITSET (%x::'q_42709::type. Suc) u 0)"
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)))))))"
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"
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)"
by (import hollight CARD_DELETE)
lemma CARD_UNION_EQ: "ALL (s::'q_42954::type => bool) (t::'q_42954::type => bool)
u::'q_42954::type => bool.
FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
CARD s + CARD t = CARD u"
by (import hollight CARD_UNION_EQ)
constdefs
HAS_SIZE :: "('q_42990 => bool) => nat => bool"
"HAS_SIZE == %(u::'q_42990::type => bool) ua::nat. FINITE u & CARD u = ua"
lemma DEF_HAS_SIZE: "HAS_SIZE = (%(u::'q_42990::type => bool) ua::nat. FINITE u & CARD u = ua)"
by (import hollight DEF_HAS_SIZE)
lemma HAS_SIZE_CARD: "ALL (x::'q_43009::type => bool) xa::nat. HAS_SIZE x xa --> CARD x = xa"
by (import hollight HAS_SIZE_CARD)
lemma HAS_SIZE_0: "ALL (s::'A::type => bool) n::'q_43025::type. HAS_SIZE s 0 = (s = EMPTY)"
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))"
by (import hollight HAS_SIZE_SUC)
lemma HAS_SIZE_UNION: "ALL (x::'q_43147::type => bool) (xa::'q_43147::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)"
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)"
by (import hollight HAS_SIZE_UNIONS)
lemma HAS_SIZE_CLAUSES: "HAS_SIZE (s::'q_43395::type => bool) 0 = (s = EMPTY) &
HAS_SIZE s (Suc (n::nat)) =
(EX (a::'q_43395::type) t::'q_43395::type => bool.
HAS_SIZE t n & ~ IN 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"
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)"
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"
by (import hollight CARD_SUBSET_LE)
lemma CARD_EQ_0: "ALL s::'q_43711::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)"
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)"
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)"
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"
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"
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)"
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))"
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)"
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)))"
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)))"
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"
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)"
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)"
by (import hollight HAS_SIZE_FUNSPACE)
lemma CARD_FUNSPACE: "ALL (s::'q_45066::type => bool) t::'q_45063::type => bool.
FINITE s & FINITE t -->
CARD
(GSPEC
(%u::'q_45066::type => 'q_45063::type.
EX f::'q_45066::type => 'q_45063::type.
SETSPEC u
((ALL x::'q_45066::type. IN x s --> IN (f x) t) &
(ALL x::'q_45066::type.
~ IN x s --> f x = (d::'q_45063::type)))
f)) =
EXP (CARD t) (CARD s)"
by (import hollight CARD_FUNSPACE)
lemma FINITE_FUNSPACE: "ALL (s::'q_45132::type => bool) t::'q_45129::type => bool.
FINITE s & FINITE t -->
FINITE
(GSPEC
(%u::'q_45132::type => 'q_45129::type.
EX f::'q_45132::type => 'q_45129::type.
SETSPEC u
((ALL x::'q_45132::type. IN x s --> IN (f x) t) &
(ALL x::'q_45132::type.
~ IN x s --> f x = (d::'q_45129::type)))
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)"
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)"
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))"
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"
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"
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))"
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)"
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))"
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"
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)"
by (import hollight num_FINITE)
lemma num_FINITE_AVOID: "ALL s::nat => bool. FINITE s --> (EX a::nat. ~ IN 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)))"
by (import hollight HAS_SIZE_INDEX)
constdefs
set_of_list :: "'q_45759 hollight.list => 'q_45759 => bool"
"set_of_list ==
SOME set_of_list::'q_45759::type hollight.list => 'q_45759::type => bool.
set_of_list NIL = EMPTY &
(ALL (h::'q_45759::type) t::'q_45759::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_45759::type hollight.list => 'q_45759::type => bool.
set_of_list NIL = EMPTY &
(ALL (h::'q_45759::type) t::'q_45759::type hollight.list.
set_of_list (CONS h t) = INSERT h (set_of_list t)))"
by (import hollight DEF_set_of_list)
constdefs
list_of_set :: "('q_45777 => bool) => 'q_45777 hollight.list"
"list_of_set ==
%u::'q_45777::type => bool.
SOME l::'q_45777::type hollight.list.
set_of_list l = u & LENGTH l = CARD u"
lemma DEF_list_of_set: "list_of_set =
(%u::'q_45777::type => bool.
SOME l::'q_45777::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_45826::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_45842::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_45887::type hollight.list. FINITE (set_of_list l)"
by (import hollight FINITE_SET_OF_LIST)
lemma IN_SET_OF_LIST: "ALL (x::'q_45905::type) l::'q_45905::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_45930::type hollight.list) xa::'q_45930::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)
constdefs
pairwise :: "('q_45989 => 'q_45989 => bool) => ('q_45989 => bool) => bool"
"pairwise ==
%(u::'q_45989::type => 'q_45989::type => bool) ua::'q_45989::type => bool.
ALL (x::'q_45989::type) y::'q_45989::type.
IN x ua & IN y ua & x ~= y --> u x y"
lemma DEF_pairwise: "pairwise =
(%(u::'q_45989::type => 'q_45989::type => bool) ua::'q_45989::type => bool.
ALL (x::'q_45989::type) y::'q_45989::type.
IN x ua & IN y ua & x ~= y --> u x y)"
by (import hollight DEF_pairwise)
constdefs
PAIRWISE :: "('q_46011 => 'q_46011 => bool) => 'q_46011 hollight.list => bool"
"PAIRWISE ==
SOME PAIRWISE::('q_46011::type => 'q_46011::type => bool)
=> 'q_46011::type hollight.list => bool.
(ALL r::'q_46011::type => 'q_46011::type => bool.
PAIRWISE r NIL = True) &
(ALL (h::'q_46011::type) (r::'q_46011::type => 'q_46011::type => bool)
t::'q_46011::type hollight.list.
PAIRWISE r (CONS h t) = (ALL_list (r h) t & PAIRWISE r t))"
lemma DEF_PAIRWISE: "PAIRWISE =
(SOME PAIRWISE::('q_46011::type => 'q_46011::type => bool)
=> 'q_46011::type hollight.list => bool.
(ALL r::'q_46011::type => 'q_46011::type => bool.
PAIRWISE r NIL = True) &
(ALL (h::'q_46011::type) (r::'q_46011::type => 'q_46011::type => bool)
t::'q_46011::type hollight.list.
PAIRWISE r (CONS h t) = (ALL_list (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)))"])
by (import hollight TYDEF_finite_image)
syntax
dest_finite_image :: _
syntax
mk_finite_image :: _
lemmas "TYDEF_finite_image_@intern" = typedef_hol2hollight
[where a="a :: 'A finite_image" and r=r ,
OF type_definition_finite_image]
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))))"
by (import hollight FINITE_IMAGE_IMAGE)
constdefs
dimindex :: "('A => bool) => nat"
"(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))"
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))"
by (import hollight CARD_FINITE_IMAGE)
lemma FINITE_FINITE_IMAGE: "(FINITE::('A::type finite_image => bool) => bool)
(hollight.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"
by (import hollight DIMINDEX_FINITE_IMAGE)
constdefs
finite_index :: "nat => 'A"
"(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))))"
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))))"
by (import hollight FINITE_INDEX_INJ)
lemma FORALL_FINITE_INDEX: "(op =::bool => bool => bool)
((All::('N::type finite_image => bool) => bool)
(P::'N::type finite_image => 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::('N::type => bool) => nat)
(hollight.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)"])
by (import hollight TYDEF_cart)
syntax
dest_cart :: _
syntax
mk_cart :: _
lemmas "TYDEF_cart_@intern" = typedef_hol2hollight
[where a="a :: ('A, 'B) cart" and r=r ,
OF type_definition_cart]
consts
"$" :: "('q_46418, 'q_46425) cart => nat => 'q_46418" ("$")
defs
"$_def": "$ ==
%(u::('q_46418::type, 'q_46425::type) cart) ua::nat.
dest_cart u (finite_index ua)"
lemma "DEF_$": "$ =
(%(u::('q_46418::type, 'q_46425::type) 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)
(($::('A::type, 'B::type) cart => nat => 'A::type) y
xa))))))"
by (import hollight CART_EQ)
constdefs
lambda :: "(nat => 'A) => ('A, 'B) cart"
"(op ==::((nat => 'A::type) => ('A::type, 'B::type) cart)
=> ((nat => 'A::type) => ('A::type, 'B::type) cart) => prop)
(lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
(%u::nat => 'A::type.
(Eps::(('A::type, 'B::type) cart => bool) => ('A::type, 'B::type) cart)
(%f::('A::type, 'B::type) cart.
(All::(nat => bool) => bool)
(%i::nat.
(op -->::bool => bool => bool)
((op &::bool => bool => bool)
((<=::nat => nat => bool)
((NUMERAL_BIT1::nat => nat) (0::nat)) i)
((<=::nat => nat => bool) i
((dimindex::('B::type => bool) => nat)
(hollight.UNIV::'B::type => bool))))
((op =::'A::type => 'A::type => bool)
(($::('A::type, 'B::type) cart => nat => 'A::type) f i)
(u i)))))"
lemma DEF_lambda: "(op =::((nat => 'A::type) => ('A::type, 'B::type) cart)
=> ((nat => 'A::type) => ('A::type, 'B::type) cart) => bool)
(lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
(%u::nat => 'A::type.
(Eps::(('A::type, 'B::type) cart => bool) => ('A::type, 'B::type) cart)
(%f::('A::type, 'B::type) cart.
(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)
(($::('A::type, 'B::type) cart => nat => 'A::type) f i)
(u i)))))"
by (import hollight DEF_lambda)
lemma LAMBDA_BETA: "(All::(nat => bool) => bool)
(%x::nat.
(op -->::bool => bool => bool)
((op &::bool => bool => bool)
((<=::nat => nat => bool) ((NUMERAL_BIT1::nat => nat) (0::nat)) x)
((<=::nat => nat => bool) x
((dimindex::('B::type => bool) => nat)
(hollight.UNIV::'B::type => bool))))
((op =::'A::type => 'A::type => bool)
(($::('A::type, 'B::type) cart => nat => 'A::type)
((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart)
(g::nat => 'A::type))
x)
(g x)))"
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)
(($::('A::type, 'B::type) cart => nat => 'A::type) x i)
(xa i))))
((op =::('A::type, 'B::type) cart
=> ('A::type, 'B::type) cart => bool)
((lambda::(nat => 'A::type) => ('A::type, 'B::type) cart) xa)
x)))"
by (import hollight LAMBDA_UNIQUE)
lemma LAMBDA_ETA: "ALL x::('q_46616::type, 'q_46620::type) cart. lambda ($ x) = x"
by (import hollight LAMBDA_ETA)
typedef (open) ('A, 'B) finite_sum = "(Collect::(('A::type finite_image, 'B::type finite_image) sum => bool)
=> ('A::type finite_image, 'B::type finite_image) sum set)
(%x::('A::type finite_image, 'B::type finite_image) sum. 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) sum => bool)
=> ('A::type finite_image, 'B::type finite_image) sum)
(%x::('A::type finite_image, 'B::type finite_image) sum. True::bool)"])
by (import hollight TYDEF_finite_sum)
syntax
dest_finite_sum :: _
syntax
mk_finite_sum :: _
lemmas "TYDEF_finite_sum_@intern" = typedef_hol2hollight
[where a="a :: ('A, 'B) finite_sum" and r=r ,
OF type_definition_finite_sum]
constdefs
pastecart :: "('A, 'M) cart => ('A, 'N) cart => ('A, ('M, 'N) finite_sum) cart"
"(op ==::(('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
=> (('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
=> prop)
(pastecart::('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
(%(u::('A::type, 'M::type) cart) ua::('A::type, 'N::type) cart.
(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
((dimindex::('M::type => bool) => nat)
(hollight.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))))))"
lemma DEF_pastecart: "(op =::(('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
=> (('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
=> bool)
(pastecart::('A::type, 'M::type) cart
=> ('A::type, 'N::type) cart
=> ('A::type, ('M::type, 'N::type) finite_sum) cart)
(%(u::('A::type, 'M::type) cart) ua::('A::type, 'N::type) cart.
(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
((dimindex::('M::type => bool) => nat)
(hollight.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))))))"
by (import hollight DEF_pastecart)
constdefs
fstcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'M) cart"
"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))"
by (import hollight DEF_fstcart)
constdefs
sndcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'N) cart"
"(op ==::(('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
=> (('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
=> prop)
(sndcart::('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
(%u::('A::type, ('M::type, 'N::type) finite_sum) cart.
(lambda::(nat => 'A::type) => ('A::type, 'N::type) cart)
(%i::nat.
($::('A::type, ('M::type, 'N::type) finite_sum) cart
=> nat => 'A::type)
u ((op +::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
(hollight.UNIV::'M::type => bool)))))"
lemma DEF_sndcart: "(op =::(('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
=> (('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
=> bool)
(sndcart::('A::type, ('M::type, 'N::type) finite_sum) cart
=> ('A::type, 'N::type) cart)
(%u::('A::type, ('M::type, 'N::type) finite_sum) cart.
(lambda::(nat => 'A::type) => ('A::type, 'N::type) cart)
(%i::nat.
($::('A::type, ('M::type, 'N::type) finite_sum) cart
=> nat => 'A::type)
u ((op +::nat => nat => nat) i
((dimindex::('M::type => bool) => nat)
(hollight.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)
((op +::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))
((op +::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_46940::type, ('q_46937::type, 'q_46935::type) finite_sum) cart.
pastecart (fstcart x) (sndcart x) = x"
by (import hollight PASTECART_FST_SND)
lemma PASTECART_EQ: "ALL (x::('q_46978::type, ('q_46968::type, 'q_46979::type) finite_sum) cart)
y::('q_46978::type, ('q_46968::type, 'q_46979::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_46999::type, ('q_47000::type, 'q_47001::type) finite_sum) cart
=> bool) =
(ALL (x::('q_46999::type, 'q_47000::type) cart)
y::('q_46999::type, 'q_47001::type) cart. P (pastecart x y))"
by (import hollight FORALL_PASTECART)
lemma EXISTS_PASTECART: "Ex (P::('q_47021::type, ('q_47022::type, 'q_47023::type) finite_sum) cart
=> bool) =
(EX (x::('q_47021::type, 'q_47022::type) cart)
y::('q_47021::type, 'q_47023::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_47348::type => bool) f::'q_47348::type => 'q_47348::type.
FINITE s & IMAGE f s = s -->
(ALL (x::'q_47348::type) y::'q_47348::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_47454::type => bool.
(ALL xa::'q_47454::type. IN xa EMPTY --> x xa) = True) &
(ALL (x::'q_47494::type => bool) (xa::'q_47494::type)
xb::'q_47494::type => bool.
(ALL xc::'q_47494::type. IN xc (INSERT xa xb) --> x xc) =
(x xa & (ALL xa::'q_47494::type. IN xa xb --> x xa)))"
by (import hollight FORALL_IN_CLAUSES)
lemma EXISTS_IN_CLAUSES: "(ALL x::'q_47514::type => bool.
(EX xa::'q_47514::type. IN xa EMPTY & x xa) = False) &
(ALL (x::'q_47554::type => bool) (xa::'q_47554::type)
xb::'q_47554::type => bool.
(EX xc::'q_47554::type. IN xc (INSERT xa xb) & x xc) =
(x xa | (EX xa::'q_47554::type. IN xa xb & x xa)))"
by (import hollight EXISTS_IN_CLAUSES)
lemma SURJECTIVE_ON_RIGHT_INVERSE: "ALL (x::'q_47610::type => 'q_47611::type) xa::'q_47611::type => bool.
(ALL xb::'q_47611::type.
IN xb xa -->
(EX xa::'q_47610::type.
IN xa (s::'q_47610::type => bool) & x xa = xb)) =
(EX g::'q_47611::type => 'q_47610::type.
ALL y::'q_47611::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_47704::type => 'q_47707::type) xa::'q_47704::type => bool.
(ALL (xb::'q_47704::type) y::'q_47704::type.
IN xb xa & IN y xa & x xb = x y --> xb = y) =
(EX xb::'q_47707::type => 'q_47704::type.
ALL xc::'q_47704::type. IN xc xa --> xb (x xc) = xc)"
by (import hollight INJECTIVE_ON_LEFT_INVERSE)
lemma SURJECTIVE_RIGHT_INVERSE: "(ALL y::'q_47732::type.
EX x::'q_47735::type. (f::'q_47735::type => 'q_47732::type) x = y) =
(EX g::'q_47732::type => 'q_47735::type. ALL y::'q_47732::type. f (g y) = y)"
by (import hollight SURJECTIVE_RIGHT_INVERSE)
lemma INJECTIVE_LEFT_INVERSE: "(ALL (x::'q_47769::type) xa::'q_47769::type.
(f::'q_47769::type => 'q_47772::type) x = f xa --> x = xa) =
(EX g::'q_47772::type => 'q_47769::type. ALL x::'q_47769::type. g (f x) = x)"
by (import hollight INJECTIVE_LEFT_INVERSE)
lemma FUNCTION_FACTORS_RIGHT: "ALL (x::'q_47808::type => 'q_47809::type)
xa::'q_47796::type => 'q_47809::type.
(ALL xb::'q_47808::type. EX y::'q_47796::type. xa y = x xb) =
(EX xb::'q_47808::type => 'q_47796::type. x = xa o xb)"
by (import hollight FUNCTION_FACTORS_RIGHT)
lemma FUNCTION_FACTORS_LEFT: "ALL (x::'q_47881::type => 'q_47882::type)
xa::'q_47881::type => 'q_47861::type.
(ALL (xb::'q_47881::type) y::'q_47881::type.
xa xb = xa y --> x xb = x y) =
(EX xb::'q_47861::type => 'q_47882::type. x = xb o xa)"
by (import hollight FUNCTION_FACTORS_LEFT)
constdefs
dotdot :: "nat => nat => nat => bool"
"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_47957::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_47995::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)
constdefs
neutral :: "('q_48776 => 'q_48776 => 'q_48776) => 'q_48776"
"neutral ==
%u::'q_48776::type => 'q_48776::type => 'q_48776::type.
SOME x::'q_48776::type. ALL y::'q_48776::type. u x y = y & u y x = y"
lemma DEF_neutral: "neutral =
(%u::'q_48776::type => 'q_48776::type => 'q_48776::type.
SOME x::'q_48776::type. ALL y::'q_48776::type. u x y = y & u y x = y)"
by (import hollight DEF_neutral)
constdefs
monoidal :: "('A => 'A => 'A) => bool"
"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)
constdefs
support :: "('B => 'B => 'B) => ('A => 'B) => ('A => bool) => 'A => bool"
"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)
constdefs
iterate :: "('q_48881 => 'q_48881 => 'q_48881)
=> ('A => bool) => ('A => 'q_48881) => 'q_48881"
"iterate ==
%(u::'q_48881::type => 'q_48881::type => 'q_48881::type)
(ua::'A::type => bool) ub::'A::type => 'q_48881::type.
ITSET (%x::'A::type. u (ub x)) (support u ub ua) (neutral u)"
lemma DEF_iterate: "iterate =
(%(u::'q_48881::type => 'q_48881::type => 'q_48881::type)
(ua::'A::type => bool) ub::'A::type => 'q_48881::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_48924::type => 'q_48924::type => 'q_48924::type)
(xa::'q_48927::type => 'q_48924::type) (xb::'q_48927::type)
xc::'q_48927::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_48949::type => 'q_48949::type => 'q_48949::type)
(xa::'q_48960::type => 'q_48949::type) xb::'q_48960::type => bool.
support x xa (support x xa xb) = support x xa xb"
by (import hollight SUPPORT_SUPPORT)
lemma SUPPORT_EMPTY: "ALL (x::'q_48985::type => 'q_48985::type => 'q_48985::type)
(xa::'q_48999::type => 'q_48985::type) xb::'q_48999::type => bool.
(ALL xc::'q_48999::type. IN xc xb --> xa xc = neutral x) =
(support x xa xb = EMPTY)"
by (import hollight SUPPORT_EMPTY)
lemma SUPPORT_SUBSET: "ALL (x::'q_49019::type => 'q_49019::type => 'q_49019::type)
(xa::'q_49020::type => 'q_49019::type) xb::'q_49020::type => bool.
SUBSET (support x xa xb) xb"
by (import hollight SUPPORT_SUBSET)
lemma FINITE_SUPPORT: "ALL (u::'q_49043::type => 'q_49043::type => 'q_49043::type)
(f::'q_49037::type => 'q_49043::type) s::'q_49037::type => bool.
FINITE s --> FINITE (support u f s)"
by (import hollight FINITE_SUPPORT)
lemma SUPPORT_CLAUSES: "(ALL x::'q_49061::type => 'q_49091::type.
support (u_4215::'q_49091::type => 'q_49091::type => 'q_49091::type) x
EMPTY =
EMPTY) &
(ALL (x::'q_49109::type => 'q_49091::type) (xa::'q_49109::type)
xb::'q_49109::type => bool.
support u_4215 x (INSERT xa xb) =
COND (x xa = neutral u_4215) (support u_4215 x xb)
(INSERT xa (support u_4215 x xb))) &
(ALL (x::'q_49142::type => 'q_49091::type) (xa::'q_49142::type)
xb::'q_49142::type => bool.
support u_4215 x (DELETE xb xa) = DELETE (support u_4215 x xb) xa) &
(ALL (x::'q_49180::type => 'q_49091::type) (xa::'q_49180::type => bool)
xb::'q_49180::type => bool.
support u_4215 x (hollight.UNION xa xb) =
hollight.UNION (support u_4215 x xa) (support u_4215 x xb)) &
(ALL (x::'q_49218::type => 'q_49091::type) (xa::'q_49218::type => bool)
xb::'q_49218::type => bool.
support u_4215 x (hollight.INTER xa xb) =
hollight.INTER (support u_4215 x xa) (support u_4215 x xb)) &
(ALL (x::'q_49254::type => 'q_49091::type) (xa::'q_49254::type => bool)
xb::'q_49254::type => bool.
support u_4215 x (DIFF xa xb) =
DIFF (support u_4215 x xa) (support u_4215 x xb))"
by (import hollight SUPPORT_CLAUSES)
lemma ITERATE_SUPPORT: "ALL (x::'q_49275::type => 'q_49275::type => 'q_49275::type)
(xa::'q_49276::type => 'q_49275::type) xb::'q_49276::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_49320::type => 'q_49320::type => 'q_49320::type)
(xa::'q_49348::type => bool) (xb::'q_49348::type => 'q_49320::type)
xc::'q_49348::type.
support x (%xa::'q_49348::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_49369::type => 'q_49369::type => 'q_49369::type)
(xa::'q_49378::type => 'q_49369::type) xb::'q_49378::type.
FINITE
(support x (%xc::'q_49378::type. COND (xc = xb) (xa xc) (neutral x))
(s::'q_49378::type => bool))"
by (import hollight FINITE_SUPPORT_DELTA)
lemma ITERATE_CLAUSES_GEN: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL f::'A::type => 'B::type. iterate u_4215 EMPTY f = neutral u_4215) &
(ALL (f::'A::type => 'B::type) (x::'A::type) s::'A::type => bool.
monoidal u_4215 & FINITE (support u_4215 f s) -->
iterate u_4215 (INSERT x s) f =
COND (IN x s) (iterate u_4215 s f)
(u_4215 (f x) (iterate u_4215 s f)))"
by (import hollight ITERATE_CLAUSES_GEN)
lemma ITERATE_CLAUSES: "ALL x::'q_49546::type => 'q_49546::type => 'q_49546::type.
monoidal x -->
(ALL f::'q_49504::type => 'q_49546::type.
iterate x EMPTY f = neutral x) &
(ALL (f::'q_49548::type => 'q_49546::type) (xa::'q_49548::type)
s::'q_49548::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_4215::'q_49634::type => 'q_49634::type => 'q_49634::type.
monoidal u_4215 -->
(ALL (f::'q_49619::type => 'q_49634::type) (s::'q_49619::type => bool)
x::'q_49619::type => bool.
FINITE s & FINITE x & DISJOINT s x -->
iterate u_4215 (hollight.UNION s x) f =
u_4215 (iterate u_4215 s f) (iterate u_4215 x f))"
by (import hollight ITERATE_UNION)
lemma ITERATE_UNION_GEN: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'A::type => bool.
FINITE (support u_4215 f s) &
FINITE (support u_4215 f t) &
DISJOINT (support u_4215 f s) (support u_4215 f t) -->
iterate u_4215 (hollight.UNION s t) f =
u_4215 (iterate u_4215 s f) (iterate u_4215 t f))"
by (import hollight ITERATE_UNION_GEN)
lemma ITERATE_DIFF: "ALL u::'q_49777::type => 'q_49777::type => 'q_49777::type.
monoidal u -->
(ALL (f::'q_49773::type => 'q_49777::type) (s::'q_49773::type => bool)
t::'q_49773::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_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL (f::'A::type => 'B::type) (s::'A::type => bool) t::'A::type => bool.
FINITE (support u_4215 f s) &
SUBSET (support u_4215 f t) (support u_4215 f s) -->
u_4215 (iterate u_4215 (DIFF s t) f) (iterate u_4215 t f) =
iterate u_4215 s f)"
by (import hollight ITERATE_DIFF_GEN)
lemma ITERATE_CLOSED: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL P::'B::type => bool.
P (neutral u_4215) &
(ALL (x::'B::type) y::'B::type. P x & P y --> P (u_4215 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_4215 x f)))"
by (import hollight ITERATE_CLOSED)
lemma ITERATE_CLOSED_GEN: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL P::'B::type => bool.
P (neutral u_4215) &
(ALL (x::'B::type) y::'B::type. P x & P y --> P (u_4215 x y)) -->
(ALL (f::'A::type => 'B::type) s::'A::type => bool.
FINITE (support u_4215 f s) &
(ALL x::'A::type. IN x s & f x ~= neutral u_4215 --> P (f x)) -->
P (iterate u_4215 s f)))"
by (import hollight ITERATE_CLOSED_GEN)
lemma ITERATE_RELATED: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL R::'B::type => 'B::type => bool.
R (neutral u_4215) (neutral u_4215) &
(ALL (x1::'B::type) (y1::'B::type) (x2::'B::type) y2::'B::type.
R x1 x2 & R y1 y2 --> R (u_4215 x1 y1) (u_4215 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_4215 x f) (iterate u_4215 x g)))"
by (import hollight ITERATE_RELATED)
lemma ITERATE_EQ_NEUTRAL: "ALL u_4215::'B::type => 'B::type => 'B::type.
monoidal u_4215 -->
(ALL (f::'A::type => 'B::type) s::'A::type => bool.
(ALL x::'A::type. IN x s --> f x = neutral u_4215) -->
iterate u_4215 s f = neutral u_4215)"
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_4215::'q_50229::type => 'q_50229::type => 'q_50229::type.
monoidal u_4215 -->
(ALL (x::'q_50248::type => 'q_50229::type) (xa::'q_50248::type)
xb::'q_50248::type => bool.
iterate u_4215 xb
(%xb::'q_50248::type. COND (xb = xa) (x xb) (neutral u_4215)) =
COND (IN xa xb) (x xa) (neutral u_4215))"
by (import hollight ITERATE_DELTA)
lemma ITERATE_IMAGE: "ALL u_4215::'q_50327::type => 'q_50327::type => 'q_50327::type.
monoidal u_4215 -->
(ALL (f::'q_50326::type => 'q_50298::type)
(g::'q_50298::type => 'q_50327::type) x::'q_50326::type => bool.
FINITE x &
(ALL (xa::'q_50326::type) y::'q_50326::type.
IN xa x & IN y x & f xa = f y --> xa = y) -->
iterate u_4215 (IMAGE f x) g = iterate u_4215 x (g o f))"
by (import hollight ITERATE_IMAGE)
constdefs
nsum :: "('q_50348 => bool) => ('q_50348 => nat) => nat"
"(op ==::(('q_50348::type => bool) => ('q_50348::type => nat) => nat)
=> (('q_50348::type => bool) => ('q_50348::type => nat) => nat)
=> prop)
(nsum::('q_50348::type => bool) => ('q_50348::type => nat) => nat)
((iterate::(nat => nat => nat)
=> ('q_50348::type => bool) => ('q_50348::type => nat) => nat)
(op +::nat => nat => nat))"
lemma DEF_nsum: "(op =::(('q_50348::type => bool) => ('q_50348::type => nat) => nat)
=> (('q_50348::type => bool) => ('q_50348::type => nat) => nat)
=> bool)
(nsum::('q_50348::type => bool) => ('q_50348::type => nat) => nat)
((iterate::(nat => nat => nat)
=> ('q_50348::type => bool) => ('q_50348::type => nat) => nat)
(op +::nat => nat => nat))"
by (import hollight DEF_nsum)
lemma NEUTRAL_ADD: "(op =::nat => nat => bool)
((neutral::(nat => nat => nat) => nat) (op +::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) (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_50386::type => nat. nsum EMPTY x = 0) &
(ALL (x::'q_50425::type) (xa::'q_50425::type => nat)
xb::'q_50425::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_50451::type => nat) (xa::'q_50451::type => bool)
xb::'q_50451::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_50506::type => nat) (s::'q_50506::type => bool)
t::'q_50506::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_50545::type => nat) xa::'q_50545::type => bool.
FINITE (support op + x xa) --> nsum (support op + x xa) x = nsum xa x"
by (import hollight NSUM_SUPPORT)
lemma NSUM_ADD: "ALL (f::'q_50591::type => nat) (g::'q_50591::type => nat)
s::'q_50591::type => bool.
FINITE s --> nsum s (%x::'q_50591::type. f x + g x) = nsum s f + nsum s g"
by (import hollight NSUM_ADD)
lemma NSUM_CMUL: "ALL (f::'q_50629::type => nat) (c::nat) s::'q_50629::type => bool.
FINITE s --> nsum s (%x::'q_50629::type. c * f x) = c * nsum s f"
by (import hollight NSUM_CMUL)
lemma NSUM_LE: "ALL (x::'q_50668::type => nat) (xa::'q_50668::type => nat)
xb::'q_50668::type => bool.
FINITE xb & (ALL xc::'q_50668::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_50790::type => nat) (g::'q_50790::type => nat)
s::'q_50790::type => bool.
FINITE s &
s ~= EMPTY & (ALL x::'q_50790::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_50832::type => nat) (xa::'q_50832::type => nat)
xb::'q_50832::type => bool.
FINITE xb & (ALL xc::'q_50832::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_50862::type => bool.
FINITE s --> nsum s (%n::'q_50862::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_50941::type => nat) xa::'q_50941::type => bool.
FINITE xa & (ALL xb::'q_50941::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_51076::type => nat) xa::'q_51076::type => bool.
FINITE xa &
(ALL xb::'q_51076::type. IN xb xa --> <= 0 (x xb)) & nsum xa x = 0 -->
(ALL xb::'q_51076::type. IN xb xa --> x xb = 0)"
by (import hollight NSUM_POS_EQ_0)
lemma NSUM_SING: "ALL (x::'q_51096::type => nat) xa::'q_51096::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_51236::type => 'q_51212::type) (xa::'q_51212::type => nat)
xb::'q_51236::type => bool.
FINITE xb &
(ALL (xa::'q_51236::type) y::'q_51236::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_51457::type => nat) s::'q_51457::type => bool.
FINITE s -->
nsum s (%x::'q_51457::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_51532::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_51675::type => bool) (f::'q_51675::type => nat) b::nat.
FINITE s & s ~= EMPTY & (ALL x::'q_51675::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_51717::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_51776::type => bool) (t::'q_51776::type => bool)
u::'q_51776::type => bool.
FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
nsum s (f::'q_51776::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_51887::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_52016::type => 'q_52015::type => bool)
(f::'q_52016::type => 'q_52015::type => nat) (s::'q_52016::type => bool)
t::'q_52015::type => bool.
FINITE s & FINITE t -->
nsum s
(%x::'q_52016::type.
nsum
(GSPEC
(%u::'q_52015::type.
EX y::'q_52015::type. SETSPEC u (IN y t & R x y) y))
(f x)) =
nsum t
(%y::'q_52015::type.
nsum
(GSPEC
(%u::'q_52016::type.
EX x::'q_52016::type. SETSPEC u (IN x s & R x y) x))
(%x::'q_52016::type. f x y))"
by (import hollight NSUM_NSUM_RESTRICT)
lemma CARD_EQ_NSUM: "ALL x::'q_52035::type => bool.
FINITE x --> CARD x = nsum x (%x::'q_52035::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_52495::type => bool) (v::'q_52495::type => bool)
f::'q_52495::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_52734::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)
consts
sum :: "('q_53311 => bool) => ('q_53311 => hollight.real) => hollight.real"
defs
sum_def: "(op ==::(('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
=> (('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
=> prop)
(hollight.sum::('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
((iterate::(hollight.real => hollight.real => hollight.real)
=> ('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
(real_add::hollight.real => hollight.real => hollight.real))"
lemma DEF_sum: "(op =::(('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
=> (('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
=> bool)
(hollight.sum::('q_53311::type => bool)
=> ('q_53311::type => hollight.real) => hollight.real)
((iterate::(hollight.real => hollight.real => hollight.real)
=> ('q_53311::type => bool)
=> ('q_53311::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_53353::type => hollight.real.
hollight.sum EMPTY x = real_of_num 0) &
(ALL (x::'q_53394::type) (xa::'q_53394::type => hollight.real)
xb::'q_53394::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_53420::type => hollight.real) (xa::'q_53420::type => bool)
xb::'q_53420::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_SUPPORT: "ALL (x::'q_53499::type => hollight.real) xa::'q_53499::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_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_53825::type => hollight.real)
(g::'q_53825::type => hollight.real) s::'q_53825::type => bool.
FINITE s &
s ~= EMPTY & (ALL x::'q_53825::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_POS_LE: "ALL (x::'q_54040::type => hollight.real) xa::'q_54040::type => bool.
FINITE xa &
(ALL xb::'q_54040::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_54187::type => hollight.real) xa::'q_54187::type => bool.
FINITE xa &
(ALL xb::'q_54187::type. IN xb xa --> real_le (real_of_num 0) (x xb)) &
hollight.sum xa x = real_of_num 0 -->
(ALL xb::'q_54187::type. IN xb xa --> x xb = real_of_num 0)"
by (import hollight SUM_POS_EQ_0)
lemma SUM_SING: "ALL (x::'q_54209::type => hollight.real) xa::'q_54209::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_IMAGE: "ALL (x::'q_54353::type => 'q_54329::type)
(xa::'q_54329::type => hollight.real) xb::'q_54353::type => bool.
FINITE xb &
(ALL (xa::'q_54353::type) y::'q_54353::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_54580::type => hollight.real) s::'q_54580::type => bool.
FINITE s -->
hollight.sum s
(%x::'q_54580::type. COND (IN x s) (f x) (real_of_num 0)) =
hollight.sum s f"
by (import hollight SUM_RESTRICT)
lemma SUM_BOUND_GEN: "ALL (s::'A::type => bool) (t::'q_54639::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_54844::type => bool) (f::'q_54844::type => hollight.real)
b::hollight.real.
FINITE s &
s ~= EMPTY & (ALL x::'q_54844::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_54866::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_54927::type => bool) (t::'q_54927::type => bool)
u::'q_54927::type => bool.
FINITE u & hollight.INTER s t = EMPTY & hollight.UNION s t = u -->
real_add (hollight.sum s (f::'q_54927::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_55040::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_55171::type => 'q_55170::type => bool)
(f::'q_55171::type => 'q_55170::type => hollight.real)
(s::'q_55171::type => bool) t::'q_55170::type => bool.
FINITE s & FINITE t -->
hollight.sum s
(%x::'q_55171::type.
hollight.sum
(GSPEC
(%u::'q_55170::type.
EX y::'q_55170::type. SETSPEC u (IN y t & R x y) y))
(f x)) =
hollight.sum t
(%y::'q_55170::type.
hollight.sum
(GSPEC
(%u::'q_55171::type.
EX x::'q_55171::type. SETSPEC u (IN x s & R x y) x))
(%x::'q_55171::type. f x y))"
by (import hollight SUM_SUM_RESTRICT)
lemma CARD_EQ_SUM: "ALL x::'q_55192::type => bool.
FINITE x -->
real_of_num (CARD x) =
hollight.sum x (%x::'q_55192::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_55589::type => nat) s::'q_55589::type => bool.
FINITE s -->
real_of_num (nsum s f) =
hollight.sum s (%x::'q_55589::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_56115::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_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)
constdefs
CASEWISE :: "(('q_56787 => 'q_56791) * ('q_56792 => 'q_56787 => 'q_56751)) hollight.list
=> 'q_56792 => 'q_56791 => 'q_56751"
"CASEWISE ==
SOME CASEWISE::(('q_56787::type => 'q_56791::type) *
('q_56792::type
=> 'q_56787::type => 'q_56751::type)) hollight.list
=> 'q_56792::type => 'q_56791::type => 'q_56751::type.
(ALL (f::'q_56792::type) x::'q_56791::type.
CASEWISE NIL f x = (SOME y::'q_56751::type. True)) &
(ALL (h::('q_56787::type => 'q_56791::type) *
('q_56792::type => 'q_56787::type => 'q_56751::type))
(t::(('q_56787::type => 'q_56791::type) *
('q_56792::type
=> 'q_56787::type => 'q_56751::type)) hollight.list)
(f::'q_56792::type) x::'q_56791::type.
CASEWISE (CONS h t) f x =
COND (EX y::'q_56787::type. fst h y = x)
(snd h f (SOME y::'q_56787::type. fst h y = x)) (CASEWISE t f x))"
lemma DEF_CASEWISE: "CASEWISE =
(SOME CASEWISE::(('q_56787::type => 'q_56791::type) *
('q_56792::type
=> 'q_56787::type => 'q_56751::type)) hollight.list
=> 'q_56792::type => 'q_56791::type => 'q_56751::type.
(ALL (f::'q_56792::type) x::'q_56791::type.
CASEWISE NIL f x = (SOME y::'q_56751::type. True)) &
(ALL (h::('q_56787::type => 'q_56791::type) *
('q_56792::type => 'q_56787::type => 'q_56751::type))
(t::(('q_56787::type => 'q_56791::type) *
('q_56792::type
=> 'q_56787::type => 'q_56751::type)) hollight.list)
(f::'q_56792::type) x::'q_56791::type.
CASEWISE (CONS h t) f x =
COND (EX y::'q_56787::type. fst h y = x)
(snd h f (SOME y::'q_56787::type. fst h y = x)) (CASEWISE t f x)))"
by (import hollight DEF_CASEWISE)
lemma CASEWISE: "(op &::bool => bool => bool)
((op =::'q_56811::type => 'q_56811::type => bool)
((CASEWISE::(('q_56803::type => 'q_56851::type) *
('q_56852::type
=> 'q_56803::type => 'q_56811::type)) hollight.list
=> 'q_56852::type => 'q_56851::type => 'q_56811::type)
(NIL::(('q_56803::type => 'q_56851::type) *
('q_56852::type
=> 'q_56803::type => 'q_56811::type)) hollight.list)
(f::'q_56852::type) (x::'q_56851::type))
((Eps::('q_56811::type => bool) => 'q_56811::type)
(%y::'q_56811::type. True::bool)))
((op =::'q_56812::type => 'q_56812::type => bool)
((CASEWISE::(('q_56854::type => 'q_56851::type) *
('q_56852::type
=> 'q_56854::type => 'q_56812::type)) hollight.list
=> 'q_56852::type => 'q_56851::type => 'q_56812::type)
((CONS::('q_56854::type => 'q_56851::type) *
('q_56852::type => 'q_56854::type => 'q_56812::type)
=> (('q_56854::type => 'q_56851::type) *
('q_56852::type
=> 'q_56854::type => 'q_56812::type)) hollight.list
=> (('q_56854::type => 'q_56851::type) *
('q_56852::type
=> 'q_56854::type => 'q_56812::type)) hollight.list)
((Pair::('q_56854::type => 'q_56851::type)
=> ('q_56852::type => 'q_56854::type => 'q_56812::type)
=> ('q_56854::type => 'q_56851::type) *
('q_56852::type => 'q_56854::type => 'q_56812::type))
(s::'q_56854::type => 'q_56851::type)
(t::'q_56852::type => 'q_56854::type => 'q_56812::type))
(clauses::(('q_56854::type => 'q_56851::type) *
('q_56852::type
=> 'q_56854::type => 'q_56812::type)) hollight.list))
f x)
((COND::bool => 'q_56812::type => 'q_56812::type => 'q_56812::type)
((Ex::('q_56854::type => bool) => bool)
(%y::'q_56854::type.
(op =::'q_56851::type => 'q_56851::type => bool) (s y) x))
(t f ((Eps::('q_56854::type => bool) => 'q_56854::type)
(%y::'q_56854::type.
(op =::'q_56851::type => 'q_56851::type => bool) (s y) x)))
((CASEWISE::(('q_56854::type => 'q_56851::type) *
('q_56852::type
=> 'q_56854::type => 'q_56812::type)) hollight.list
=> 'q_56852::type => 'q_56851::type => 'q_56812::type)
clauses f x)))"
by (import hollight CASEWISE)
lemma CASEWISE_CASES: "ALL (clauses::(('q_56946::type => 'q_56943::type) *
('q_56944::type
=> 'q_56946::type => 'q_56953::type)) hollight.list)
(c::'q_56944::type) x::'q_56943::type.
(EX (s::'q_56946::type => 'q_56943::type)
(t::'q_56944::type => 'q_56946::type => 'q_56953::type)
a::'q_56946::type.
MEM (s, t) clauses & s a = x & CASEWISE clauses c x = t c a) |
~ (EX (s::'q_56946::type => 'q_56943::type)
(t::'q_56944::type => 'q_56946::type => 'q_56953::type)
a::'q_56946::type. MEM (s, t) clauses & s a = x) &
CASEWISE clauses c x = (SOME y::'q_56953::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)
constdefs
admissible :: "('q_57089 => 'q_57082 => bool)
=> (('q_57089 => 'q_57085) => 'q_57095 => bool)
=> ('q_57095 => 'q_57082)
=> (('q_57089 => 'q_57085) => 'q_57095 => 'q_57090) => bool"
"admissible ==
%(u::'q_57089::type => 'q_57082::type => bool)
(ua::('q_57089::type => 'q_57085::type) => 'q_57095::type => bool)
(ub::'q_57095::type => 'q_57082::type)
uc::('q_57089::type => 'q_57085::type)
=> 'q_57095::type => 'q_57090::type.
ALL (f::'q_57089::type => 'q_57085::type)
(g::'q_57089::type => 'q_57085::type) a::'q_57095::type.
ua f a &
ua g a & (ALL z::'q_57089::type. u z (ub a) --> f z = g z) -->
uc f a = uc g a"
lemma DEF_admissible: "admissible =
(%(u::'q_57089::type => 'q_57082::type => bool)
(ua::('q_57089::type => 'q_57085::type) => 'q_57095::type => bool)
(ub::'q_57095::type => 'q_57082::type)
uc::('q_57089::type => 'q_57085::type)
=> 'q_57095::type => 'q_57090::type.
ALL (f::'q_57089::type => 'q_57085::type)
(g::'q_57089::type => 'q_57085::type) a::'q_57095::type.
ua f a &
ua g a & (ALL z::'q_57089::type. u z (ub a) --> f z = g z) -->
uc f a = uc g a)"
by (import hollight DEF_admissible)
constdefs
tailadmissible :: "('A => 'A => bool)
=> (('A => 'B) => 'P => bool)
=> ('P => 'A) => (('A => 'B) => 'P => 'B) => bool"
"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.
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) -->
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))"
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.
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) -->
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)))"
by (import hollight DEF_tailadmissible)
constdefs
superadmissible :: "('q_57239 => 'q_57239 => bool)
=> (('q_57239 => 'q_57241) => 'q_57247 => bool)
=> ('q_57247 => 'q_57239)
=> (('q_57239 => 'q_57241) => 'q_57247 => 'q_57241) => bool"
"superadmissible ==
%(u::'q_57239::type => 'q_57239::type => bool)
(ua::('q_57239::type => 'q_57241::type) => 'q_57247::type => bool)
(ub::'q_57247::type => 'q_57239::type)
uc::('q_57239::type => 'q_57241::type)
=> 'q_57247::type => 'q_57241::type.
admissible u
(%(f::'q_57239::type => 'q_57241::type) a::'q_57247::type. True) ub
ua -->
tailadmissible u ua ub uc"
lemma DEF_superadmissible: "superadmissible =
(%(u::'q_57239::type => 'q_57239::type => bool)
(ua::('q_57239::type => 'q_57241::type) => 'q_57247::type => bool)
(ub::'q_57247::type => 'q_57239::type)
uc::('q_57239::type => 'q_57241::type)
=> 'q_57247::type => 'q_57241::type.
admissible u
(%(f::'q_57239::type => 'q_57241::type) a::'q_57247::type. 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_354::'A::type => 'q_57627::type => bool)
(p::('A::type => 'B::type) => 'P::type => bool)
(P::('A::type => 'B::type) => 'P::type => bool)
(s::'P::type => 'q_57627::type)
(h::('A::type => 'B::type) => 'P::type => 'B::type)
k::('A::type => 'B::type) => 'P::type => 'B::type.
admissible u_354 p s P &
admissible u_354 (%(f::'A::type => 'B::type) x::'P::type. p f x & P f x)
s h &
admissible u_354
(%(f::'A::type => 'B::type) x::'P::type. p f x & ~ P f x) s k -->
admissible u_354 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_354::'q_57702::type => 'q_57701::type => bool)
(p::('q_57702::type => 'q_57703::type) => 'q_57704::type => bool)
(s::'q_57704::type => 'q_57701::type)
(%f::'q_57702::type => 'q_57703::type. c::'q_57704::type => 'q_57705::type)"
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))"
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))"
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 WF_REC_CASES: "ALL (u_354::'A::type => 'A::type => bool)
clauses::(('P::type => 'A::type) *
(('A::type => 'B::type)
=> 'P::type => 'B::type)) hollight.list.
WF u_354 &
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_354 y (G f a) --> u_354 y (s a)) &
(ALL (f::'A::type => 'B::type) (g::'A::type => 'B::type)
a::'P::type.
(ALL z::'A::type. u_354 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_58634::type)
(p2::'q_58645::type => 'A::type => 'q_58639::type)
(p1'::'A::type => 'q_58634::type)
p2'::'q_58645::type => 'A::type => 'q_58639::type.
(ALL (c::'q_58645::type) (x::'A::type) y::'A::type.
p1 x = p1' y --> p2 c x = p2' c y) -->
(ALL (c::'q_58645::type) (x::'A::type) y::'A::type.
p1' x = p1 y --> p2' c x = p2 c y)"
by (import hollight cth)
lemma RECURSION_CASEWISE_PAIRWISE: "ALL x::(('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)) hollight.list.
(EX u::'q_58662::type => 'q_58662::type => bool.
WF u &
ALL_list
(GABS
(%f::('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> bool.
ALL (s::'q_58682::type => 'q_58662::type)
t::('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type.
GEQ (f (s, t))
(tailadmissible u
(%(f::'q_58662::type => 'q_58678::type)
a::'q_58682::type. True)
s t)))
x) &
ALL_list
(GABS
(%f::('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> bool.
ALL (a::'q_58682::type => 'q_58662::type)
b::('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type.
GEQ (f (a, b))
(ALL (c::'q_58662::type => 'q_58678::type) (x::'q_58682::type)
y::'q_58682::type. a x = a y --> b c x = b c y)))
x &
PAIRWISE
(GABS
(%f::('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> ('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> bool.
ALL (s::'q_58682::type => 'q_58662::type)
t::('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type.
GEQ (f (s, t))
(GABS
(%f::('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> bool.
ALL (s'::'q_58682::type => 'q_58662::type)
t'::('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type.
GEQ (f (s', t'))
(ALL (c::'q_58662::type => 'q_58678::type)
(x::'q_58682::type) y::'q_58682::type.
s x = s' y --> t c x = t' c y)))))
x -->
(EX f::'q_58662::type => 'q_58678::type.
ALL_list
(GABS
(%fa::('q_58682::type => 'q_58662::type) *
(('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type)
=> bool.
ALL (s::'q_58682::type => 'q_58662::type)
t::('q_58662::type => 'q_58678::type)
=> 'q_58682::type => 'q_58678::type.
GEQ (fa (s, t)) (ALL x::'q_58682::type. f (s x) = t f x)))
x)"
by (import hollight RECURSION_CASEWISE_PAIRWISE)
lemma SUPERADMISSIBLE_T: "superadmissible (u_354::'q_58792::type => 'q_58792::type => bool)
(%(f::'q_58792::type => 'q_58794::type) x::'q_58798::type. True)
(s::'q_58798::type => 'q_58792::type)
(t::('q_58792::type => 'q_58794::type)
=> 'q_58798::type => 'q_58794::type) =
tailadmissible u_354
(%(f::'q_58792::type => 'q_58794::type) x::'q_58798::type. True) s t"
by (import hollight SUPERADMISSIBLE_T)
lemma SUB_SUB: "ALL (x::nat) xa::nat. <= xa x --> (ALL a::nat. a - (x - xa) = a + xa - x)"
by (import hollight SUB_SUB)
lemma SUB_OLD: "(ALL m::nat. 0 - m = 0) &
(ALL (m::nat) n::nat. Suc m - n = COND (< m n) 0 (Suc (m - n)))"
by (import hollight SUB_OLD)
lemma real_le: "ALL (x::hollight.real) xa::hollight.real. real_le x xa = (~ real_lt xa x)"
by (import hollight real_le)
lemma REAL_MUL_RID: "ALL x::hollight.real. real_mul x (real_of_num (NUMERAL_BIT1 0)) = x"
by (import hollight REAL_MUL_RID)
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)"
by (import hollight REAL_MUL_RINV)
lemma REAL_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)"
by (import hollight REAL_RDISTRIB)
lemma REAL_EQ_LADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
(real_add x y = real_add x z) = (y = z)"
by (import hollight REAL_EQ_LADD)
lemma REAL_EQ_RADD: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
(real_add x z = real_add y z) = (x = y)"
by (import hollight REAL_EQ_RADD)
lemma REAL_ADD_LID_UNIQ: "ALL (x::hollight.real) y::hollight.real.
(real_add x y = y) = (x = real_of_num 0)"
by (import hollight REAL_ADD_LID_UNIQ)
lemma REAL_ADD_RID_UNIQ: "ALL (x::hollight.real) y::hollight.real.
(real_add x y = x) = (y = real_of_num 0)"
by (import hollight REAL_ADD_RID_UNIQ)
lemma REAL_LNEG_UNIQ: "ALL (x::hollight.real) y::hollight.real.
(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)"
by (import hollight REAL_RNEG_UNIQ)
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)"
by (import hollight REAL_NEG_ADD)
lemma REAL_MUL_LZERO: "ALL x::hollight.real. real_mul (real_of_num 0) x = real_of_num 0"
by (import hollight REAL_MUL_LZERO)
lemma REAL_MUL_RZERO: "ALL x::hollight.real. real_mul x (real_of_num 0) = real_of_num 0"
by (import hollight REAL_MUL_RZERO)
lemma REAL_NEG_LMUL: "ALL (x::hollight.real) y::hollight.real.
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)"
by (import hollight REAL_NEG_RMUL)
lemma REAL_NEGNEG: "ALL x::hollight.real. real_neg (real_neg x) = x"
by (import hollight REAL_NEGNEG)
lemma REAL_NEG_MUL2: "ALL (x::hollight.real) xa::hollight.real.
real_mul (real_neg x) (real_neg xa) = real_mul x xa"
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"
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"
by (import hollight REAL_LT_RADD)
lemma REAL_NOT_LT: "ALL (x::hollight.real) y::hollight.real. (~ real_lt x y) = real_le y x"
by (import hollight REAL_NOT_LT)
lemma REAL_LT_ANTISYM: "ALL (x::hollight.real) y::hollight.real. ~ (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"
by (import hollight REAL_LT_GT)
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_LT: "ALL (x::hollight.real) y::hollight.real. real_le x y = (real_lt x y | x = y)"
by (import hollight REAL_LE_LT)
lemma REAL_LT_LE: "ALL (x::hollight.real) y::hollight.real.
real_lt x y = (real_le x y & x ~= y)"
by (import hollight REAL_LT_LE)
lemma REAL_LT_IMP_LE: "ALL (x::hollight.real) y::hollight.real. 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"
by (import hollight REAL_LTE_TRANS)
lemma REAL_LE_TRANS: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_le x y & real_le y z --> real_le x z"
by (import hollight REAL_LE_TRANS)
lemma REAL_NEG_LT0: "ALL x::hollight.real.
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)"
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"
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)"
by (import hollight REAL_NEG_GE0)
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)"
by (import hollight REAL_LT_NEGTOTAL)
lemma REAL_LE_NEGTOTAL: "ALL x::hollight.real.
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_MUL: "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_mul x y)"
by (import hollight REAL_LE_MUL)
lemma REAL_LE_SQUARE: "ALL x::hollight.real. real_le (real_of_num 0) (real_mul x x)"
by (import hollight REAL_LE_SQUARE)
lemma REAL_LT_01: "real_lt (real_of_num 0) (real_of_num (NUMERAL_BIT1 0))"
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"
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"
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)"
by (import hollight REAL_LT_ADD2)
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)"
by (import hollight REAL_LT_ADD)
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"
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)"
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)))"
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"
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"
by (import hollight REAL_SUB_ADD2)
lemma REAL_SUB_REFL: "ALL x::hollight.real. real_sub x x = real_of_num 0"
by (import hollight REAL_SUB_REFL)
lemma REAL_SUB_0: "ALL (x::hollight.real) y::hollight.real.
(real_sub x y = real_of_num 0) = (x = y)"
by (import hollight REAL_SUB_0)
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"
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"
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)"
by (import hollight REAL_LE_NEGR)
lemma REAL_NEG_EQ0: "ALL x::hollight.real. (real_neg x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight REAL_NEG_EQ0)
lemma REAL_NEG_0: "real_neg (real_of_num 0) = real_of_num 0"
by (import hollight REAL_NEG_0)
lemma REAL_NEG_SUB: "ALL (x::hollight.real) y::hollight.real.
real_neg (real_sub x y) = real_sub y x"
by (import hollight REAL_NEG_SUB)
lemma REAL_SUB_LT: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) (real_sub x y) = real_lt y x"
by (import hollight REAL_SUB_LT)
lemma REAL_SUB_LE: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) (real_sub x y) = real_le y x"
by (import hollight REAL_SUB_LE)
lemma REAL_EQ_LMUL: "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)"
by (import hollight REAL_EQ_LMUL)
lemma REAL_EQ_RMUL: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
(real_mul x z = real_mul y z) = (z = real_of_num 0 | x = y)"
by (import hollight REAL_EQ_RMUL)
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)"
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)"
by (import hollight REAL_SUB_RDISTRIB)
lemma REAL_NEG_EQ: "ALL (x::hollight.real) y::hollight.real. (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"
by (import hollight REAL_NEG_MINUS1)
lemma REAL_INV_NZ: "ALL x::hollight.real. x ~= real_of_num 0 --> real_inv x ~= real_of_num 0"
by (import hollight REAL_INV_NZ)
lemma REAL_INVINV: "ALL x::hollight.real. x ~= real_of_num 0 --> real_inv (real_inv x) = x"
by (import hollight REAL_INVINV)
lemma REAL_LT_IMP_NE: "ALL (x::hollight.real) y::hollight.real. real_lt x y --> x ~= y"
by (import hollight REAL_LT_IMP_NE)
lemma REAL_INV_POS: "ALL x::hollight.real.
real_lt (real_of_num 0) x --> real_lt (real_of_num 0) (real_inv x)"
by (import hollight REAL_INV_POS)
lemma REAL_LT_LMUL_0: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) x -->
real_lt (real_of_num 0) (real_mul x y) = real_lt (real_of_num 0) y"
by (import hollight REAL_LT_LMUL_0)
lemma REAL_LT_RMUL_0: "ALL (x::hollight.real) y::hollight.real.
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_RMUL_0)
lemma REAL_LT_LMUL_EQ: "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"
by (import hollight REAL_LT_LMUL_EQ)
lemma REAL_LT_RMUL_EQ: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt (real_of_num 0) z -->
real_lt (real_mul x z) (real_mul y z) = real_lt x y"
by (import hollight REAL_LT_RMUL_EQ)
lemma REAL_LT_RMUL_IMP: "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)"
by (import hollight REAL_LT_RMUL_IMP)
lemma REAL_LT_LMUL_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt y z & real_lt (real_of_num 0) x -->
real_lt (real_mul x y) (real_mul x z)"
by (import hollight REAL_LT_LMUL_IMP)
lemma REAL_LINV_UNIQ: "ALL (x::hollight.real) y::hollight.real.
real_mul x y = real_of_num (NUMERAL_BIT1 0) --> x = real_inv y"
by (import hollight REAL_LINV_UNIQ)
lemma REAL_RINV_UNIQ: "ALL (x::hollight.real) y::hollight.real.
real_mul x y = real_of_num (NUMERAL_BIT1 0) --> y = real_inv x"
by (import hollight REAL_RINV_UNIQ)
lemma REAL_NEG_INV: "ALL x::hollight.real.
x ~= real_of_num 0 --> real_neg (real_inv x) = real_inv (real_neg x)"
by (import hollight REAL_NEG_INV)
lemma REAL_INV_1OVER: "ALL x::hollight.real. real_inv x = real_div (real_of_num (NUMERAL_BIT1 0)) x"
by (import hollight REAL_INV_1OVER)
lemma REAL: "ALL x::nat.
real_of_num (Suc x) =
real_add (real_of_num x) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight REAL)
lemma REAL_POS: "ALL n::nat. real_le (real_of_num 0) (real_of_num n)"
by (import hollight REAL_POS)
lemma REAL_LE: "ALL (m::nat) n::nat. real_le (real_of_num m) (real_of_num n) = <= m n"
by (import hollight REAL_LE)
lemma REAL_LT: "ALL (m::nat) n::nat. real_lt (real_of_num m) (real_of_num n) = < m n"
by (import hollight REAL_LT)
lemma th: "((m::nat) = (n::nat)) = (<= m n & <= n m)"
by (import hollight th)
lemma REAL_INJ: "ALL (m::nat) n::nat. (real_of_num m = real_of_num n) = (m = n)"
by (import hollight REAL_INJ)
lemma REAL_ADD: "ALL (m::nat) n::nat.
real_add (real_of_num m) (real_of_num n) = real_of_num (m + n)"
by (import hollight REAL_ADD)
lemma REAL_MUL: "ALL (m::nat) n::nat.
real_mul (real_of_num m) (real_of_num n) = real_of_num (m * n)"
by (import hollight REAL_MUL)
lemma REAL_INV1: "real_inv (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight REAL_INV1)
lemma REAL_DIV_LZERO: "ALL x::hollight.real. real_div (real_of_num 0) x = real_of_num 0"
by (import hollight REAL_DIV_LZERO)
lemma REAL_LT_NZ: "ALL n::nat.
(real_of_num n ~= real_of_num 0) =
real_lt (real_of_num 0) (real_of_num n)"
by (import hollight REAL_LT_NZ)
lemma REAL_NZ_IMP_LT: "ALL n::nat. n ~= 0 --> real_lt (real_of_num 0) (real_of_num n)"
by (import hollight REAL_NZ_IMP_LT)
lemma REAL_LT_RDIV_0: "ALL (y::hollight.real) z::hollight.real.
real_lt (real_of_num 0) z -->
real_lt (real_of_num 0) (real_div y z) = real_lt (real_of_num 0) y"
by (import hollight REAL_LT_RDIV_0)
lemma REAL_LT_RDIV: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt (real_of_num 0) z -->
real_lt (real_div x z) (real_div y z) = real_lt x y"
by (import hollight REAL_LT_RDIV)
lemma REAL_LT_FRACTION_0: "ALL (n::nat) d::hollight.real.
n ~= 0 -->
real_lt (real_of_num 0) (real_div d (real_of_num n)) =
real_lt (real_of_num 0) d"
by (import hollight REAL_LT_FRACTION_0)
lemma REAL_LT_MULTIPLE: "ALL (x::nat) xa::hollight.real.
< (NUMERAL_BIT1 0) x -->
real_lt xa (real_mul (real_of_num x) xa) = real_lt (real_of_num 0) xa"
by (import hollight REAL_LT_MULTIPLE)
lemma REAL_LT_FRACTION: "ALL (n::nat) d::hollight.real.
< (NUMERAL_BIT1 0) n -->
real_lt (real_div d (real_of_num n)) d = real_lt (real_of_num 0) d"
by (import hollight REAL_LT_FRACTION)
lemma REAL_LT_HALF1: "ALL d::hollight.real.
real_lt (real_of_num 0)
(real_div d (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) =
real_lt (real_of_num 0) d"
by (import hollight REAL_LT_HALF1)
lemma REAL_LT_HALF2: "ALL d::hollight.real.
real_lt (real_div d (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) d =
real_lt (real_of_num 0) d"
by (import hollight REAL_LT_HALF2)
lemma REAL_DOUBLE: "ALL x::hollight.real.
real_add x x = real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x"
by (import hollight REAL_DOUBLE)
lemma REAL_HALF_DOUBLE: "ALL x::hollight.real.
real_add (real_div x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_div x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) =
x"
by (import hollight REAL_HALF_DOUBLE)
lemma REAL_SUB_SUB: "ALL (x::hollight.real) y::hollight.real.
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)"
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)"
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"
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"
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)"
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"
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"
by (import hollight REAL_LE_NEG)
lemma REAL_SUB_LZERO: "ALL x::hollight.real. 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"
by (import hollight REAL_SUB_RZERO)
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)"
by (import hollight REAL_LTE_ADD2)
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)"
by (import hollight REAL_LTE_ADD)
lemma REAL_LT_MUL2_ALT: "ALL (x1::hollight.real) (x2::hollight.real) (y1::hollight.real)
y2::hollight.real.
real_le (real_of_num 0) x1 &
real_le (real_of_num 0) y1 & real_lt x1 x2 & real_lt y1 y2 -->
real_lt (real_mul x1 y1) (real_mul x2 y2)"
by (import hollight REAL_LT_MUL2_ALT)
lemma REAL_SUB_LNEG: "ALL (x::hollight.real) y::hollight.real.
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"
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"
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"
by (import hollight REAL_SUB_TRIANGLE)
lemma REAL_INV_MUL_WEAK: "ALL (x::hollight.real) xa::hollight.real.
x ~= real_of_num 0 & xa ~= real_of_num 0 -->
real_inv (real_mul x xa) = real_mul (real_inv x) (real_inv xa)"
by (import hollight REAL_INV_MUL_WEAK)
lemma REAL_LE_LMUL_LOCAL: "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"
by (import hollight REAL_LE_LMUL_LOCAL)
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"
by (import hollight REAL_LE_RMUL_EQ)
lemma REAL_SUB_INV2: "ALL (x::hollight.real) y::hollight.real.
x ~= real_of_num 0 & y ~= real_of_num 0 -->
real_sub (real_inv x) (real_inv y) =
real_div (real_sub y x) (real_mul x y)"
by (import hollight REAL_SUB_INV2)
lemma REAL_SUB_SUB2: "ALL (x::hollight.real) y::hollight.real. real_sub x (real_sub x y) = y"
by (import hollight REAL_SUB_SUB2)
lemma REAL_MEAN: "ALL (x::hollight.real) y::hollight.real.
real_lt x y --> (EX z::hollight.real. real_lt x z & real_lt z y)"
by (import hollight REAL_MEAN)
lemma REAL_EQ_LMUL2: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
x ~= real_of_num 0 --> (y = z) = (real_mul x y = real_mul x z)"
by (import hollight REAL_EQ_LMUL2)
lemma REAL_LE_MUL2V: "ALL (x1::hollight.real) (x2::hollight.real) (y1::hollight.real)
y2::hollight.real.
real_le (real_of_num 0) x1 &
real_le (real_of_num 0) y1 & real_le x1 x2 & real_le y1 y2 -->
real_le (real_mul x1 y1) (real_mul x2 y2)"
by (import hollight REAL_LE_MUL2V)
lemma REAL_LE_LDIV: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt (real_of_num 0) x & real_le y (real_mul z x) -->
real_le (real_div y x) z"
by (import hollight REAL_LE_LDIV)
lemma REAL_LE_RDIV: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt (real_of_num 0) x & real_le (real_mul y x) z -->
real_le y (real_div z x)"
by (import hollight REAL_LE_RDIV)
lemma REAL_LT_1: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_lt x y -->
real_lt (real_div x y) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight REAL_LT_1)
lemma REAL_LE_LMUL_IMP: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_le (real_of_num 0) x & real_le y z -->
real_le (real_mul x y) (real_mul x z)"
by (import hollight REAL_LE_LMUL_IMP)
lemma REAL_LE_RMUL_IMP: "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 xa x) (real_mul xb x)"
by (import hollight REAL_LE_RMUL_IMP)
lemma REAL_INV_LT1: "ALL x::hollight.real.
real_lt (real_of_num 0) x & real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
real_lt (real_of_num (NUMERAL_BIT1 0)) (real_inv x)"
by (import hollight REAL_INV_LT1)
lemma REAL_POS_NZ: "ALL x::hollight.real. real_lt (real_of_num 0) x --> x ~= real_of_num 0"
by (import hollight REAL_POS_NZ)
lemma REAL_EQ_RMUL_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"
by (import hollight REAL_EQ_RMUL_IMP)
lemma REAL_EQ_LMUL_IMP: "ALL (x::hollight.real) (xa::hollight.real) xb::hollight.real.
x ~= real_of_num 0 & real_mul x xa = real_mul x xb --> xa = xb"
by (import hollight REAL_EQ_LMUL_IMP)
lemma REAL_FACT_NZ: "ALL n::nat. real_of_num (FACT n) ~= real_of_num 0"
by (import hollight REAL_FACT_NZ)
lemma REAL_POSSQ: "ALL x::hollight.real.
real_lt (real_of_num 0) (real_mul x x) = (x ~= real_of_num 0)"
by (import hollight REAL_POSSQ)
lemma REAL_SUMSQ: "ALL (x::hollight.real) y::hollight.real.
(real_add (real_mul x x) (real_mul y y) = real_of_num 0) =
(x = real_of_num 0 & y = real_of_num 0)"
by (import hollight REAL_SUMSQ)
lemma REAL_EQ_NEG: "ALL (x::hollight.real) y::hollight.real. (real_neg x = real_neg y) = (x = y)"
by (import hollight REAL_EQ_NEG)
lemma REAL_DIV_MUL2: "ALL (x::hollight.real) z::hollight.real.
x ~= real_of_num 0 & z ~= real_of_num 0 -->
(ALL y::hollight.real.
real_div y z = real_div (real_mul x y) (real_mul x z))"
by (import hollight REAL_DIV_MUL2)
lemma REAL_MIDDLE1: "ALL (a::hollight.real) b::hollight.real.
real_le a b -->
real_le a
(real_div (real_add a b) (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight REAL_MIDDLE1)
lemma REAL_MIDDLE2: "ALL (a::hollight.real) b::hollight.real.
real_le a b -->
real_le
(real_div (real_add a b) (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
b"
by (import hollight REAL_MIDDLE2)
lemma ABS_ZERO: "ALL x::hollight.real. (real_abs x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight ABS_ZERO)
lemma ABS_0: "real_abs (real_of_num 0) = real_of_num 0"
by (import hollight ABS_0)
lemma ABS_1: "real_abs (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight ABS_1)
lemma ABS_NEG: "ALL x::hollight.real. real_abs (real_neg x) = real_abs x"
by (import hollight ABS_NEG)
lemma 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))"
by (import hollight ABS_TRIANGLE)
lemma ABS_POS: "ALL x::hollight.real. real_le (real_of_num 0) (real_abs x)"
by (import hollight ABS_POS)
lemma ABS_MUL: "ALL (x::hollight.real) y::hollight.real.
real_abs (real_mul x y) = real_mul (real_abs x) (real_abs y)"
by (import hollight ABS_MUL)
lemma ABS_LT_MUL2: "ALL (w::hollight.real) (x::hollight.real) (y::hollight.real)
z::hollight.real.
real_lt (real_abs w) y & real_lt (real_abs x) z -->
real_lt (real_abs (real_mul w x)) (real_mul y z)"
by (import hollight ABS_LT_MUL2)
lemma ABS_SUB: "ALL (x::hollight.real) y::hollight.real.
real_abs (real_sub x y) = real_abs (real_sub y x)"
by (import hollight ABS_SUB)
lemma ABS_NZ: "ALL x::hollight.real.
(x ~= real_of_num 0) = real_lt (real_of_num 0) (real_abs x)"
by (import hollight ABS_NZ)
lemma ABS_INV: "ALL x::hollight.real.
x ~= real_of_num 0 --> real_abs (real_inv x) = real_inv (real_abs x)"
by (import hollight ABS_INV)
lemma ABS_ABS: "ALL x::hollight.real. real_abs (real_abs x) = real_abs x"
by (import hollight ABS_ABS)
lemma ABS_LE: "ALL x::hollight.real. real_le x (real_abs x)"
by (import hollight ABS_LE)
lemma ABS_REFL: "ALL x::hollight.real. (real_abs x = x) = real_le (real_of_num 0) x"
by (import hollight ABS_REFL)
lemma ABS_N: "ALL n::nat. real_abs (real_of_num n) = real_of_num n"
by (import hollight ABS_N)
lemma 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"
by (import hollight ABS_BETWEEN)
lemma 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)"
by (import hollight ABS_BOUND)
lemma 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"
by (import hollight ABS_STILLNZ)
lemma ABS_CASES: "ALL x::hollight.real.
x = real_of_num 0 | real_lt (real_of_num 0) (real_abs x)"
by (import hollight ABS_CASES)
lemma 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"
by (import hollight ABS_BETWEEN1)
lemma 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"
by (import hollight ABS_SIGN)
lemma 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)"
by (import hollight ABS_SIGN2)
lemma ABS_DIV: "ALL y::hollight.real.
y ~= real_of_num 0 -->
(ALL x::hollight.real.
real_abs (real_div x y) = real_div (real_abs x) (real_abs y))"
by (import hollight ABS_DIV)
lemma 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)"
by (import hollight 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))"
by (import hollight REAL_SUB_ABS)
lemma 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))"
by (import hollight ABS_SUB_ABS)
lemma ABS_BETWEEN2: "ALL (x0::hollight.real) (x::hollight.real) (y0::hollight.real)
y::hollight.real.
real_lt x0 y0 &
real_lt (real_abs (real_sub x x0))
(real_div (real_sub y0 x0)
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
real_lt (real_abs (real_sub y y0))
(real_div (real_sub y0 x0)
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt x y"
by (import hollight ABS_BETWEEN2)
lemma 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)"
by (import hollight ABS_BOUNDS)
lemma POW_0: "ALL n::nat. real_pow (real_of_num 0) (Suc n) = real_of_num 0"
by (import hollight POW_0)
lemma POW_NZ: "ALL (c::hollight.real) n::nat.
c ~= real_of_num 0 --> real_pow c n ~= real_of_num 0"
by (import hollight POW_NZ)
lemma POW_INV: "ALL (c::hollight.real) x::nat.
c ~= real_of_num 0 --> real_inv (real_pow c x) = real_pow (real_inv c) x"
by (import hollight POW_INV)
lemma POW_ABS: "ALL (c::hollight.real) n::nat.
real_pow (real_abs c) n = real_abs (real_pow c n)"
by (import hollight POW_ABS)
lemma POW_PLUS1: "ALL (e::hollight.real) x::nat.
real_lt (real_of_num 0) e -->
real_le
(real_add (real_of_num (NUMERAL_BIT1 0)) (real_mul (real_of_num x) e))
(real_pow (real_add (real_of_num (NUMERAL_BIT1 0)) e) x)"
by (import hollight POW_PLUS1)
lemma POW_ADD: "ALL (c::hollight.real) (m::nat) n::nat.
real_pow c (m + n) = real_mul (real_pow c m) (real_pow c n)"
by (import hollight POW_ADD)
lemma POW_1: "ALL x::hollight.real. real_pow x (NUMERAL_BIT1 0) = x"
by (import hollight POW_1)
lemma POW_2: "ALL x::hollight.real.
real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = real_mul x x"
by (import hollight POW_2)
lemma POW_POS: "ALL (x::hollight.real) xa::nat.
real_le (real_of_num 0) x --> real_le (real_of_num 0) (real_pow x xa)"
by (import hollight POW_POS)
lemma POW_LE: "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)"
by (import hollight POW_LE)
lemma POW_M1: "ALL n::nat.
real_abs (real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) n) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight POW_M1)
lemma POW_MUL: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_pow (real_mul x y) n = real_mul (real_pow x n) (real_pow y n)"
by (import hollight POW_MUL)
lemma REAL_LE_SQUARE_POW: "ALL x::hollight.real.
real_le (real_of_num 0) (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight REAL_LE_SQUARE_POW)
lemma ABS_POW2: "ALL x::hollight.real.
real_abs (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))"
by (import hollight ABS_POW2)
lemma REAL_LE1_POW2: "ALL x::hollight.real.
real_le (real_of_num (NUMERAL_BIT1 0)) x -->
real_le (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight REAL_LE1_POW2)
lemma REAL_LT1_POW2: "ALL x::hollight.real.
real_lt (real_of_num (NUMERAL_BIT1 0)) x -->
real_lt (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight REAL_LT1_POW2)
lemma POW_POS_LT: "ALL (x::hollight.real) n::nat.
real_lt (real_of_num 0) x -->
real_lt (real_of_num 0) (real_pow x (Suc n))"
by (import hollight POW_POS_LT)
lemma POW_2_LE1: "ALL n::nat.
real_le (real_of_num (NUMERAL_BIT1 0))
(real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) n)"
by (import hollight POW_2_LE1)
lemma POW_2_LT: "ALL n::nat.
real_lt (real_of_num n)
(real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) n)"
by (import hollight POW_2_LT)
lemma POW_MINUS1: "ALL n::nat.
real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(NUMERAL_BIT0 (NUMERAL_BIT1 0) * n) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight POW_MINUS1)
lemma REAL_SUP_EXISTS: "ALL P::hollight.real => bool.
Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_lt x z) -->
(EX s::hollight.real.
ALL y::hollight.real.
(EX x::hollight.real. P x & real_lt y x) = real_lt y s)"
by (import hollight REAL_SUP_EXISTS)
constdefs
sup :: "(hollight.real => bool) => hollight.real"
"sup ==
%u::hollight.real => bool.
SOME a::hollight.real.
(ALL x::hollight.real. IN x u --> real_le x a) &
(ALL b::hollight.real.
(ALL x::hollight.real. IN x u --> real_le x b) --> real_le a b)"
lemma DEF_sup: "sup =
(%u::hollight.real => bool.
SOME a::hollight.real.
(ALL x::hollight.real. IN x u --> real_le x a) &
(ALL b::hollight.real.
(ALL x::hollight.real. IN x u --> real_le x b) --> real_le a b))"
by (import hollight DEF_sup)
lemma sup: "sup (P::hollight.real => bool) =
(SOME s::hollight.real.
ALL y::hollight.real.
(EX x::hollight.real. P x & real_lt y x) = real_lt y s)"
by (import hollight sup)
lemma REAL_SUP: "ALL P::hollight.real => bool.
Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_lt x z) -->
(ALL y::hollight.real.
(EX x::hollight.real. P x & real_lt y x) = real_lt y (sup P))"
by (import hollight REAL_SUP)
lemma REAL_SUP_UBOUND: "ALL P::hollight.real => bool.
Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_lt x z) -->
(ALL y::hollight.real. P y --> real_le y (sup P))"
by (import hollight REAL_SUP_UBOUND)
lemma SETOK_LE_LT: "ALL P::hollight.real => bool.
(Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_le x z)) =
(Ex P &
(EX x::hollight.real. ALL xa::hollight.real. P xa --> real_lt xa x))"
by (import hollight SETOK_LE_LT)
lemma REAL_SUP_LE: "ALL P::hollight.real => bool.
Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_le x z) -->
(ALL y::hollight.real.
(EX x::hollight.real. P x & real_lt y x) = real_lt y (sup P))"
by (import hollight REAL_SUP_LE)
lemma REAL_SUP_UBOUND_LE: "ALL P::hollight.real => bool.
Ex P &
(EX z::hollight.real. ALL x::hollight.real. P x --> real_le x z) -->
(ALL y::hollight.real. P y --> real_le y (sup P))"
by (import hollight REAL_SUP_UBOUND_LE)
lemma REAL_ARCH_SIMPLE: "ALL x::hollight.real. EX n::nat. real_le x (real_of_num n)"
by (import hollight REAL_ARCH_SIMPLE)
lemma REAL_ARCH: "ALL x::hollight.real.
real_lt (real_of_num 0) x -->
(ALL y::hollight.real. EX n::nat. real_lt y (real_mul (real_of_num n) x))"
by (import hollight REAL_ARCH)
lemma REAL_ARCH_LEAST: "ALL y::hollight.real.
real_lt (real_of_num 0) y -->
(ALL x::hollight.real.
real_le (real_of_num 0) x -->
(EX n::nat.
real_le (real_mul (real_of_num n) y) x &
real_lt x (real_mul (real_of_num (Suc n)) y)))"
by (import hollight REAL_ARCH_LEAST)
lemma sum_EXISTS: "EX x::nat * nat => (nat => hollight.real) => hollight.real.
(ALL (f::nat => hollight.real) n::nat. x (n, 0) f = real_of_num 0) &
(ALL (f::nat => hollight.real) (m::nat) n::nat.
x (n, Suc m) f = real_add (x (n, m) f) (f (n + m)))"
by (import hollight sum_EXISTS)
constdefs
psum :: "nat * nat => (nat => hollight.real) => hollight.real"
"psum ==
SOME sum::nat * nat => (nat => hollight.real) => hollight.real.
(ALL (f::nat => hollight.real) n::nat. sum (n, 0) f = real_of_num 0) &
(ALL (f::nat => hollight.real) (m::nat) n::nat.
sum (n, Suc m) f = real_add (sum (n, m) f) (f (n + m)))"
lemma DEF_psum: "psum =
(SOME sum::nat * nat => (nat => hollight.real) => hollight.real.
(ALL (f::nat => hollight.real) n::nat. sum (n, 0) f = real_of_num 0) &
(ALL (f::nat => hollight.real) (m::nat) n::nat.
sum (n, Suc m) f = real_add (sum (n, m) f) (f (n + m))))"
by (import hollight DEF_psum)
lemma sum: "psum (n::nat, 0) (f::nat => hollight.real) = real_of_num 0 &
psum (n, Suc (m::nat)) f = real_add (psum (n, m) f) (f (n + m))"
by (import hollight sum)
lemma PSUM_SUM: "ALL (f::nat => hollight.real) (m::nat) n::nat.
psum (m, n) f =
hollight.sum
(GSPEC (%u::nat. EX i::nat. SETSPEC u (<= m i & < i (m + n)) i)) f"
by (import hollight PSUM_SUM)
lemma PSUM_SUM_NUMSEG: "ALL (f::nat => hollight.real) (m::nat) n::nat.
~ (m = 0 & n = 0) -->
psum (m, n) f = hollight.sum (dotdot m (m + n - NUMERAL_BIT1 0)) f"
by (import hollight PSUM_SUM_NUMSEG)
lemma SUM_TWO: "ALL (f::nat => hollight.real) (n::nat) p::nat.
real_add (psum (0, n) f) (psum (n, p) f) = psum (0, n + p) f"
by (import hollight SUM_TWO)
lemma SUM_DIFF: "ALL (f::nat => hollight.real) (m::nat) n::nat.
psum (m, n) f = real_sub (psum (0, m + n) f) (psum (0, m) f)"
by (import hollight SUM_DIFF)
lemma ABS_SUM: "ALL (f::nat => hollight.real) (m::nat) n::nat.
real_le (real_abs (psum (m, n) f))
(psum (m, n) (%n::nat. real_abs (f n)))"
by (import hollight ABS_SUM)
lemma SUM_LE: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
(ALL r::nat. <= m r & < r (n + m) --> real_le (f r) (g r)) -->
real_le (psum (m, n) f) (psum (m, n) g)"
by (import hollight SUM_LE)
lemma SUM_EQ: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
(ALL r::nat. <= m r & < r (n + m) --> f r = g r) -->
psum (m, n) f = psum (m, n) g"
by (import hollight SUM_EQ)
lemma SUM_POS: "ALL f::nat => hollight.real.
(ALL n::nat. real_le (real_of_num 0) (f n)) -->
(ALL (m::nat) n::nat. real_le (real_of_num 0) (psum (m, n) f))"
by (import hollight SUM_POS)
lemma SUM_POS_GEN: "ALL (f::nat => hollight.real) (m::nat) n::nat.
(ALL n::nat. <= m n --> real_le (real_of_num 0) (f n)) -->
real_le (real_of_num 0) (psum (m, n) f)"
by (import hollight SUM_POS_GEN)
lemma SUM_ABS: "ALL (f::nat => hollight.real) (m::nat) x::nat.
real_abs (psum (m, x) (%m::nat. real_abs (f m))) =
psum (m, x) (%m::nat. real_abs (f m))"
by (import hollight SUM_ABS)
lemma SUM_ABS_LE: "ALL (f::nat => hollight.real) (m::nat) n::nat.
real_le (real_abs (psum (m, n) f))
(psum (m, n) (%n::nat. real_abs (f n)))"
by (import hollight SUM_ABS_LE)
lemma SUM_ZERO: "ALL (f::nat => hollight.real) N::nat.
(ALL n::nat. >= n N --> f n = real_of_num 0) -->
(ALL (m::nat) n::nat. >= m N --> psum (m, n) f = real_of_num 0)"
by (import hollight SUM_ZERO)
lemma SUM_ADD: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
psum (m, n) (%n::nat. real_add (f n) (g n)) =
real_add (psum (m, n) f) (psum (m, n) g)"
by (import hollight SUM_ADD)
lemma SUM_CMUL: "ALL (f::nat => hollight.real) (c::hollight.real) (m::nat) n::nat.
psum (m, n) (%n::nat. real_mul c (f n)) = real_mul c (psum (m, n) f)"
by (import hollight SUM_CMUL)
lemma SUM_NEG: "ALL (f::nat => hollight.real) (n::nat) d::nat.
psum (n, d) (%n::nat. real_neg (f n)) = real_neg (psum (n, d) f)"
by (import hollight SUM_NEG)
lemma SUM_SUB: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
psum (m, n) (%x::nat. real_sub (f x) (g x)) =
real_sub (psum (m, n) f) (psum (m, n) g)"
by (import hollight SUM_SUB)
lemma SUM_SUBST: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (m::nat) n::nat.
(ALL p::nat. <= m p & < p (m + n) --> f p = g p) -->
psum (m, n) f = psum (m, n) g"
by (import hollight SUM_SUBST)
lemma SUM_NSUB: "ALL (n::nat) (f::nat => hollight.real) c::hollight.real.
real_sub (psum (0, n) f) (real_mul (real_of_num n) c) =
psum (0, n) (%p::nat. real_sub (f p) c)"
by (import hollight SUM_NSUB)
lemma SUM_BOUND: "ALL (f::nat => hollight.real) (K::hollight.real) (m::nat) n::nat.
(ALL p::nat. <= m p & < p (m + n) --> real_le (f p) K) -->
real_le (psum (m, n) f) (real_mul (real_of_num n) K)"
by (import hollight SUM_BOUND)
lemma SUM_GROUP: "ALL (n::nat) (k::nat) f::nat => hollight.real.
psum (0, n) (%m::nat. psum (m * k, k) f) = psum (0, n * k) f"
by (import hollight SUM_GROUP)
lemma SUM_1: "ALL (f::nat => hollight.real) n::nat. psum (n, NUMERAL_BIT1 0) f = f n"
by (import hollight SUM_1)
lemma SUM_2: "ALL (f::nat => hollight.real) n::nat.
psum (n, NUMERAL_BIT0 (NUMERAL_BIT1 0)) f =
real_add (f n) (f (n + NUMERAL_BIT1 0))"
by (import hollight SUM_2)
lemma SUM_OFFSET: "ALL (f::nat => hollight.real) (n::nat) k::nat.
psum (0, n) (%m::nat. f (m + k)) =
real_sub (psum (0, n + k) f) (psum (0, k) f)"
by (import hollight SUM_OFFSET)
lemma SUM_REINDEX: "ALL (f::nat => hollight.real) (m::nat) (k::nat) n::nat.
psum (m + k, n) f = psum (m, n) (%r::nat. f (r + k))"
by (import hollight SUM_REINDEX)
lemma SUM_0: "ALL (m::nat) n::nat. psum (m, n) (%r::nat. real_of_num 0) = real_of_num 0"
by (import hollight SUM_0)
lemma SUM_CANCEL: "ALL (f::nat => hollight.real) (n::nat) d::nat.
psum (n, d) (%n::nat. real_sub (f (Suc n)) (f n)) =
real_sub (f (n + d)) (f n)"
by (import hollight SUM_CANCEL)
lemma SUM_HORNER: "ALL (f::nat => hollight.real) (n::nat) x::hollight.real.
psum (0, Suc n) (%i::nat. real_mul (f i) (real_pow x i)) =
real_add (f 0)
(real_mul x
(psum (0, n) (%i::nat. real_mul (f (Suc i)) (real_pow x i))))"
by (import hollight SUM_HORNER)
lemma SUM_CONST: "ALL (c::hollight.real) n::nat.
psum (0, n) (%m::nat. c) = real_mul (real_of_num n) c"
by (import hollight SUM_CONST)
lemma SUM_SPLIT: "ALL (f::nat => hollight.real) (n::nat) p::nat.
real_add (psum (m::nat, n) f) (psum (m + n, p) f) = psum (m, n + p) f"
by (import hollight SUM_SPLIT)
lemma SUM_SWAP: "ALL (f::nat => nat => hollight.real) (m1::nat) (n1::nat) (m2::nat) n2::nat.
psum (m1, n1) (%a::nat. psum (m2, n2) (f a)) =
psum (m2, n2) (%b::nat. psum (m1, n1) (%a::nat. f a b))"
by (import hollight SUM_SWAP)
lemma SUM_EQ_0: "(ALL r::nat.
<= (m::nat) r & < r (m + (n::nat)) -->
(f::nat => hollight.real) r = real_of_num 0) -->
psum (m, n) f = real_of_num 0"
by (import hollight SUM_EQ_0)
lemma SUM_MORETERMS_EQ: "ALL (m::nat) (n::nat) p::nat.
<= n p &
(ALL r::nat.
<= (m + n) r & < r (m + p) -->
(f::nat => hollight.real) r = real_of_num 0) -->
psum (m, p) f = psum (m, n) f"
by (import hollight SUM_MORETERMS_EQ)
lemma SUM_DIFFERENCES_EQ: "ALL (x::nat) (xa::nat) xb::nat.
<= xa xb &
(ALL r::nat.
<= (x + xa) r & < r (x + xb) -->
(f::nat => hollight.real) r = (g::nat => hollight.real) r) -->
real_sub (psum (x, xb) f) (psum (x, xa) f) =
real_sub (psum (x, xb) g) (psum (x, xa) g)"
by (import hollight SUM_DIFFERENCES_EQ)
constdefs
re_Union :: "(('A => bool) => bool) => 'A => bool"
"re_Union ==
%(u::('A::type => bool) => bool) x::'A::type.
EX s::'A::type => bool. u s & s x"
lemma DEF_re_Union: "re_Union =
(%(u::('A::type => bool) => bool) x::'A::type.
EX s::'A::type => bool. u s & s x)"
by (import hollight DEF_re_Union)
constdefs
re_union :: "('A => bool) => ('A => bool) => 'A => bool"
"re_union ==
%(u::'A::type => bool) (ua::'A::type => bool) x::'A::type. u x | ua x"
lemma DEF_re_union: "re_union =
(%(u::'A::type => bool) (ua::'A::type => bool) x::'A::type. u x | ua x)"
by (import hollight DEF_re_union)
constdefs
re_intersect :: "('A => bool) => ('A => bool) => 'A => bool"
"re_intersect ==
%(u::'A::type => bool) (ua::'A::type => bool) x::'A::type. u x & ua x"
lemma DEF_re_intersect: "re_intersect =
(%(u::'A::type => bool) (ua::'A::type => bool) x::'A::type. u x & ua x)"
by (import hollight DEF_re_intersect)
constdefs
re_null :: "'A => bool"
"re_null == %x::'A::type. False"
lemma DEF_re_null: "re_null = (%x::'A::type. False)"
by (import hollight DEF_re_null)
constdefs
re_universe :: "'A => bool"
"re_universe == %x::'A::type. True"
lemma DEF_re_universe: "re_universe = (%x::'A::type. True)"
by (import hollight DEF_re_universe)
constdefs
re_subset :: "('A => bool) => ('A => bool) => bool"
"re_subset ==
%(u::'A::type => bool) ua::'A::type => bool. ALL x::'A::type. u x --> ua x"
lemma DEF_re_subset: "re_subset =
(%(u::'A::type => bool) ua::'A::type => bool. ALL x::'A::type. u x --> ua x)"
by (import hollight DEF_re_subset)
constdefs
re_compl :: "('A => bool) => 'A => bool"
"re_compl == %(u::'A::type => bool) x::'A::type. ~ u x"
lemma DEF_re_compl: "re_compl = (%(u::'A::type => bool) x::'A::type. ~ u x)"
by (import hollight DEF_re_compl)
lemma SUBSETA_REFL: "ALL S::'A::type => bool. re_subset S S"
by (import hollight SUBSETA_REFL)
lemma COMPL_MEM: "ALL (S::'A::type => bool) x::'A::type. S x = (~ re_compl S x)"
by (import hollight COMPL_MEM)
lemma SUBSETA_ANTISYM: "ALL (P::'A::type => bool) Q::'A::type => bool.
(re_subset P Q & re_subset Q P) = (P = Q)"
by (import hollight SUBSETA_ANTISYM)
lemma SUBSETA_TRANS: "ALL (P::'A::type => bool) (Q::'A::type => bool) R::'A::type => bool.
re_subset P Q & re_subset Q R --> re_subset P R"
by (import hollight SUBSETA_TRANS)
constdefs
istopology :: "(('A => bool) => bool) => bool"
"istopology ==
%u::('A::type => bool) => bool.
u re_null &
u re_universe &
(ALL (a::'A::type => bool) b::'A::type => bool.
u a & u b --> u (re_intersect a b)) &
(ALL P::('A::type => bool) => bool. re_subset P u --> u (re_Union P))"
lemma DEF_istopology: "istopology =
(%u::('A::type => bool) => bool.
u re_null &
u re_universe &
(ALL (a::'A::type => bool) b::'A::type => bool.
u a & u b --> u (re_intersect a b)) &
(ALL P::('A::type => bool) => bool. re_subset P u --> u (re_Union P)))"
by (import hollight DEF_istopology)
typedef (open) ('A) topology = "(Collect::((('A::type => bool) => bool) => bool)
=> (('A::type => bool) => bool) set)
(istopology::(('A::type => bool) => bool) => bool)" morphisms "open" "topology"
apply (rule light_ex_imp_nonempty[where t="(Eps::((('A::type => bool) => bool) => bool) => ('A::type => bool) => bool)
(istopology::(('A::type => bool) => bool) => bool)"])
by (import hollight TYDEF_topology)
syntax
"open" :: _
syntax
topology :: _
lemmas "TYDEF_topology_@intern" = typedef_hol2hollight
[where a="a :: 'A topology" and r=r ,
OF type_definition_topology]
lemma TOPOLOGY: "ALL L::'A::type topology.
open L re_null &
open L re_universe &
(ALL (a::'A::type => bool) b::'A::type => bool.
open L a & open L b --> open L (re_intersect a b)) &
(ALL P::('A::type => bool) => bool.
re_subset P (open L) --> open L (re_Union P))"
by (import hollight TOPOLOGY)
lemma TOPOLOGY_UNION: "ALL (x::'A::type topology) xa::('A::type => bool) => bool.
re_subset xa (open x) --> open x (re_Union xa)"
by (import hollight TOPOLOGY_UNION)
constdefs
neigh :: "'A topology => ('A => bool) * 'A => bool"
"neigh ==
%(u::'A::type topology) ua::('A::type => bool) * 'A::type.
EX P::'A::type => bool. open u P & re_subset P (fst ua) & P (snd ua)"
lemma DEF_neigh: "neigh =
(%(u::'A::type topology) ua::('A::type => bool) * 'A::type.
EX P::'A::type => bool. open u P & re_subset P (fst ua) & P (snd ua))"
by (import hollight DEF_neigh)
lemma OPEN_OWN_NEIGH: "ALL (S::'A::type => bool) (top::'A::type topology) x::'A::type.
open top S & S x --> neigh top (S, x)"
by (import hollight OPEN_OWN_NEIGH)
lemma OPEN_UNOPEN: "ALL (S::'A::type => bool) top::'A::type topology.
open top S =
(re_Union (%P::'A::type => bool. open top P & re_subset P S) = S)"
by (import hollight OPEN_UNOPEN)
lemma OPEN_SUBOPEN: "ALL (S::'A::type => bool) top::'A::type topology.
open top S =
(ALL x::'A::type.
S x -->
(EX xa::'A::type => bool. xa x & open top xa & re_subset xa S))"
by (import hollight OPEN_SUBOPEN)
lemma OPEN_NEIGH: "ALL (S::'A::type => bool) top::'A::type topology.
open top S =
(ALL x::'A::type.
S x -->
(EX xa::'A::type => bool. neigh top (xa, x) & re_subset xa S))"
by (import hollight OPEN_NEIGH)
constdefs
closed :: "'A topology => ('A => bool) => bool"
"closed == %(u::'A::type topology) ua::'A::type => bool. open u (re_compl ua)"
lemma DEF_closed: "closed =
(%(u::'A::type topology) ua::'A::type => bool. open u (re_compl ua))"
by (import hollight DEF_closed)
constdefs
limpt :: "'A topology => 'A => ('A => bool) => bool"
"limpt ==
%(u::'A::type topology) (ua::'A::type) ub::'A::type => bool.
ALL N::'A::type => bool.
neigh u (N, ua) --> (EX y::'A::type. ua ~= y & ub y & N y)"
lemma DEF_limpt: "limpt =
(%(u::'A::type topology) (ua::'A::type) ub::'A::type => bool.
ALL N::'A::type => bool.
neigh u (N, ua) --> (EX y::'A::type. ua ~= y & ub y & N y))"
by (import hollight DEF_limpt)
lemma CLOSED_LIMPT: "ALL (top::'A::type topology) S::'A::type => bool.
closed top S = (ALL x::'A::type. limpt top x S --> S x)"
by (import hollight CLOSED_LIMPT)
constdefs
ismet :: "('A * 'A => hollight.real) => bool"
"ismet ==
%u::'A::type * 'A::type => hollight.real.
(ALL (x::'A::type) y::'A::type. (u (x, y) = real_of_num 0) = (x = y)) &
(ALL (x::'A::type) (y::'A::type) z::'A::type.
real_le (u (y, z)) (real_add (u (x, y)) (u (x, z))))"
lemma DEF_ismet: "ismet =
(%u::'A::type * 'A::type => hollight.real.
(ALL (x::'A::type) y::'A::type. (u (x, y) = real_of_num 0) = (x = y)) &
(ALL (x::'A::type) (y::'A::type) z::'A::type.
real_le (u (y, z)) (real_add (u (x, y)) (u (x, z)))))"
by (import hollight DEF_ismet)
typedef (open) ('A) metric = "(Collect::(('A::type * 'A::type => hollight.real) => bool)
=> ('A::type * 'A::type => hollight.real) set)
(ismet::('A::type * 'A::type => hollight.real) => bool)" morphisms "mdist" "metric"
apply (rule light_ex_imp_nonempty[where t="(Eps::(('A::type * 'A::type => hollight.real) => bool)
=> 'A::type * 'A::type => hollight.real)
(ismet::('A::type * 'A::type => hollight.real) => bool)"])
by (import hollight TYDEF_metric)
syntax
mdist :: _
syntax
metric :: _
lemmas "TYDEF_metric_@intern" = typedef_hol2hollight
[where a="a :: 'A metric" and r=r ,
OF type_definition_metric]
lemma METRIC_ISMET: "ALL m::'A::type metric. ismet (mdist m)"
by (import hollight METRIC_ISMET)
lemma METRIC_ZERO: "ALL (m::'A::type metric) (x::'A::type) y::'A::type.
(mdist m (x, y) = real_of_num 0) = (x = y)"
by (import hollight METRIC_ZERO)
lemma METRIC_SAME: "ALL (m::'A::type metric) x::'A::type. mdist m (x, x) = real_of_num 0"
by (import hollight METRIC_SAME)
lemma METRIC_POS: "ALL (m::'A::type metric) (x::'A::type) y::'A::type.
real_le (real_of_num 0) (mdist m (x, y))"
by (import hollight METRIC_POS)
lemma METRIC_SYM: "ALL (m::'A::type metric) (x::'A::type) y::'A::type.
mdist m (x, y) = mdist m (y, x)"
by (import hollight METRIC_SYM)
lemma METRIC_TRIANGLE: "ALL (m::'A::type metric) (x::'A::type) (y::'A::type) z::'A::type.
real_le (mdist m (x, z)) (real_add (mdist m (x, y)) (mdist m (y, z)))"
by (import hollight METRIC_TRIANGLE)
lemma METRIC_NZ: "ALL (m::'A::type metric) (x::'A::type) y::'A::type.
x ~= y --> real_lt (real_of_num 0) (mdist m (x, y))"
by (import hollight METRIC_NZ)
constdefs
mtop :: "'A metric => 'A topology"
"mtop ==
%u::'A::type metric.
topology
(%S::'A::type => bool.
ALL x::'A::type.
S x -->
(EX e::hollight.real.
real_lt (real_of_num 0) e &
(ALL y::'A::type. real_lt (mdist u (x, y)) e --> S y)))"
lemma DEF_mtop: "mtop =
(%u::'A::type metric.
topology
(%S::'A::type => bool.
ALL x::'A::type.
S x -->
(EX e::hollight.real.
real_lt (real_of_num 0) e &
(ALL y::'A::type. real_lt (mdist u (x, y)) e --> S y))))"
by (import hollight DEF_mtop)
lemma mtop_istopology: "ALL m::'A::type metric.
istopology
(%S::'A::type => bool.
ALL x::'A::type.
S x -->
(EX e::hollight.real.
real_lt (real_of_num 0) e &
(ALL y::'A::type. real_lt (mdist m (x, y)) e --> S y)))"
by (import hollight mtop_istopology)
lemma MTOP_OPEN: "ALL m::'A::type metric.
open (mtop m) (S::'A::type => bool) =
(ALL x::'A::type.
S x -->
(EX e::hollight.real.
real_lt (real_of_num 0) e &
(ALL y::'A::type. real_lt (mdist m (x, y)) e --> S y)))"
by (import hollight MTOP_OPEN)
constdefs
ball :: "'A metric => 'A * hollight.real => 'A => bool"
"ball ==
%(u::'A::type metric) (ua::'A::type * hollight.real) y::'A::type.
real_lt (mdist u (fst ua, y)) (snd ua)"
lemma DEF_ball: "ball =
(%(u::'A::type metric) (ua::'A::type * hollight.real) y::'A::type.
real_lt (mdist u (fst ua, y)) (snd ua))"
by (import hollight DEF_ball)
lemma BALL_OPEN: "ALL (m::'A::type metric) (x::'A::type) e::hollight.real.
real_lt (real_of_num 0) e --> open (mtop m) (ball m (x, e))"
by (import hollight BALL_OPEN)
lemma BALL_NEIGH: "ALL (m::'A::type metric) (x::'A::type) e::hollight.real.
real_lt (real_of_num 0) e --> neigh (mtop m) (ball m (x, e), x)"
by (import hollight BALL_NEIGH)
lemma MTOP_LIMPT: "ALL (m::'A::type metric) (x::'A::type) S::'A::type => bool.
limpt (mtop m) x S =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX y::'A::type. x ~= y & S y & real_lt (mdist m (x, y)) e))"
by (import hollight MTOP_LIMPT)
lemma ISMET_R1: "ismet
(GABS
(%f::hollight.real * hollight.real => hollight.real.
ALL (x::hollight.real) y::hollight.real.
GEQ (f (x, y)) (real_abs (real_sub y x))))"
by (import hollight ISMET_R1)
constdefs
mr1 :: "hollight.real metric"
"mr1 ==
metric
(GABS
(%f::hollight.real * hollight.real => hollight.real.
ALL (x::hollight.real) y::hollight.real.
GEQ (f (x, y)) (real_abs (real_sub y x))))"
lemma DEF_mr1: "mr1 =
metric
(GABS
(%f::hollight.real * hollight.real => hollight.real.
ALL (x::hollight.real) y::hollight.real.
GEQ (f (x, y)) (real_abs (real_sub y x))))"
by (import hollight DEF_mr1)
lemma MR1_DEF: "ALL (x::hollight.real) y::hollight.real.
mdist mr1 (x, y) = real_abs (real_sub y x)"
by (import hollight MR1_DEF)
lemma MR1_ADD: "ALL (x::hollight.real) d::hollight.real.
mdist mr1 (x, real_add x d) = real_abs d"
by (import hollight MR1_ADD)
lemma MR1_SUB: "ALL (x::hollight.real) d::hollight.real.
mdist mr1 (x, real_sub x d) = real_abs d"
by (import hollight MR1_SUB)
lemma MR1_ADD_LE: "ALL (x::hollight.real) d::hollight.real.
real_le (real_of_num 0) d --> mdist mr1 (x, real_add x d) = d"
by (import hollight MR1_ADD_LE)
lemma MR1_SUB_LE: "ALL (x::hollight.real) d::hollight.real.
real_le (real_of_num 0) d --> mdist mr1 (x, real_sub x d) = d"
by (import hollight MR1_SUB_LE)
lemma MR1_ADD_LT: "ALL (x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d --> mdist mr1 (x, real_add x d) = d"
by (import hollight MR1_ADD_LT)
lemma MR1_SUB_LT: "ALL (x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d --> mdist mr1 (x, real_sub x d) = d"
by (import hollight MR1_SUB_LT)
lemma MR1_BETWEEN1: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
real_lt x z & real_lt (mdist mr1 (x, y)) (real_sub z x) --> real_lt y z"
by (import hollight MR1_BETWEEN1)
lemma MR1_LIMPT: "ALL x::hollight.real. limpt (mtop mr1) x re_universe"
by (import hollight MR1_LIMPT)
constdefs
dorder :: "('A => 'A => bool) => bool"
"dorder ==
%u::'A::type => 'A::type => bool.
ALL (x::'A::type) y::'A::type.
u x x & u y y -->
(EX z::'A::type. u z z & (ALL w::'A::type. u w z --> u w x & u w y))"
lemma DEF_dorder: "dorder =
(%u::'A::type => 'A::type => bool.
ALL (x::'A::type) y::'A::type.
u x x & u y y -->
(EX z::'A::type. u z z & (ALL w::'A::type. u w z --> u w x & u w y)))"
by (import hollight DEF_dorder)
constdefs
tends :: "('B => 'A) => 'A => 'A topology * ('B => 'B => bool) => bool"
"tends ==
%(u::'B::type => 'A::type) (ua::'A::type)
ub::'A::type topology * ('B::type => 'B::type => bool).
ALL N::'A::type => bool.
neigh (fst ub) (N, ua) -->
(EX n::'B::type.
snd ub n n & (ALL m::'B::type. snd ub m n --> N (u m)))"
lemma DEF_tends: "tends =
(%(u::'B::type => 'A::type) (ua::'A::type)
ub::'A::type topology * ('B::type => 'B::type => bool).
ALL N::'A::type => bool.
neigh (fst ub) (N, ua) -->
(EX n::'B::type.
snd ub n n & (ALL m::'B::type. snd ub m n --> N (u m))))"
by (import hollight DEF_tends)
constdefs
bounded :: "'A metric * ('B => 'B => bool) => ('B => 'A) => bool"
"bounded ==
%(u::'A::type metric * ('B::type => 'B::type => bool))
ua::'B::type => 'A::type.
EX (k::hollight.real) (x::'A::type) N::'B::type.
snd u N N &
(ALL n::'B::type. snd u n N --> real_lt (mdist (fst u) (ua n, x)) k)"
lemma DEF_bounded: "bounded =
(%(u::'A::type metric * ('B::type => 'B::type => bool))
ua::'B::type => 'A::type.
EX (k::hollight.real) (x::'A::type) N::'B::type.
snd u N N &
(ALL n::'B::type. snd u n N --> real_lt (mdist (fst u) (ua n, x)) k))"
by (import hollight DEF_bounded)
constdefs
tendsto :: "'A metric * 'A => 'A => 'A => bool"
"tendsto ==
%(u::'A::type metric * 'A::type) (ua::'A::type) ub::'A::type.
real_lt (real_of_num 0) (mdist (fst u) (snd u, ua)) &
real_le (mdist (fst u) (snd u, ua)) (mdist (fst u) (snd u, ub))"
lemma DEF_tendsto: "tendsto =
(%(u::'A::type metric * 'A::type) (ua::'A::type) ub::'A::type.
real_lt (real_of_num 0) (mdist (fst u) (snd u, ua)) &
real_le (mdist (fst u) (snd u, ua)) (mdist (fst u) (snd u, ub)))"
by (import hollight DEF_tendsto)
lemma DORDER_LEMMA: "ALL g::'A::type => 'A::type => bool.
dorder g -->
(ALL (P::'A::type => bool) Q::'A::type => bool.
(EX n::'A::type. g n n & (ALL m::'A::type. g m n --> P m)) &
(EX n::'A::type. g n n & (ALL m::'A::type. g m n --> Q m)) -->
(EX n::'A::type. g n n & (ALL m::'A::type. g m n --> P m & Q m)))"
by (import hollight DORDER_LEMMA)
lemma DORDER_NGE: "dorder >="
by (import hollight DORDER_NGE)
lemma DORDER_TENDSTO: "ALL (m::'A::type metric) x::'A::type. dorder (tendsto (m, x))"
by (import hollight DORDER_TENDSTO)
lemma MTOP_TENDS: "ALL (d::'A::type metric) (g::'B::type => 'B::type => bool)
(x::'B::type => 'A::type) x0::'A::type.
tends x x0 (mtop d, g) =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX n::'B::type.
g n n &
(ALL m::'B::type. g m n --> real_lt (mdist d (x m, x0)) e)))"
by (import hollight MTOP_TENDS)
lemma MTOP_TENDS_UNIQ: "ALL (g::'B::type => 'B::type => bool) d::'A::type metric.
dorder g -->
tends (x::'B::type => 'A::type) (x0::'A::type) (mtop d, g) &
tends x (x1::'A::type) (mtop d, g) -->
x0 = x1"
by (import hollight MTOP_TENDS_UNIQ)
lemma SEQ_TENDS: "ALL (d::'A::type metric) (x::nat => 'A::type) x0::'A::type.
tends x x0 (mtop d, >=) =
(ALL xa::hollight.real.
real_lt (real_of_num 0) xa -->
(EX xb::nat.
ALL xc::nat. >= xc xb --> real_lt (mdist d (x xc, x0)) xa))"
by (import hollight SEQ_TENDS)
lemma LIM_TENDS: "ALL (m1::'A::type metric) (m2::'B::type metric) (f::'A::type => 'B::type)
(x0::'A::type) y0::'B::type.
limpt (mtop m1) x0 re_universe -->
tends f y0 (mtop m2, tendsto (m1, x0)) =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL x::'A::type.
real_lt (real_of_num 0) (mdist m1 (x, x0)) &
real_le (mdist m1 (x, x0)) d -->
real_lt (mdist m2 (f x, y0)) e)))"
by (import hollight LIM_TENDS)
lemma LIM_TENDS2: "ALL (m1::'A::type metric) (m2::'B::type metric) (f::'A::type => 'B::type)
(x0::'A::type) y0::'B::type.
limpt (mtop m1) x0 re_universe -->
tends f y0 (mtop m2, tendsto (m1, x0)) =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL x::'A::type.
real_lt (real_of_num 0) (mdist m1 (x, x0)) &
real_lt (mdist m1 (x, x0)) d -->
real_lt (mdist m2 (f x, y0)) e)))"
by (import hollight LIM_TENDS2)
lemma MR1_BOUNDED: "ALL (g::'A::type => 'A::type => bool) f::'A::type => hollight.real.
bounded (mr1, g) f =
(EX (k::hollight.real) N::'A::type.
g N N & (ALL n::'A::type. g n N --> real_lt (real_abs (f n)) k))"
by (import hollight MR1_BOUNDED)
lemma NET_NULL: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real.
tends x x0 (mtop mr1, g) =
tends (%n::'A::type. real_sub (x n) x0) (real_of_num 0) (mtop mr1, g)"
by (import hollight NET_NULL)
lemma NET_CONV_BOUNDED: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real. tends x x0 (mtop mr1, g) --> bounded (mr1, g) x"
by (import hollight NET_CONV_BOUNDED)
lemma NET_CONV_NZ: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real.
tends x x0 (mtop mr1, g) & x0 ~= real_of_num 0 -->
(EX N::'A::type.
g N N & (ALL n::'A::type. g n N --> x n ~= real_of_num 0))"
by (import hollight NET_CONV_NZ)
lemma NET_CONV_IBOUNDED: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real.
tends x x0 (mtop mr1, g) & x0 ~= real_of_num 0 -->
bounded (mr1, g) (%n::'A::type. real_inv (x n))"
by (import hollight NET_CONV_IBOUNDED)
lemma NET_NULL_ADD: "ALL g::'A::type => 'A::type => bool.
dorder g -->
(ALL (x::'A::type => hollight.real) y::'A::type => hollight.real.
tends x (real_of_num 0) (mtop mr1, g) &
tends y (real_of_num 0) (mtop mr1, g) -->
tends (%n::'A::type. real_add (x n) (y n)) (real_of_num 0)
(mtop mr1, g))"
by (import hollight NET_NULL_ADD)
lemma NET_NULL_MUL: "ALL g::'A::type => 'A::type => bool.
dorder g -->
(ALL (x::'A::type => hollight.real) y::'A::type => hollight.real.
bounded (mr1, g) x & tends y (real_of_num 0) (mtop mr1, g) -->
tends (%n::'A::type. real_mul (x n) (y n)) (real_of_num 0)
(mtop mr1, g))"
by (import hollight NET_NULL_MUL)
lemma NET_NULL_CMUL: "ALL (g::'A::type => 'A::type => bool) (k::hollight.real)
x::'A::type => hollight.real.
tends x (real_of_num 0) (mtop mr1, g) -->
tends (%n::'A::type. real_mul k (x n)) (real_of_num 0) (mtop mr1, g)"
by (import hollight NET_NULL_CMUL)
lemma NET_ADD: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
(x0::hollight.real) (y::'A::type => hollight.real) y0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
tends (%n::'A::type. real_add (x n) (y n)) (real_add x0 y0) (mtop mr1, g)"
by (import hollight NET_ADD)
lemma NET_NEG: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) =
tends (%n::'A::type. real_neg (x n)) (real_neg x0) (mtop mr1, g)"
by (import hollight NET_NEG)
lemma NET_SUB: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
(x0::hollight.real) (y::'A::type => hollight.real) y0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
tends (%xa::'A::type. real_sub (x xa) (y xa)) (real_sub x0 y0)
(mtop mr1, g)"
by (import hollight NET_SUB)
lemma NET_MUL: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
(y::'A::type => hollight.real) (x0::hollight.real) y0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
tends (%n::'A::type. real_mul (x n) (y n)) (real_mul x0 y0) (mtop mr1, g)"
by (import hollight NET_MUL)
lemma NET_INV: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
x0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) & x0 ~= real_of_num 0 -->
tends (%n::'A::type. real_inv (x n)) (real_inv x0) (mtop mr1, g)"
by (import hollight NET_INV)
lemma NET_DIV: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
(x0::hollight.real) (y::'A::type => hollight.real) y0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) &
tends y y0 (mtop mr1, g) & y0 ~= real_of_num 0 -->
tends (%xa::'A::type. real_div (x xa) (y xa)) (real_div x0 y0)
(mtop mr1, g)"
by (import hollight NET_DIV)
lemma NET_ABS: "ALL (x::'A::type => hollight.real) x0::hollight.real.
tends x x0 (mtop mr1, g::'A::type => 'A::type => bool) -->
tends (%n::'A::type. real_abs (x n)) (real_abs x0) (mtop mr1, g)"
by (import hollight NET_ABS)
lemma NET_SUM: "ALL g::'q_71813::type => 'q_71813::type => bool.
dorder g &
tends (%x::'q_71813::type. real_of_num 0) (real_of_num 0)
(mtop mr1, g) -->
(ALL (x::nat) n::nat.
(ALL r::nat.
<= x r & < r (x + n) -->
tends ((f::nat => 'q_71813::type => hollight.real) r)
((l::nat => hollight.real) r) (mtop mr1, g)) -->
tends (%xa::'q_71813::type. psum (x, n) (%r::nat. f r xa))
(psum (x, n) l) (mtop mr1, g))"
by (import hollight NET_SUM)
lemma NET_LE: "ALL (g::'A::type => 'A::type => bool) (x::'A::type => hollight.real)
(x0::hollight.real) (y::'A::type => hollight.real) y0::hollight.real.
dorder g -->
tends x x0 (mtop mr1, g) &
tends y y0 (mtop mr1, g) &
(EX N::'A::type.
g N N & (ALL n::'A::type. g n N --> real_le (x n) (y n))) -->
real_le x0 y0"
by (import hollight NET_LE)
constdefs
tends_num_real :: "(nat => hollight.real) => hollight.real => bool"
"tends_num_real ==
%(u::nat => hollight.real) ua::hollight.real. tends u ua (mtop mr1, >=)"
lemma DEF_tends_num_real: "tends_num_real =
(%(u::nat => hollight.real) ua::hollight.real. tends u ua (mtop mr1, >=))"
by (import hollight DEF_tends_num_real)
lemma SEQ: "ALL (x::nat => hollight.real) x0::hollight.real.
tends_num_real x x0 =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX N::nat.
ALL n::nat. >= n N --> real_lt (real_abs (real_sub (x n) x0)) e))"
by (import hollight SEQ)
lemma SEQ_CONST: "ALL k::hollight.real. tends_num_real (%x::nat. k) k"
by (import hollight SEQ_CONST)
lemma SEQ_ADD: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
tends_num_real x x0 & tends_num_real y y0 -->
tends_num_real (%n::nat. real_add (x n) (y n)) (real_add x0 y0)"
by (import hollight SEQ_ADD)
lemma SEQ_MUL: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
tends_num_real x x0 & tends_num_real y y0 -->
tends_num_real (%n::nat. real_mul (x n) (y n)) (real_mul x0 y0)"
by (import hollight SEQ_MUL)
lemma SEQ_NEG: "ALL (x::nat => hollight.real) x0::hollight.real.
tends_num_real x x0 =
tends_num_real (%n::nat. real_neg (x n)) (real_neg x0)"
by (import hollight SEQ_NEG)
lemma SEQ_INV: "ALL (x::nat => hollight.real) x0::hollight.real.
tends_num_real x x0 & x0 ~= real_of_num 0 -->
tends_num_real (%n::nat. real_inv (x n)) (real_inv x0)"
by (import hollight SEQ_INV)
lemma SEQ_SUB: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
tends_num_real x x0 & tends_num_real y y0 -->
tends_num_real (%n::nat. real_sub (x n) (y n)) (real_sub x0 y0)"
by (import hollight SEQ_SUB)
lemma SEQ_DIV: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
tends_num_real x x0 & tends_num_real y y0 & y0 ~= real_of_num 0 -->
tends_num_real (%n::nat. real_div (x n) (y n)) (real_div x0 y0)"
by (import hollight SEQ_DIV)
lemma SEQ_UNIQ: "ALL (x::nat => hollight.real) (x1::hollight.real) x2::hollight.real.
tends_num_real x x1 & tends_num_real x x2 --> x1 = x2"
by (import hollight SEQ_UNIQ)
lemma SEQ_NULL: "ALL (s::nat => hollight.real) l::hollight.real.
tends_num_real s l =
tends_num_real (%n::nat. real_sub (s n) l) (real_of_num 0)"
by (import hollight SEQ_NULL)
lemma SEQ_SUM: "ALL (f::nat => nat => hollight.real) (l::nat => hollight.real) (m::nat)
n::nat.
(ALL x::nat. <= m x & < x (m + n) --> tends_num_real (f x) (l x)) -->
tends_num_real (%k::nat. psum (m, n) (%r::nat. f r k)) (psum (m, n) l)"
by (import hollight SEQ_SUM)
lemma SEQ_TRANSFORM: "ALL (x::nat => hollight.real) (xa::nat => hollight.real) (xb::hollight.real)
xc::nat.
(ALL n::nat. <= xc n --> x n = xa n) & tends_num_real x xb -->
tends_num_real xa xb"
by (import hollight SEQ_TRANSFORM)
constdefs
convergent :: "(nat => hollight.real) => bool"
"convergent == %u::nat => hollight.real. Ex (tends_num_real u)"
lemma DEF_convergent: "convergent = (%u::nat => hollight.real. Ex (tends_num_real u))"
by (import hollight DEF_convergent)
constdefs
cauchy :: "(nat => hollight.real) => bool"
"cauchy ==
%u::nat => hollight.real.
ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (m::nat) n::nat.
>= m N & >= n N -->
real_lt (real_abs (real_sub (u m) (u n))) e)"
lemma DEF_cauchy: "cauchy =
(%u::nat => hollight.real.
ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (m::nat) n::nat.
>= m N & >= n N -->
real_lt (real_abs (real_sub (u m) (u n))) e))"
by (import hollight DEF_cauchy)
constdefs
lim :: "(nat => hollight.real) => hollight.real"
"lim == %u::nat => hollight.real. Eps (tends_num_real u)"
lemma DEF_lim: "lim = (%u::nat => hollight.real. Eps (tends_num_real u))"
by (import hollight DEF_lim)
lemma SEQ_LIM: "ALL f::nat => hollight.real. convergent f = tends_num_real f (lim f)"
by (import hollight SEQ_LIM)
constdefs
subseq :: "(nat => nat) => bool"
"subseq == %u::nat => nat. ALL (m::nat) n::nat. < m n --> < (u m) (u n)"
lemma DEF_subseq: "subseq = (%u::nat => nat. ALL (m::nat) n::nat. < m n --> < (u m) (u n))"
by (import hollight DEF_subseq)
lemma SUBSEQ_SUC: "ALL f::nat => nat. subseq f = (ALL n::nat. < (f n) (f (Suc n)))"
by (import hollight SUBSEQ_SUC)
consts
mono :: "(nat => hollight.real) => bool"
defs
mono_def: "hollight.mono ==
%u::nat => hollight.real.
(ALL (m::nat) n::nat. <= m n --> real_le (u m) (u n)) |
(ALL (m::nat) n::nat. <= m n --> hollight.real_ge (u m) (u n))"
lemma DEF_mono: "hollight.mono =
(%u::nat => hollight.real.
(ALL (m::nat) n::nat. <= m n --> real_le (u m) (u n)) |
(ALL (m::nat) n::nat. <= m n --> hollight.real_ge (u m) (u n)))"
by (import hollight DEF_mono)
lemma MONO_SUC: "ALL f::nat => hollight.real.
hollight.mono f =
((ALL x::nat. hollight.real_ge (f (Suc x)) (f x)) |
(ALL n::nat. real_le (f (Suc n)) (f n)))"
by (import hollight MONO_SUC)
lemma MAX_LEMMA: "ALL (s::nat => hollight.real) N::nat.
EX k::hollight.real. ALL n::nat. < n N --> real_lt (real_abs (s n)) k"
by (import hollight MAX_LEMMA)
lemma SEQ_BOUNDED: "ALL s::nat => hollight.real.
bounded (mr1, >=) s =
(EX k::hollight.real. ALL n::nat. real_lt (real_abs (s n)) k)"
by (import hollight SEQ_BOUNDED)
lemma SEQ_BOUNDED_2: "ALL (f::nat => hollight.real) (k::hollight.real) K::hollight.real.
(ALL n::nat. real_le k (f n) & real_le (f n) K) --> bounded (mr1, >=) f"
by (import hollight SEQ_BOUNDED_2)
lemma SEQ_CBOUNDED: "ALL f::nat => hollight.real. cauchy f --> bounded (mr1, >=) f"
by (import hollight SEQ_CBOUNDED)
lemma SEQ_ICONV: "ALL f::nat => hollight.real.
bounded (mr1, >=) f &
(ALL (m::nat) n::nat. >= m n --> hollight.real_ge (f m) (f n)) -->
convergent f"
by (import hollight SEQ_ICONV)
lemma SEQ_NEG_CONV: "ALL f::nat => hollight.real.
convergent f = convergent (%n::nat. real_neg (f n))"
by (import hollight SEQ_NEG_CONV)
lemma SEQ_NEG_BOUNDED: "ALL f::nat => hollight.real.
bounded (mr1, >=) (%n::nat. real_neg (f n)) = bounded (mr1, >=) f"
by (import hollight SEQ_NEG_BOUNDED)
lemma SEQ_BCONV: "ALL f::nat => hollight.real.
bounded (mr1, >=) f & hollight.mono f --> convergent f"
by (import hollight SEQ_BCONV)
lemma SEQ_MONOSUB: "ALL s::nat => hollight.real.
EX f::nat => nat. subseq f & hollight.mono (%n::nat. s (f n))"
by (import hollight SEQ_MONOSUB)
lemma SEQ_SBOUNDED: "ALL (s::nat => hollight.real) f::nat => nat.
bounded (mr1, >=) s --> bounded (mr1, >=) (%n::nat. s (f n))"
by (import hollight SEQ_SBOUNDED)
lemma SEQ_SUBLE: "ALL (f::nat => nat) x::nat. subseq f --> <= x (f x)"
by (import hollight SEQ_SUBLE)
lemma SEQ_DIRECT: "ALL f::nat => nat.
subseq f --> (ALL (N1::nat) N2::nat. EX x::nat. >= x N1 & >= (f x) N2)"
by (import hollight SEQ_DIRECT)
lemma SEQ_CAUCHY: "ALL f::nat => hollight.real. cauchy f = convergent f"
by (import hollight SEQ_CAUCHY)
lemma SEQ_LE: "ALL (f::nat => hollight.real) (g::nat => hollight.real) (l::hollight.real)
m::hollight.real.
tends_num_real f l &
tends_num_real g m &
(EX N::nat. ALL n::nat. >= n N --> real_le (f n) (g n)) -->
real_le l m"
by (import hollight SEQ_LE)
lemma SEQ_LE_0: "ALL (x::nat => hollight.real) xa::nat => hollight.real.
tends_num_real x (real_of_num 0) &
(EX xb::nat.
ALL xc::nat.
>= xc xb --> real_le (real_abs (xa xc)) (real_abs (x xc))) -->
tends_num_real xa (real_of_num 0)"
by (import hollight SEQ_LE_0)
lemma SEQ_SUC: "ALL (f::nat => hollight.real) l::hollight.real.
tends_num_real f l = tends_num_real (%n::nat. f (Suc n)) l"
by (import hollight SEQ_SUC)
lemma SEQ_ABS: "ALL f::nat => hollight.real.
tends_num_real (%n::nat. real_abs (f n)) (real_of_num 0) =
tends_num_real f (real_of_num 0)"
by (import hollight SEQ_ABS)
lemma SEQ_ABS_IMP: "ALL (f::nat => hollight.real) l::hollight.real.
tends_num_real f l -->
tends_num_real (%n::nat. real_abs (f n)) (real_abs l)"
by (import hollight SEQ_ABS_IMP)
lemma SEQ_INV0: "ALL f::nat => hollight.real.
(ALL y::hollight.real.
EX N::nat. ALL n::nat. >= n N --> hollight.real_gt (f n) y) -->
tends_num_real (%n::nat. real_inv (f n)) (real_of_num 0)"
by (import hollight SEQ_INV0)
lemma SEQ_POWER_ABS: "ALL c::hollight.real.
real_lt (real_abs c) (real_of_num (NUMERAL_BIT1 0)) -->
tends_num_real (real_pow (real_abs c)) (real_of_num 0)"
by (import hollight SEQ_POWER_ABS)
lemma SEQ_POWER: "ALL c::hollight.real.
real_lt (real_abs c) (real_of_num (NUMERAL_BIT1 0)) -->
tends_num_real (real_pow c) (real_of_num 0)"
by (import hollight SEQ_POWER)
lemma NEST_LEMMA: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(ALL n::nat. hollight.real_ge (f (Suc n)) (f n)) &
(ALL n::nat. real_le (g (Suc n)) (g n)) &
(ALL n::nat. real_le (f n) (g n)) -->
(EX (l::hollight.real) m::hollight.real.
real_le l m &
((ALL n::nat. real_le (f n) l) & tends_num_real f l) &
(ALL n::nat. real_le m (g n)) & tends_num_real g m)"
by (import hollight NEST_LEMMA)
lemma NEST_LEMMA_UNIQ: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(ALL n::nat. hollight.real_ge (f (Suc n)) (f n)) &
(ALL n::nat. real_le (g (Suc n)) (g n)) &
(ALL n::nat. real_le (f n) (g n)) &
tends_num_real (%n::nat. real_sub (f n) (g n)) (real_of_num 0) -->
(EX l::hollight.real.
((ALL n::nat. real_le (f n) l) & tends_num_real f l) &
(ALL n::nat. real_le l (g n)) & tends_num_real g l)"
by (import hollight NEST_LEMMA_UNIQ)
lemma BOLZANO_LEMMA: "ALL P::hollight.real * hollight.real => bool.
(ALL (a::hollight.real) (b::hollight.real) c::hollight.real.
real_le a b & real_le b c & P (a, b) & P (b, c) --> P (a, c)) &
(ALL x::hollight.real.
EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL (a::hollight.real) b::hollight.real.
real_le a x & real_le x b & real_lt (real_sub b a) d -->
P (a, b))) -->
(ALL (a::hollight.real) b::hollight.real. real_le a b --> P (a, b))"
by (import hollight BOLZANO_LEMMA)
constdefs
sums :: "(nat => hollight.real) => hollight.real => bool"
"sums == %u::nat => hollight.real. tends_num_real (%n::nat. psum (0, n) u)"
lemma DEF_sums: "sums = (%u::nat => hollight.real. tends_num_real (%n::nat. psum (0, n) u))"
by (import hollight DEF_sums)
constdefs
summable :: "(nat => hollight.real) => bool"
"summable == %u::nat => hollight.real. Ex (sums u)"
lemma DEF_summable: "summable = (%u::nat => hollight.real. Ex (sums u))"
by (import hollight DEF_summable)
constdefs
suminf :: "(nat => hollight.real) => hollight.real"
"suminf == %u::nat => hollight.real. Eps (sums u)"
lemma DEF_suminf: "suminf = (%u::nat => hollight.real. Eps (sums u))"
by (import hollight DEF_suminf)
lemma SUM_SUMMABLE: "ALL (f::nat => hollight.real) l::hollight.real. sums f l --> summable f"
by (import hollight SUM_SUMMABLE)
lemma SUMMABLE_SUM: "ALL f::nat => hollight.real. summable f --> sums f (suminf f)"
by (import hollight SUMMABLE_SUM)
lemma SUM_UNIQ: "ALL (f::nat => hollight.real) x::hollight.real. sums f x --> x = suminf f"
by (import hollight SUM_UNIQ)
lemma SER_UNIQ: "ALL (f::nat => hollight.real) (x::hollight.real) y::hollight.real.
sums f x & sums f y --> x = y"
by (import hollight SER_UNIQ)
lemma SER_0: "ALL (f::nat => hollight.real) n::nat.
(ALL m::nat. <= n m --> f m = real_of_num 0) --> sums f (psum (0, n) f)"
by (import hollight SER_0)
lemma SER_POS_LE: "ALL (f::nat => hollight.real) n::nat.
summable f & (ALL m::nat. <= n m --> real_le (real_of_num 0) (f m)) -->
real_le (psum (0, n) f) (suminf f)"
by (import hollight SER_POS_LE)
lemma SER_POS_LT: "ALL (f::nat => hollight.real) n::nat.
summable f & (ALL m::nat. <= n m --> real_lt (real_of_num 0) (f m)) -->
real_lt (psum (0, n) f) (suminf f)"
by (import hollight SER_POS_LT)
lemma SER_GROUP: "ALL (f::nat => hollight.real) k::nat.
summable f & < 0 k --> sums (%n::nat. psum (n * k, k) f) (suminf f)"
by (import hollight SER_GROUP)
lemma SER_PAIR: "ALL f::nat => hollight.real.
summable f -->
sums
(%n::nat.
psum
(NUMERAL_BIT0 (NUMERAL_BIT1 0) * n, NUMERAL_BIT0 (NUMERAL_BIT1 0))
f)
(suminf f)"
by (import hollight SER_PAIR)
lemma SER_OFFSET: "ALL f::nat => hollight.real.
summable f -->
(ALL k::nat.
sums (%n::nat. f (n + k)) (real_sub (suminf f) (psum (0, k) f)))"
by (import hollight SER_OFFSET)
lemma SER_OFFSET_REV: "ALL (f::nat => hollight.real) k::nat.
summable (%n::nat. f (n + k)) -->
sums f (real_add (psum (0, k) f) (suminf (%n::nat. f (n + k))))"
by (import hollight SER_OFFSET_REV)
lemma SER_POS_LT_PAIR: "ALL (f::nat => hollight.real) n::nat.
summable f &
(ALL d::nat.
real_lt (real_of_num 0)
(real_add (f (n + NUMERAL_BIT0 (NUMERAL_BIT1 0) * d))
(f (n +
(NUMERAL_BIT0 (NUMERAL_BIT1 0) * d + NUMERAL_BIT1 0))))) -->
real_lt (psum (0, n) f) (suminf f)"
by (import hollight SER_POS_LT_PAIR)
lemma SER_ADD: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
sums x x0 & sums y y0 -->
sums (%n::nat. real_add (x n) (y n)) (real_add x0 y0)"
by (import hollight SER_ADD)
lemma SER_CMUL: "ALL (x::nat => hollight.real) (x0::hollight.real) c::hollight.real.
sums x x0 --> sums (%n::nat. real_mul c (x n)) (real_mul c x0)"
by (import hollight SER_CMUL)
lemma SER_NEG: "ALL (x::nat => hollight.real) x0::hollight.real.
sums x x0 --> sums (%xa::nat. real_neg (x xa)) (real_neg x0)"
by (import hollight SER_NEG)
lemma SER_SUB: "ALL (x::nat => hollight.real) (x0::hollight.real) (y::nat => hollight.real)
y0::hollight.real.
sums x x0 & sums y y0 -->
sums (%n::nat. real_sub (x n) (y n)) (real_sub x0 y0)"
by (import hollight SER_SUB)
lemma SER_CDIV: "ALL (x::nat => hollight.real) (x0::hollight.real) c::hollight.real.
sums x x0 --> sums (%xa::nat. real_div (x xa) c) (real_div x0 c)"
by (import hollight SER_CDIV)
lemma SER_CAUCHY: "ALL f::nat => hollight.real.
summable f =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (m::nat) n::nat.
>= m N --> real_lt (real_abs (psum (m, n) f)) e))"
by (import hollight SER_CAUCHY)
lemma SER_ZERO: "ALL f::nat => hollight.real. summable f --> tends_num_real f (real_of_num 0)"
by (import hollight SER_ZERO)
lemma SER_COMPAR: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(EX x::nat. ALL xa::nat. >= xa x --> real_le (real_abs (f xa)) (g xa)) &
summable g -->
summable f"
by (import hollight SER_COMPAR)
lemma SER_COMPARA: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(EX x::nat. ALL xa::nat. >= xa x --> real_le (real_abs (f xa)) (g xa)) &
summable g -->
summable (%k::nat. real_abs (f k))"
by (import hollight SER_COMPARA)
lemma SER_LE: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(ALL n::nat. real_le (f n) (g n)) & summable f & summable g -->
real_le (suminf f) (suminf g)"
by (import hollight SER_LE)
lemma SER_LE2: "ALL (f::nat => hollight.real) g::nat => hollight.real.
(ALL n::nat. real_le (real_abs (f n)) (g n)) & summable g -->
summable f & real_le (suminf f) (suminf g)"
by (import hollight SER_LE2)
lemma SER_ACONV: "ALL f::nat => hollight.real.
summable (%n::nat. real_abs (f n)) --> summable f"
by (import hollight SER_ACONV)
lemma SER_ABS: "ALL f::nat => hollight.real.
summable (%n::nat. real_abs (f n)) -->
real_le (real_abs (suminf f)) (suminf (%n::nat. real_abs (f n)))"
by (import hollight SER_ABS)
lemma GP_FINITE: "ALL x::hollight.real.
x ~= real_of_num (NUMERAL_BIT1 0) -->
(ALL n::nat.
psum (0, n) (real_pow x) =
real_div (real_sub (real_pow x n) (real_of_num (NUMERAL_BIT1 0)))
(real_sub x (real_of_num (NUMERAL_BIT1 0))))"
by (import hollight GP_FINITE)
lemma GP: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
sums (real_pow x) (real_inv (real_sub (real_of_num (NUMERAL_BIT1 0)) x))"
by (import hollight GP)
lemma ABS_NEG_LEMMA: "ALL (c::hollight.real) (x::hollight.real) y::hollight.real.
real_le c (real_of_num 0) -->
real_le (real_abs x) (real_mul c (real_abs y)) --> x = real_of_num 0"
by (import hollight ABS_NEG_LEMMA)
lemma SER_RATIO: "ALL (f::nat => hollight.real) (c::hollight.real) N::nat.
real_lt c (real_of_num (NUMERAL_BIT1 0)) &
(ALL n::nat.
>= n N -->
real_le (real_abs (f (Suc n))) (real_mul c (real_abs (f n)))) -->
summable f"
by (import hollight SER_RATIO)
lemma SEQ_TRUNCATION: "ALL (f::nat => hollight.real) (l::hollight.real) (n::nat) b::hollight.real.
sums f l & (ALL m::nat. real_le (real_abs (psum (n, m) f)) b) -->
real_le (real_abs (real_sub l (psum (0, n) f))) b"
by (import hollight SEQ_TRUNCATION)
constdefs
tends_real_real :: "(hollight.real => hollight.real) => hollight.real => hollight.real => bool"
"tends_real_real ==
%(u::hollight.real => hollight.real) (ua::hollight.real) ub::hollight.real.
tends u ua (mtop mr1, tendsto (mr1, ub))"
lemma DEF_tends_real_real: "tends_real_real =
(%(u::hollight.real => hollight.real) (ua::hollight.real) ub::hollight.real.
tends u ua (mtop mr1, tendsto (mr1, ub)))"
by (import hollight DEF_tends_real_real)
lemma LIM: "ALL (f::hollight.real => hollight.real) (y0::hollight.real)
x0::hollight.real.
tends_real_real f y0 x0 =
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL x::hollight.real.
real_lt (real_of_num 0) (real_abs (real_sub x x0)) &
real_lt (real_abs (real_sub x x0)) d -->
real_lt (real_abs (real_sub (f x) y0)) e)))"
by (import hollight LIM)
lemma LIM_CONST: "ALL k::hollight.real. All (tends_real_real (%x::hollight.real. k) k)"
by (import hollight LIM_CONST)
lemma LIM_ADD: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) m::hollight.real.
tends_real_real f l (x::hollight.real) & tends_real_real g m x -->
tends_real_real (%x::hollight.real. real_add (f x) (g x)) (real_add l m)
x"
by (import hollight LIM_ADD)
lemma LIM_MUL: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) m::hollight.real.
tends_real_real f l (x::hollight.real) & tends_real_real g m x -->
tends_real_real (%x::hollight.real. real_mul (f x) (g x)) (real_mul l m)
x"
by (import hollight LIM_MUL)
lemma LIM_NEG: "ALL (f::hollight.real => hollight.real) l::hollight.real.
tends_real_real f l (x::hollight.real) =
tends_real_real (%x::hollight.real. real_neg (f x)) (real_neg l) x"
by (import hollight LIM_NEG)
lemma LIM_INV: "ALL (f::hollight.real => hollight.real) l::hollight.real.
tends_real_real f l (x::hollight.real) & l ~= real_of_num 0 -->
tends_real_real (%x::hollight.real. real_inv (f x)) (real_inv l) x"
by (import hollight LIM_INV)
lemma LIM_SUB: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) m::hollight.real.
tends_real_real f l (x::hollight.real) & tends_real_real g m x -->
tends_real_real (%x::hollight.real. real_sub (f x) (g x)) (real_sub l m)
x"
by (import hollight LIM_SUB)
lemma LIM_DIV: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) m::hollight.real.
tends_real_real f l (x::hollight.real) &
tends_real_real g m x & m ~= real_of_num 0 -->
tends_real_real (%x::hollight.real. real_div (f x) (g x)) (real_div l m)
x"
by (import hollight LIM_DIV)
lemma LIM_NULL: "ALL (f::hollight.real => hollight.real) (l::hollight.real) x::hollight.real.
tends_real_real f l x =
tends_real_real (%x::hollight.real. real_sub (f x) l) (real_of_num 0) x"
by (import hollight LIM_NULL)
lemma LIM_SUM: "ALL (f::nat => hollight.real => hollight.real) (l::nat => hollight.real)
(m::nat) (n::nat) x::hollight.real.
(ALL xa::nat.
<= m xa & < xa (m + n) --> tends_real_real (f xa) (l xa) x) -->
tends_real_real (%x::hollight.real. psum (m, n) (%r::nat. f r x))
(psum (m, n) l) x"
by (import hollight LIM_SUM)
lemma LIM_X: "ALL x0::hollight.real. tends_real_real (%x::hollight.real. x) x0 x0"
by (import hollight LIM_X)
lemma LIM_UNIQ: "ALL (f::hollight.real => hollight.real) (l::hollight.real)
(m::hollight.real) x::hollight.real.
tends_real_real f l x & tends_real_real f m x --> l = m"
by (import hollight LIM_UNIQ)
lemma LIM_EQUAL: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) x0::hollight.real.
(ALL x::hollight.real. x ~= x0 --> f x = g x) -->
tends_real_real f l x0 = tends_real_real g l x0"
by (import hollight LIM_EQUAL)
lemma LIM_TRANSFORM: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(x0::hollight.real) l::hollight.real.
tends_real_real (%x::hollight.real. real_sub (f x) (g x)) (real_of_num 0)
x0 &
tends_real_real g l x0 -->
tends_real_real f l x0"
by (import hollight LIM_TRANSFORM)
constdefs
diffl :: "(hollight.real => hollight.real) => hollight.real => hollight.real => bool"
"diffl ==
%(u::hollight.real => hollight.real) (ua::hollight.real) ub::hollight.real.
tends_real_real
(%h::hollight.real. real_div (real_sub (u (real_add ub h)) (u ub)) h) ua
(real_of_num 0)"
lemma DEF_diffl: "diffl =
(%(u::hollight.real => hollight.real) (ua::hollight.real) ub::hollight.real.
tends_real_real
(%h::hollight.real. real_div (real_sub (u (real_add ub h)) (u ub)) h)
ua (real_of_num 0))"
by (import hollight DEF_diffl)
constdefs
contl :: "(hollight.real => hollight.real) => hollight.real => bool"
"contl ==
%(u::hollight.real => hollight.real) ua::hollight.real.
tends_real_real (%h::hollight.real. u (real_add ua h)) (u ua)
(real_of_num 0)"
lemma DEF_contl: "contl =
(%(u::hollight.real => hollight.real) ua::hollight.real.
tends_real_real (%h::hollight.real. u (real_add ua h)) (u ua)
(real_of_num 0))"
by (import hollight DEF_contl)
constdefs
differentiable :: "(hollight.real => hollight.real) => hollight.real => bool"
"differentiable ==
%(u::hollight.real => hollight.real) ua::hollight.real.
EX l::hollight.real. diffl u l ua"
lemma DEF_differentiable: "differentiable =
(%(u::hollight.real => hollight.real) ua::hollight.real.
EX l::hollight.real. diffl u l ua)"
by (import hollight DEF_differentiable)
lemma DIFF_UNIQ: "ALL (f::hollight.real => hollight.real) (l::hollight.real)
(m::hollight.real) x::hollight.real. diffl f l x & diffl f m x --> l = m"
by (import hollight DIFF_UNIQ)
lemma DIFF_CONT: "ALL (f::hollight.real => hollight.real) (l::hollight.real) x::hollight.real.
diffl f l x --> contl f x"
by (import hollight DIFF_CONT)
lemma CONTL_LIM: "ALL (f::hollight.real => hollight.real) x::hollight.real.
contl f x = tends_real_real f (f x) x"
by (import hollight CONTL_LIM)
lemma CONT_X: "All (contl (%x::hollight.real. x))"
by (import hollight CONT_X)
lemma CONT_CONST: "All (contl (%x::hollight.real. k::hollight.real))"
by (import hollight CONT_CONST)
lemma CONT_ADD: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x &
contl (g::hollight.real => hollight.real) x -->
contl (%x::hollight.real. real_add (f x) (g x)) x"
by (import hollight CONT_ADD)
lemma CONT_MUL: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x &
contl (g::hollight.real => hollight.real) x -->
contl (%x::hollight.real. real_mul (f x) (g x)) x"
by (import hollight CONT_MUL)
lemma CONT_NEG: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x -->
contl (%x::hollight.real. real_neg (f x)) x"
by (import hollight CONT_NEG)
lemma CONT_INV: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x & f x ~= real_of_num 0 -->
contl (%x::hollight.real. real_inv (f x)) x"
by (import hollight CONT_INV)
lemma CONT_SUB: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x &
contl (g::hollight.real => hollight.real) x -->
contl (%x::hollight.real. real_sub (f x) (g x)) x"
by (import hollight CONT_SUB)
lemma CONT_DIV: "ALL x::hollight.real.
contl (f::hollight.real => hollight.real) x &
contl (g::hollight.real => hollight.real) x & g x ~= real_of_num 0 -->
contl (%x::hollight.real. real_div (f x) (g x)) x"
by (import hollight CONT_DIV)
lemma CONT_COMPOSE: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
x::hollight.real.
contl f x & contl g (f x) --> contl (%x::hollight.real. g (f x)) x"
by (import hollight CONT_COMPOSE)
lemma IVT: "ALL (f::hollight.real => hollight.real) (a::hollight.real)
(b::hollight.real) y::hollight.real.
real_le a b &
(real_le (f a) y & real_le y (f b)) &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX x::hollight.real. real_le a x & real_le x b & f x = y)"
by (import hollight IVT)
lemma IVT2: "ALL (f::hollight.real => hollight.real) (a::hollight.real)
(b::hollight.real) y::hollight.real.
real_le a b &
(real_le (f b) y & real_le y (f a)) &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX x::hollight.real. real_le a x & real_le x b & f x = y)"
by (import hollight IVT2)
lemma DIFF_CONST: "ALL k::hollight.real. All (diffl (%x::hollight.real. k) (real_of_num 0))"
by (import hollight DIFF_CONST)
lemma DIFF_ADD: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (m::hollight.real) x::hollight.real.
diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_add (f x) (g x)) (real_add l m) x"
by (import hollight DIFF_ADD)
lemma DIFF_MUL: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (m::hollight.real) x::hollight.real.
diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_mul (f x) (g x))
(real_add (real_mul l (g x)) (real_mul m (f x))) x"
by (import hollight DIFF_MUL)
lemma DIFF_CMUL: "ALL (f::hollight.real => hollight.real) (c::hollight.real)
(l::hollight.real) x::hollight.real.
diffl f l x -->
diffl (%x::hollight.real. real_mul c (f x)) (real_mul c l) x"
by (import hollight DIFF_CMUL)
lemma DIFF_NEG: "ALL (f::hollight.real => hollight.real) (l::hollight.real) x::hollight.real.
diffl f l x --> diffl (%x::hollight.real. real_neg (f x)) (real_neg l) x"
by (import hollight DIFF_NEG)
lemma DIFF_SUB: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (m::hollight.real) x::hollight.real.
diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_sub (f x) (g x)) (real_sub l m) x"
by (import hollight DIFF_SUB)
lemma DIFF_CARAT: "ALL (f::hollight.real => hollight.real) (l::hollight.real) x::hollight.real.
diffl f l x =
(EX xa::hollight.real => hollight.real.
(ALL z::hollight.real.
real_sub (f z) (f x) = real_mul (xa z) (real_sub z x)) &
contl xa x & xa x = l)"
by (import hollight DIFF_CARAT)
lemma DIFF_CHAIN: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (m::hollight.real) x::hollight.real.
diffl f l (g x) & diffl g m x -->
diffl (%x::hollight.real. f (g x)) (real_mul l m) x"
by (import hollight DIFF_CHAIN)
lemma DIFF_X: "All (diffl (%x::hollight.real. x) (real_of_num (NUMERAL_BIT1 0)))"
by (import hollight DIFF_X)
lemma DIFF_POW: "ALL (n::nat) x::hollight.real.
diffl (%x::hollight.real. real_pow x n)
(real_mul (real_of_num n) (real_pow x (n - NUMERAL_BIT1 0))) x"
by (import hollight DIFF_POW)
lemma DIFF_XM1: "ALL x::hollight.real.
x ~= real_of_num 0 -->
diffl real_inv
(real_neg (real_pow (real_inv x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) x"
by (import hollight DIFF_XM1)
lemma DIFF_INV: "ALL (f::hollight.real => hollight.real) (l::hollight.real) x::hollight.real.
diffl f l x & f x ~= real_of_num 0 -->
diffl (%x::hollight.real. real_inv (f x))
(real_neg (real_div l (real_pow (f x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x"
by (import hollight DIFF_INV)
lemma DIFF_DIV: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) m::hollight.real.
diffl f l (x::hollight.real) & diffl g m x & g x ~= real_of_num 0 -->
diffl (%x::hollight.real. real_div (f x) (g x))
(real_div (real_sub (real_mul l (g x)) (real_mul m (f x)))
(real_pow (g x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x"
by (import hollight DIFF_DIV)
lemma DIFF_SUM: "ALL (f::nat => hollight.real => hollight.real)
(f'::nat => hollight.real => hollight.real) (m::nat) (n::nat)
x::hollight.real.
(ALL r::nat. <= m r & < r (m + n) --> diffl (f r) (f' r x) x) -->
diffl (%x::hollight.real. psum (m, n) (%n::nat. f n x))
(psum (m, n) (%r::nat. f' r x)) x"
by (import hollight DIFF_SUM)
lemma CONT_BOUNDED: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX M::hollight.real.
ALL x::hollight.real. real_le a x & real_le x b --> real_le (f x) M)"
by (import hollight CONT_BOUNDED)
lemma CONT_BOUNDED_ABS: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX M::hollight.real.
ALL x::hollight.real.
real_le a x & real_le x b --> real_le (real_abs (f x)) M)"
by (import hollight CONT_BOUNDED_ABS)
lemma CONT_HASSUP: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX M::hollight.real.
(ALL x::hollight.real.
real_le a x & real_le x b --> real_le (f x) M) &
(ALL N::hollight.real.
real_lt N M -->
(EX x::hollight.real.
real_le a x & real_le x b & real_lt N (f x))))"
by (import hollight CONT_HASSUP)
lemma CONT_ATTAINS: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX x::hollight.real.
(ALL xa::hollight.real.
real_le a xa & real_le xa b --> real_le (f xa) x) &
(EX xa::hollight.real. real_le a xa & real_le xa b & f xa = x))"
by (import hollight CONT_ATTAINS)
lemma CONT_ATTAINS2: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX M::hollight.real.
(ALL x::hollight.real.
real_le a x & real_le x b --> real_le M (f x)) &
(EX x::hollight.real. real_le a x & real_le x b & f x = M))"
by (import hollight CONT_ATTAINS2)
lemma CONT_ATTAINS_ALL: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) -->
(EX (L::hollight.real) M::hollight.real.
(ALL x::hollight.real.
real_le a x & real_le x b -->
real_le L (f x) & real_le (f x) M) &
(ALL y::hollight.real.
real_le L y & real_le y M -->
(EX x::hollight.real. real_le a x & real_le x b & f x = y)))"
by (import hollight CONT_ATTAINS_ALL)
lemma DIFF_LINC: "ALL (f::hollight.real => hollight.real) (x::hollight.real) l::hollight.real.
diffl f l x & real_lt (real_of_num 0) l -->
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL h::hollight.real.
real_lt (real_of_num 0) h & real_lt h d -->
real_lt (f x) (f (real_add x h))))"
by (import hollight DIFF_LINC)
lemma DIFF_LDEC: "ALL (f::hollight.real => hollight.real) (x::hollight.real) l::hollight.real.
diffl f l x & real_lt l (real_of_num 0) -->
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL h::hollight.real.
real_lt (real_of_num 0) h & real_lt h d -->
real_lt (f x) (f (real_sub x h))))"
by (import hollight DIFF_LDEC)
lemma DIFF_LMAX: "ALL (f::hollight.real => hollight.real) (x::hollight.real) l::hollight.real.
diffl f l x &
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL y::hollight.real.
real_lt (real_abs (real_sub x y)) d --> real_le (f y) (f x))) -->
l = real_of_num 0"
by (import hollight DIFF_LMAX)
lemma DIFF_LMIN: "ALL (f::hollight.real => hollight.real) (x::hollight.real) l::hollight.real.
diffl f l x &
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL y::hollight.real.
real_lt (real_abs (real_sub x y)) d --> real_le (f x) (f y))) -->
l = real_of_num 0"
by (import hollight DIFF_LMIN)
lemma DIFF_LCONST: "ALL (f::hollight.real => hollight.real) (x::hollight.real) l::hollight.real.
diffl f l x &
(EX d::hollight.real.
real_lt (real_of_num 0) d &
(ALL y::hollight.real.
real_lt (real_abs (real_sub x y)) d --> f y = f x)) -->
l = real_of_num 0"
by (import hollight DIFF_LCONST)
lemma INTERVAL_LEMMA_LT: "ALL (a::hollight.real) (b::hollight.real) x::hollight.real.
real_lt a x & real_lt x b -->
(EX xa::hollight.real.
real_lt (real_of_num 0) xa &
(ALL xb::hollight.real.
real_lt (real_abs (real_sub x xb)) xa -->
real_lt a xb & real_lt xb b))"
by (import hollight INTERVAL_LEMMA_LT)
lemma INTERVAL_LEMMA: "ALL (a::hollight.real) (b::hollight.real) x::hollight.real.
real_lt a x & real_lt x b -->
(EX xa::hollight.real.
real_lt (real_of_num 0) xa &
(ALL y::hollight.real.
real_lt (real_abs (real_sub x y)) xa -->
real_le a y & real_le y b))"
by (import hollight INTERVAL_LEMMA)
lemma ROLLE: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_lt a b &
f a = f b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) &
(ALL x::hollight.real.
real_lt a x & real_lt x b --> differentiable f x) -->
(EX z::hollight.real.
real_lt a z & real_lt z b & diffl f (real_of_num 0) z)"
by (import hollight ROLLE)
lemma MVT_LEMMA: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_sub (f a)
(real_mul (real_div (real_sub (f b) (f a)) (real_sub b a)) a) =
real_sub (f b)
(real_mul (real_div (real_sub (f b) (f a)) (real_sub b a)) b)"
by (import hollight MVT_LEMMA)
lemma MVT: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_lt a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) &
(ALL x::hollight.real.
real_lt a x & real_lt x b --> differentiable f x) -->
(EX (l::hollight.real) z::hollight.real.
real_lt a z &
real_lt z b &
diffl f l z & real_sub (f b) (f a) = real_mul (real_sub b a) l)"
by (import hollight MVT)
lemma MVT_ALT: "ALL (f::hollight.real => hollight.real) (f'::hollight.real => hollight.real)
(a::hollight.real) b::hollight.real.
real_lt a b &
(ALL x::hollight.real.
real_le a x & real_le x b --> diffl f (f' x) x) -->
(EX z::hollight.real.
real_lt a z &
real_lt z b & real_sub (f b) (f a) = real_mul (real_sub b a) (f' z))"
by (import hollight MVT_ALT)
lemma DIFF_ISCONST_END: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_lt a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) &
(ALL x::hollight.real.
real_lt a x & real_lt x b --> diffl f (real_of_num 0) x) -->
f b = f a"
by (import hollight DIFF_ISCONST_END)
lemma DIFF_ISCONST: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_lt a b &
(ALL x::hollight.real. real_le a x & real_le x b --> contl f x) &
(ALL x::hollight.real.
real_lt a x & real_lt x b --> diffl f (real_of_num 0) x) -->
(ALL x::hollight.real. real_le a x & real_le x b --> f x = f a)"
by (import hollight DIFF_ISCONST)
lemma DIFF_ISCONST_END_SIMPLE: "ALL (f::hollight.real => hollight.real) (a::hollight.real) b::hollight.real.
real_lt a b &
(ALL x::hollight.real.
real_le a x & real_le x b --> diffl f (real_of_num 0) x) -->
f b = f a"
by (import hollight DIFF_ISCONST_END_SIMPLE)
lemma DIFF_ISCONST_ALL: "ALL (x::hollight.real => hollight.real) (xa::hollight.real)
xb::hollight.real. All (diffl x (real_of_num 0)) --> x xa = x xb"
by (import hollight DIFF_ISCONST_ALL)
lemma INTERVAL_ABS: "ALL (x::hollight.real) (z::hollight.real) d::hollight.real.
(real_le (real_sub x d) z & real_le z (real_add x d)) =
real_le (real_abs (real_sub z x)) d"
by (import hollight INTERVAL_ABS)
lemma CONT_INJ_LEMMA: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> contl f z) -->
~ (ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> real_le (f z) (f x))"
by (import hollight CONT_INJ_LEMMA)
lemma CONT_INJ_LEMMA2: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> contl f z) -->
~ (ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> real_le (f x) (f z))"
by (import hollight CONT_INJ_LEMMA2)
lemma CONT_INJ_RANGE: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> contl f z) -->
(EX e::hollight.real.
real_lt (real_of_num 0) e &
(ALL y::hollight.real.
real_le (real_abs (real_sub y (f x))) e -->
(EX z::hollight.real.
real_le (real_abs (real_sub z x)) d & f z = y)))"
by (import hollight CONT_INJ_RANGE)
lemma CONT_INVERSE: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> contl f z) -->
contl g (f x)"
by (import hollight CONT_INVERSE)
lemma DIFF_INVERSE: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_le (real_abs (real_sub z x)) d --> contl f z) &
diffl f l x & l ~= real_of_num 0 -->
diffl g (real_inv l) (f x)"
by (import hollight DIFF_INVERSE)
lemma DIFF_INVERSE_LT: "ALL (f::hollight.real => hollight.real) (g::hollight.real => hollight.real)
(l::hollight.real) (x::hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL z::hollight.real.
real_lt (real_abs (real_sub z x)) d --> g (f z) = z) &
(ALL z::hollight.real.
real_lt (real_abs (real_sub z x)) d --> contl f z) &
diffl f l x & l ~= real_of_num 0 -->
diffl g (real_inv l) (f x)"
by (import hollight DIFF_INVERSE_LT)
lemma IVT_DERIVATIVE_0: "ALL (f::hollight.real => hollight.real) (f'::hollight.real => hollight.real)
(a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real. real_le a x & real_le x b --> diffl f (f' x) x) &
hollight.real_gt (f' a) (real_of_num 0) &
real_lt (f' b) (real_of_num 0) -->
(EX z::hollight.real. real_lt a z & real_lt z b & f' z = real_of_num 0)"
by (import hollight IVT_DERIVATIVE_0)
lemma IVT_DERIVATIVE_POS: "ALL (x::hollight.real => hollight.real) (xa::hollight.real => hollight.real)
(xb::hollight.real) (xc::hollight.real) xd::hollight.real.
real_le xb xc &
(ALL xd::hollight.real.
real_le xb xd & real_le xd xc --> diffl x (xa xd) xd) &
hollight.real_gt (xa xb) xd & real_lt (xa xc) xd -->
(EX z::hollight.real. real_lt xb z & real_lt z xc & xa z = xd)"
by (import hollight IVT_DERIVATIVE_POS)
lemma IVT_DERIVATIVE_NEG: "ALL (x::hollight.real => hollight.real) (xa::hollight.real => hollight.real)
(xb::hollight.real) (xc::hollight.real) xd::hollight.real.
real_le xb xc &
(ALL xd::hollight.real.
real_le xb xd & real_le xd xc --> diffl x (xa xd) xd) &
real_lt (xa xb) xd & hollight.real_gt (xa xc) xd -->
(EX z::hollight.real. real_lt xb z & real_lt z xc & xa z = xd)"
by (import hollight IVT_DERIVATIVE_NEG)
lemma SEQ_CONT_UNIFORM: "ALL (s::nat => hollight.real => hollight.real)
(f::hollight.real => hollight.real) x0::hollight.real.
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX (N::nat) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL (x::hollight.real) n::nat.
real_lt (real_abs (real_sub x x0)) d & >= n N -->
real_lt (real_abs (real_sub (s n x) (f x))) e))) &
(EX N::nat. ALL n::nat. >= n N --> contl (s n) x0) -->
contl f x0"
by (import hollight SEQ_CONT_UNIFORM)
lemma SER_COMPARA_UNIFORM: "ALL (s::hollight.real => nat => hollight.real) (x0::hollight.real)
g::nat => hollight.real.
(EX (N::nat) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL (n::nat) x::hollight.real.
real_lt (real_abs (real_sub x x0)) d & >= n N -->
real_le (real_abs (s x n)) (g n))) &
summable g -->
(EX (f::hollight.real => hollight.real) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (x::hollight.real) n::nat.
real_lt (real_abs (real_sub x x0)) d & >= n N -->
real_lt (real_abs (real_sub (psum (0, n) (s x)) (f x)))
e)))"
by (import hollight SER_COMPARA_UNIFORM)
lemma SER_COMPARA_UNIFORM_WEAK: "ALL (s::hollight.real => nat => hollight.real) (x0::hollight.real)
g::nat => hollight.real.
(EX (N::nat) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL (n::nat) x::hollight.real.
real_lt (real_abs (real_sub x x0)) d & >= n N -->
real_le (real_abs (s x n)) (g n))) &
summable g -->
(EX f::hollight.real => hollight.real.
ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX (N::nat) d::hollight.real.
real_lt (real_of_num 0) d &
(ALL (x::hollight.real) n::nat.
real_lt (real_abs (real_sub x x0)) d & >= n N -->
real_lt (real_abs (real_sub (psum (0, n) (s x)) (f x)))
e)))"
by (import hollight SER_COMPARA_UNIFORM_WEAK)
lemma POWDIFF_LEMMA: "ALL (n::nat) (x::hollight.real) y::hollight.real.
psum (0, Suc n)
(%p::nat. real_mul (real_pow x p) (real_pow y (Suc n - p))) =
real_mul y
(psum (0, Suc n)
(%p::nat. real_mul (real_pow x p) (real_pow y (n - p))))"
by (import hollight POWDIFF_LEMMA)
lemma POWDIFF: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_sub (real_pow x (Suc n)) (real_pow y (Suc n)) =
real_mul (real_sub x y)
(psum (0, Suc n)
(%p::nat. real_mul (real_pow x p) (real_pow y (n - p))))"
by (import hollight POWDIFF)
lemma POWREV: "ALL (n::nat) (x::hollight.real) y::hollight.real.
psum (0, Suc n)
(%xa::nat. real_mul (real_pow x xa) (real_pow y (n - xa))) =
psum (0, Suc n)
(%xa::nat. real_mul (real_pow x (n - xa)) (real_pow y xa))"
by (import hollight POWREV)
lemma POWSER_INSIDEA: "ALL (f::nat => hollight.real) (x::hollight.real) z::hollight.real.
summable (%n::nat. real_mul (f n) (real_pow x n)) &
real_lt (real_abs z) (real_abs x) -->
summable (%n::nat. real_mul (real_abs (f n)) (real_pow z n))"
by (import hollight POWSER_INSIDEA)
lemma POWSER_INSIDE: "ALL (f::nat => hollight.real) (x::hollight.real) z::hollight.real.
summable (%n::nat. real_mul (f n) (real_pow x n)) &
real_lt (real_abs z) (real_abs x) -->
summable (%n::nat. real_mul (f n) (real_pow z n))"
by (import hollight POWSER_INSIDE)
constdefs
diffs :: "(nat => hollight.real) => nat => hollight.real"
"diffs ==
%(u::nat => hollight.real) n::nat.
real_mul (real_of_num (Suc n)) (u (Suc n))"
lemma DEF_diffs: "diffs =
(%(u::nat => hollight.real) n::nat.
real_mul (real_of_num (Suc n)) (u (Suc n)))"
by (import hollight DEF_diffs)
lemma DIFFS_NEG: "ALL c::nat => hollight.real.
diffs (%n::nat. real_neg (c n)) = (%x::nat. real_neg (diffs c x))"
by (import hollight DIFFS_NEG)
lemma DIFFS_LEMMA: "ALL (n::nat) (c::nat => hollight.real) x::hollight.real.
psum (0, n) (%n::nat. real_mul (diffs c n) (real_pow x n)) =
real_add
(psum (0, n)
(%n::nat.
real_mul (real_of_num n)
(real_mul (c n) (real_pow x (n - NUMERAL_BIT1 0)))))
(real_mul (real_of_num n)
(real_mul (c n) (real_pow x (n - NUMERAL_BIT1 0))))"
by (import hollight DIFFS_LEMMA)
lemma DIFFS_LEMMA2: "ALL (n::nat) (c::nat => hollight.real) x::hollight.real.
psum (0, n)
(%n::nat.
real_mul (real_of_num n)
(real_mul (c n) (real_pow x (n - NUMERAL_BIT1 0)))) =
real_sub (psum (0, n) (%n::nat. real_mul (diffs c n) (real_pow x n)))
(real_mul (real_of_num n)
(real_mul (c n) (real_pow x (n - NUMERAL_BIT1 0))))"
by (import hollight DIFFS_LEMMA2)
lemma DIFFS_EQUIV: "ALL (c::nat => hollight.real) x::hollight.real.
summable (%n::nat. real_mul (diffs c n) (real_pow x n)) -->
sums
(%n::nat.
real_mul (real_of_num n)
(real_mul (c n) (real_pow x (n - NUMERAL_BIT1 0))))
(suminf (%n::nat. real_mul (diffs c n) (real_pow x n)))"
by (import hollight DIFFS_EQUIV)
lemma TERMDIFF_LEMMA1: "ALL (m::nat) (z::hollight.real) h::hollight.real.
psum (0, m)
(%p::nat.
real_sub (real_mul (real_pow (real_add z h) (m - p)) (real_pow z p))
(real_pow z m)) =
psum (0, m)
(%p::nat.
real_mul (real_pow z p)
(real_sub (real_pow (real_add z h) (m - p)) (real_pow z (m - p))))"
by (import hollight TERMDIFF_LEMMA1)
lemma TERMDIFF_LEMMA2: "ALL (z::hollight.real) h::hollight.real.
h ~= real_of_num 0 -->
real_sub
(real_div (real_sub (real_pow (real_add z h) (n::nat)) (real_pow z n))
h)
(real_mul (real_of_num n) (real_pow z (n - NUMERAL_BIT1 0))) =
real_mul h
(psum (0, n - NUMERAL_BIT1 0)
(%p::nat.
real_mul (real_pow z p)
(psum (0, n - NUMERAL_BIT1 0 - p)
(%q::nat.
real_mul (real_pow (real_add z h) q)
(real_pow z
(n - NUMERAL_BIT0 (NUMERAL_BIT1 0) - p - q))))))"
by (import hollight TERMDIFF_LEMMA2)
lemma TERMDIFF_LEMMA3: "ALL (z::hollight.real) (h::hollight.real) (n::nat) K::hollight.real.
h ~= real_of_num 0 &
real_le (real_abs z) K & real_le (real_abs (real_add z h)) K -->
real_le
(real_abs
(real_sub
(real_div (real_sub (real_pow (real_add z h) n) (real_pow z n)) h)
(real_mul (real_of_num n) (real_pow z (n - NUMERAL_BIT1 0)))))
(real_mul (real_of_num n)
(real_mul (real_of_num (n - NUMERAL_BIT1 0))
(real_mul (real_pow K (n - NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_abs h))))"
by (import hollight TERMDIFF_LEMMA3)
lemma TERMDIFF_LEMMA4: "ALL (f::hollight.real => hollight.real) (K::hollight.real) k::hollight.real.
real_lt (real_of_num 0) k &
(ALL h::hollight.real.
real_lt (real_of_num 0) (real_abs h) & real_lt (real_abs h) k -->
real_le (real_abs (f h)) (real_mul K (real_abs h))) -->
tends_real_real f (real_of_num 0) (real_of_num 0)"
by (import hollight TERMDIFF_LEMMA4)
lemma TERMDIFF_LEMMA5: "ALL (f::nat => hollight.real) (g::hollight.real => nat => hollight.real)
k::hollight.real.
real_lt (real_of_num 0) k &
summable f &
(ALL h::hollight.real.
real_lt (real_of_num 0) (real_abs h) & real_lt (real_abs h) k -->
(ALL n::nat.
real_le (real_abs (g h n)) (real_mul (f n) (real_abs h)))) -->
tends_real_real (%h::hollight.real. suminf (g h)) (real_of_num 0)
(real_of_num 0)"
by (import hollight TERMDIFF_LEMMA5)
lemma TERMDIFF: "ALL (c::nat => hollight.real) K::hollight.real.
summable (%n::nat. real_mul (c n) (real_pow K n)) &
summable (%n::nat. real_mul (diffs c n) (real_pow K n)) &
summable (%n::nat. real_mul (diffs (diffs c) n) (real_pow K n)) &
real_lt (real_abs (x::hollight.real)) (real_abs K) -->
diffl
(%x::hollight.real. suminf (%n::nat. real_mul (c n) (real_pow x n)))
(suminf (%n::nat. real_mul (diffs c n) (real_pow x n))) x"
by (import hollight TERMDIFF)
lemma SEQ_NPOW: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
tends_num_real (%n::nat. real_mul (real_of_num n) (real_pow x n))
(real_of_num 0)"
by (import hollight SEQ_NPOW)
lemma TERMDIFF_CONVERGES: "ALL K::hollight.real.
(ALL x::hollight.real.
real_lt (real_abs x) K -->
summable
(%n::nat.
real_mul ((c::nat => hollight.real) n) (real_pow x n))) -->
(ALL x::hollight.real.
real_lt (real_abs x) K -->
summable (%n::nat. real_mul (diffs c n) (real_pow x n)))"
by (import hollight TERMDIFF_CONVERGES)
lemma TERMDIFF_STRONG: "ALL (c::nat => hollight.real) (K::hollight.real) x::hollight.real.
summable (%n::nat. real_mul (c n) (real_pow K n)) &
real_lt (real_abs x) (real_abs K) -->
diffl
(%x::hollight.real. suminf (%n::nat. real_mul (c n) (real_pow x n)))
(suminf (%n::nat. real_mul (diffs c n) (real_pow x n))) x"
by (import hollight TERMDIFF_STRONG)
lemma POWSER_0: "ALL a::nat => hollight.real.
sums (%n::nat. real_mul (a n) (real_pow (real_of_num 0) n)) (a 0)"
by (import hollight POWSER_0)
lemma POWSER_LIMIT_0: "ALL (f::hollight.real => hollight.real) (a::nat => hollight.real)
s::hollight.real.
real_lt (real_of_num 0) s &
(ALL x::hollight.real.
real_lt (real_abs x) s -->
sums (%n::nat. real_mul (a n) (real_pow x n)) (f x)) -->
tends_real_real f (a 0) (real_of_num 0)"
by (import hollight POWSER_LIMIT_0)
lemma POWSER_LIMIT_0_STRONG: "ALL (f::hollight.real => hollight.real) (a::nat => hollight.real)
s::hollight.real.
real_lt (real_of_num 0) s &
(ALL x::hollight.real.
real_lt (real_of_num 0) (real_abs x) & real_lt (real_abs x) s -->
sums (%n::nat. real_mul (a n) (real_pow x n)) (f x)) -->
tends_real_real f (a 0) (real_of_num 0)"
by (import hollight POWSER_LIMIT_0_STRONG)
lemma POWSER_EQUAL_0: "ALL (f::hollight.real => hollight.real) (a::nat => hollight.real)
(b::nat => hollight.real) P::hollight.real => bool.
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX x::hollight.real.
P x &
real_lt (real_of_num 0) (real_abs x) & real_lt (real_abs x) e)) &
(ALL x::hollight.real.
real_lt (real_of_num 0) (real_abs x) & P x -->
sums (%n::nat. real_mul (a n) (real_pow x n)) (f x) &
sums (%n::nat. real_mul (b n) (real_pow x n)) (f x)) -->
a 0 = b 0"
by (import hollight POWSER_EQUAL_0)
lemma POWSER_EQUAL: "ALL (f::hollight.real => hollight.real) (a::nat => hollight.real)
(b::nat => hollight.real) P::hollight.real => bool.
(ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX x::hollight.real.
P x &
real_lt (real_of_num 0) (real_abs x) & real_lt (real_abs x) e)) &
(ALL x::hollight.real.
P x -->
sums (%n::nat. real_mul (a n) (real_pow x n)) (f x) &
sums (%n::nat. real_mul (b n) (real_pow x n)) (f x)) -->
a = b"
by (import hollight POWSER_EQUAL)
lemma MULT_DIV_2: "ALL n::nat.
DIV (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n) (NUMERAL_BIT0 (NUMERAL_BIT1 0)) =
n"
by (import hollight MULT_DIV_2)
lemma EVEN_DIV2: "ALL n::nat.
~ EVEN n -->
DIV (Suc n) (NUMERAL_BIT0 (NUMERAL_BIT1 0)) =
Suc (DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight EVEN_DIV2)
lemma POW_ZERO: "ALL (n::nat) x::hollight.real.
real_pow x n = real_of_num 0 --> x = real_of_num 0"
by (import hollight POW_ZERO)
lemma POW_ZERO_EQ: "ALL (n::nat) x::hollight.real.
(real_pow x (Suc n) = real_of_num 0) = (x = real_of_num 0)"
by (import hollight POW_ZERO_EQ)
lemma POW_LT: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_lt x y -->
real_lt (real_pow x (Suc n)) (real_pow y (Suc n))"
by (import hollight POW_LT)
lemma POW_EQ: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x &
real_le (real_of_num 0) y & real_pow x (Suc n) = real_pow y (Suc n) -->
x = y"
by (import hollight POW_EQ)
constdefs
exp :: "hollight.real => hollight.real"
"exp ==
%u::hollight.real.
suminf
(%n::nat. real_mul (real_inv (real_of_num (FACT n))) (real_pow u n))"
lemma DEF_exp: "exp =
(%u::hollight.real.
suminf
(%n::nat. real_mul (real_inv (real_of_num (FACT n))) (real_pow u n)))"
by (import hollight DEF_exp)
constdefs
sin :: "hollight.real => hollight.real"
"sin ==
%u::hollight.real.
suminf
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n))))
(real_pow u n))"
lemma DEF_sin: "sin =
(%u::hollight.real.
suminf
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n))))
(real_pow u n)))"
by (import hollight DEF_sin)
constdefs
cos :: "hollight.real => hollight.real"
"cos ==
%u::hollight.real.
suminf
(%n::nat.
real_mul
(COND (EVEN n)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))
(real_of_num 0))
(real_pow u n))"
lemma DEF_cos: "cos =
(%u::hollight.real.
suminf
(%n::nat.
real_mul
(COND (EVEN n)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))
(real_of_num 0))
(real_pow u n)))"
by (import hollight DEF_cos)
lemma REAL_EXP_CONVERGES: "ALL x::hollight.real.
sums (%n::nat. real_mul (real_inv (real_of_num (FACT n))) (real_pow x n))
(exp x)"
by (import hollight REAL_EXP_CONVERGES)
lemma SIN_CONVERGES: "ALL x::hollight.real.
sums
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n))))
(real_pow x n))
(sin x)"
by (import hollight SIN_CONVERGES)
lemma COS_CONVERGES: "ALL x::hollight.real.
sums
(%n::nat.
real_mul
(COND (EVEN n)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))
(real_of_num 0))
(real_pow x n))
(cos x)"
by (import hollight COS_CONVERGES)
lemma REAL_EXP_FDIFF: "diffs (%n::nat. real_inv (real_of_num (FACT n))) =
(%n::nat. real_inv (real_of_num (FACT n)))"
by (import hollight REAL_EXP_FDIFF)
lemma SIN_FDIFF: "diffs
(%n::nat.
COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))) =
(%n::nat.
COND (EVEN n)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))
(real_of_num 0))"
by (import hollight SIN_FDIFF)
lemma COS_FDIFF: "diffs
(%n::nat.
COND (EVEN n)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))
(real_of_num 0)) =
(%n::nat.
real_neg
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n)))))"
by (import hollight COS_FDIFF)
lemma SIN_NEGLEMMA: "ALL x::hollight.real.
real_neg (sin x) =
suminf
(%n::nat.
real_neg
(real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT n))))
(real_pow x n)))"
by (import hollight SIN_NEGLEMMA)
lemma DIFF_EXP: "ALL x::hollight.real. diffl exp (exp x) x"
by (import hollight DIFF_EXP)
lemma DIFF_SIN: "ALL x::hollight.real. diffl sin (cos x) x"
by (import hollight DIFF_SIN)
lemma DIFF_COS: "ALL x::hollight.real. diffl cos (real_neg (sin x)) x"
by (import hollight DIFF_COS)
lemma DIFF_COMPOSITE: "(diffl (f::hollight.real => hollight.real) (l::hollight.real)
(x::hollight.real) &
f x ~= real_of_num 0 -->
diffl (%x::hollight.real. real_inv (f x))
(real_neg (real_div l (real_pow (f x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x) &
(diffl f l x &
diffl (g::hollight.real => hollight.real) (m::hollight.real) x &
g x ~= real_of_num 0 -->
diffl (%x::hollight.real. real_div (f x) (g x))
(real_div (real_sub (real_mul l (g x)) (real_mul m (f x)))
(real_pow (g x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x) &
(diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_add (f x) (g x)) (real_add l m) x) &
(diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_mul (f x) (g x))
(real_add (real_mul l (g x)) (real_mul m (f x))) x) &
(diffl f l x & diffl g m x -->
diffl (%x::hollight.real. real_sub (f x) (g x)) (real_sub l m) x) &
(diffl f l x --> diffl (%x::hollight.real. real_neg (f x)) (real_neg l) x) &
(diffl g m x -->
diffl (%x::hollight.real. real_pow (g x) (n::nat))
(real_mul (real_mul (real_of_num n) (real_pow (g x) (n - NUMERAL_BIT1 0)))
m)
x) &
(diffl g m x -->
diffl (%x::hollight.real. exp (g x)) (real_mul (exp (g x)) m) x) &
(diffl g m x -->
diffl (%x::hollight.real. sin (g x)) (real_mul (cos (g x)) m) x) &
(diffl g m x -->
diffl (%x::hollight.real. cos (g x)) (real_mul (real_neg (sin (g x))) m) x)"
by (import hollight DIFF_COMPOSITE)
lemma REAL_EXP_0: "exp (real_of_num 0) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight REAL_EXP_0)
lemma REAL_EXP_LE_X: "ALL x::hollight.real.
real_le (real_of_num 0) x -->
real_le (real_add (real_of_num (NUMERAL_BIT1 0)) x) (exp x)"
by (import hollight REAL_EXP_LE_X)
lemma REAL_EXP_LT_1: "ALL x::hollight.real.
real_lt (real_of_num 0) x -->
real_lt (real_of_num (NUMERAL_BIT1 0)) (exp x)"
by (import hollight REAL_EXP_LT_1)
lemma REAL_EXP_ADD_MUL: "ALL (x::hollight.real) y::hollight.real.
real_mul (exp (real_add x y)) (exp (real_neg x)) = exp y"
by (import hollight REAL_EXP_ADD_MUL)
lemma REAL_EXP_NEG_MUL: "ALL x::hollight.real.
real_mul (exp x) (exp (real_neg x)) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight REAL_EXP_NEG_MUL)
lemma REAL_EXP_NEG_MUL2: "ALL x::hollight.real.
real_mul (exp (real_neg x)) (exp x) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight REAL_EXP_NEG_MUL2)
lemma REAL_EXP_NEG: "ALL x::hollight.real. exp (real_neg x) = real_inv (exp x)"
by (import hollight REAL_EXP_NEG)
lemma REAL_EXP_ADD: "ALL (x::hollight.real) y::hollight.real.
exp (real_add x y) = real_mul (exp x) (exp y)"
by (import hollight REAL_EXP_ADD)
lemma REAL_EXP_POS_LE: "ALL x::hollight.real. real_le (real_of_num 0) (exp x)"
by (import hollight REAL_EXP_POS_LE)
lemma REAL_EXP_NZ: "ALL x::hollight.real. exp x ~= real_of_num 0"
by (import hollight REAL_EXP_NZ)
lemma REAL_EXP_POS_LT: "ALL x::hollight.real. real_lt (real_of_num 0) (exp x)"
by (import hollight REAL_EXP_POS_LT)
lemma REAL_EXP_N: "ALL (n::nat) x::hollight.real.
exp (real_mul (real_of_num n) x) = real_pow (exp x) n"
by (import hollight REAL_EXP_N)
lemma REAL_EXP_SUB: "ALL (x::hollight.real) y::hollight.real.
exp (real_sub x y) = real_div (exp x) (exp y)"
by (import hollight REAL_EXP_SUB)
lemma REAL_EXP_MONO_IMP: "ALL (x::hollight.real) y::hollight.real.
real_lt x y --> real_lt (exp x) (exp y)"
by (import hollight REAL_EXP_MONO_IMP)
lemma REAL_EXP_MONO_LT: "ALL (x::hollight.real) y::hollight.real.
real_lt (exp x) (exp y) = real_lt x y"
by (import hollight REAL_EXP_MONO_LT)
lemma REAL_EXP_MONO_LE: "ALL (x::hollight.real) y::hollight.real.
real_le (exp x) (exp y) = real_le x y"
by (import hollight REAL_EXP_MONO_LE)
lemma REAL_EXP_INJ: "ALL (x::hollight.real) y::hollight.real. (exp x = exp y) = (x = y)"
by (import hollight REAL_EXP_INJ)
lemma REAL_EXP_TOTAL_LEMMA: "ALL y::hollight.real.
real_le (real_of_num (NUMERAL_BIT1 0)) y -->
(EX x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_sub y (real_of_num (NUMERAL_BIT1 0))) & exp x = y)"
by (import hollight REAL_EXP_TOTAL_LEMMA)
lemma REAL_EXP_TOTAL: "ALL y::hollight.real.
real_lt (real_of_num 0) y --> (EX x::hollight.real. exp x = y)"
by (import hollight REAL_EXP_TOTAL)
lemma REAL_EXP_BOUND_LEMMA: "ALL x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_inv (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_le (exp x)
(real_add (real_of_num (NUMERAL_BIT1 0))
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x))"
by (import hollight REAL_EXP_BOUND_LEMMA)
constdefs
ln :: "hollight.real => hollight.real"
"ln == %u::hollight.real. SOME ua::hollight.real. exp ua = u"
lemma DEF_ln: "ln = (%u::hollight.real. SOME ua::hollight.real. exp ua = u)"
by (import hollight DEF_ln)
lemma LN_EXP: "ALL x::hollight.real. ln (exp x) = x"
by (import hollight LN_EXP)
lemma REAL_EXP_LN: "ALL x::hollight.real. (exp (ln x) = x) = real_lt (real_of_num 0) x"
by (import hollight REAL_EXP_LN)
lemma LN_MUL: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
ln (real_mul x y) = real_add (ln x) (ln y)"
by (import hollight LN_MUL)
lemma LN_INJ: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
(ln x = ln y) = (x = y)"
by (import hollight LN_INJ)
lemma LN_1: "ln (real_of_num (NUMERAL_BIT1 0)) = real_of_num 0"
by (import hollight LN_1)
lemma LN_INV: "ALL x::hollight.real.
real_lt (real_of_num 0) x --> ln (real_inv x) = real_neg (ln x)"
by (import hollight LN_INV)
lemma LN_DIV: "ALL x::hollight.real.
real_lt (real_of_num 0) x &
real_lt (real_of_num 0) (y::hollight.real) -->
ln (real_div x y) = real_sub (ln x) (ln y)"
by (import hollight LN_DIV)
lemma LN_MONO_LT: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
real_lt (ln x) (ln y) = real_lt x y"
by (import hollight LN_MONO_LT)
lemma LN_MONO_LE: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_of_num 0) x & real_lt (real_of_num 0) y -->
real_le (ln x) (ln y) = real_le x y"
by (import hollight LN_MONO_LE)
lemma LN_POW: "ALL (n::nat) x::hollight.real.
real_lt (real_of_num 0) x -->
ln (real_pow x n) = real_mul (real_of_num n) (ln x)"
by (import hollight LN_POW)
lemma LN_LE: "ALL x::hollight.real.
real_le (real_of_num 0) x -->
real_le (ln (real_add (real_of_num (NUMERAL_BIT1 0)) x)) x"
by (import hollight LN_LE)
lemma LN_LT_X: "ALL x::hollight.real. real_lt (real_of_num 0) x --> real_lt (ln x) x"
by (import hollight LN_LT_X)
lemma LN_POS: "ALL x::hollight.real.
real_le (real_of_num (NUMERAL_BIT1 0)) x -->
real_le (real_of_num 0) (ln x)"
by (import hollight LN_POS)
lemma LN_POS_LT: "ALL x::hollight.real.
real_lt (real_of_num (NUMERAL_BIT1 0)) x -->
real_lt (real_of_num 0) (ln x)"
by (import hollight LN_POS_LT)
lemma DIFF_LN: "ALL x::hollight.real. real_lt (real_of_num 0) x --> diffl ln (real_inv x) x"
by (import hollight DIFF_LN)
constdefs
root :: "nat => hollight.real => hollight.real"
"root ==
%(u::nat) ua::hollight.real.
SOME ub::hollight.real.
(real_lt (real_of_num 0) ua --> real_lt (real_of_num 0) ub) &
real_pow ub u = ua"
lemma DEF_root: "root =
(%(u::nat) ua::hollight.real.
SOME ub::hollight.real.
(real_lt (real_of_num 0) ua --> real_lt (real_of_num 0) ub) &
real_pow ub u = ua)"
by (import hollight DEF_root)
constdefs
sqrt :: "hollight.real => hollight.real"
"sqrt ==
%u::hollight.real.
SOME y::hollight.real.
real_le (real_of_num 0) y &
real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = u"
lemma DEF_sqrt: "sqrt =
(%u::hollight.real.
SOME y::hollight.real.
real_le (real_of_num 0) y &
real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = u)"
by (import hollight DEF_sqrt)
lemma sqrt: "sqrt (x::hollight.real) = root (NUMERAL_BIT0 (NUMERAL_BIT1 0)) x"
by (import hollight sqrt)
lemma ROOT_LT_LEMMA: "ALL (n::nat) x::hollight.real.
real_lt (real_of_num 0) x -->
real_pow (exp (real_div (ln x) (real_of_num (Suc n)))) (Suc n) = x"
by (import hollight ROOT_LT_LEMMA)
lemma ROOT_LN: "ALL x::hollight.real.
real_lt (real_of_num 0) x -->
(ALL n::nat.
root (Suc n) x = exp (real_div (ln x) (real_of_num (Suc n))))"
by (import hollight ROOT_LN)
lemma ROOT_0: "ALL n::nat. root (Suc n) (real_of_num 0) = real_of_num 0"
by (import hollight ROOT_0)
lemma ROOT_1: "ALL n::nat.
root (Suc n) (real_of_num (NUMERAL_BIT1 0)) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight ROOT_1)
lemma ROOT_POW_POS: "ALL (n::nat) x::hollight.real.
real_le (real_of_num 0) x --> real_pow (root (Suc n) x) (Suc n) = x"
by (import hollight ROOT_POW_POS)
lemma POW_ROOT_POS: "ALL (n::nat) x::hollight.real.
real_le (real_of_num 0) x --> root (Suc n) (real_pow x (Suc n)) = x"
by (import hollight POW_ROOT_POS)
lemma ROOT_POS_POSITIVE: "ALL (x::hollight.real) n::nat.
real_le (real_of_num 0) x --> real_le (real_of_num 0) (root (Suc n) x)"
by (import hollight ROOT_POS_POSITIVE)
lemma ROOT_POS_UNIQ: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x &
real_le (real_of_num 0) y & real_pow y (Suc n) = x -->
root (Suc n) x = y"
by (import hollight ROOT_POS_UNIQ)
lemma ROOT_MUL: "ALL (n::nat) (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) y -->
root (Suc n) (real_mul x y) = real_mul (root (Suc n) x) (root (Suc n) y)"
by (import hollight ROOT_MUL)
lemma ROOT_INV: "ALL (n::nat) x::hollight.real.
real_le (real_of_num 0) x -->
root (Suc n) (real_inv x) = real_inv (root (Suc n) x)"
by (import hollight ROOT_INV)
lemma ROOT_DIV: "ALL (x::nat) (xa::hollight.real) xb::hollight.real.
real_le (real_of_num 0) xa & real_le (real_of_num 0) xb -->
root (Suc x) (real_div xa xb) =
real_div (root (Suc x) xa) (root (Suc x) xb)"
by (import hollight ROOT_DIV)
lemma ROOT_MONO_LT: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_lt x y -->
real_lt (root (Suc (n::nat)) x) (root (Suc n) y)"
by (import hollight ROOT_MONO_LT)
lemma ROOT_MONO_LE: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_le x y -->
real_le (root (Suc (n::nat)) x) (root (Suc n) y)"
by (import hollight ROOT_MONO_LE)
lemma ROOT_MONO_LT_EQ: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) y -->
real_lt (root (Suc (n::nat)) x) (root (Suc n) y) = real_lt x y"
by (import hollight ROOT_MONO_LT_EQ)
lemma ROOT_MONO_LE_EQ: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) y -->
real_le (root (Suc (n::nat)) x) (root (Suc n) y) = real_le x y"
by (import hollight ROOT_MONO_LE_EQ)
lemma ROOT_INJ: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
(root (Suc (n::nat)) x = root (Suc n) xa) = (x = xa)"
by (import hollight ROOT_INJ)
lemma SQRT_0: "sqrt (real_of_num 0) = real_of_num 0"
by (import hollight SQRT_0)
lemma SQRT_1: "sqrt (real_of_num (NUMERAL_BIT1 0)) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight SQRT_1)
lemma SQRT_POS_LT: "ALL x::hollight.real.
real_lt (real_of_num 0) x --> real_lt (real_of_num 0) (sqrt x)"
by (import hollight SQRT_POS_LT)
lemma SQRT_POS_LE: "ALL x::hollight.real.
real_le (real_of_num 0) x --> real_le (real_of_num 0) (sqrt x)"
by (import hollight SQRT_POS_LE)
lemma SQRT_POW2: "ALL x::hollight.real.
(real_pow (sqrt x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = x) =
real_le (real_of_num 0) x"
by (import hollight SQRT_POW2)
lemma SQRT_POW_2: "ALL x::hollight.real.
real_le (real_of_num 0) x -->
real_pow (sqrt x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = x"
by (import hollight SQRT_POW_2)
lemma POW_2_SQRT: "real_le (real_of_num 0) (x::hollight.real) -->
sqrt (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))) = x"
by (import hollight POW_2_SQRT)
lemma SQRT_POS_UNIQ: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x &
real_le (real_of_num 0) xa &
real_pow xa (NUMERAL_BIT0 (NUMERAL_BIT1 0)) = x -->
sqrt x = xa"
by (import hollight SQRT_POS_UNIQ)
lemma SQRT_MUL: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
sqrt (real_mul x xa) = real_mul (sqrt x) (sqrt xa)"
by (import hollight SQRT_MUL)
lemma SQRT_INV: "ALL x::hollight.real.
real_le (real_of_num 0) x --> sqrt (real_inv x) = real_inv (sqrt x)"
by (import hollight SQRT_INV)
lemma SQRT_DIV: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
sqrt (real_div x xa) = real_div (sqrt x) (sqrt xa)"
by (import hollight SQRT_DIV)
lemma SQRT_MONO_LT: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_lt x xa --> real_lt (sqrt x) (sqrt xa)"
by (import hollight SQRT_MONO_LT)
lemma SQRT_MONO_LE: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le x xa --> real_le (sqrt x) (sqrt xa)"
by (import hollight SQRT_MONO_LE)
lemma SQRT_MONO_LT_EQ: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
real_lt (sqrt x) (sqrt xa) = real_lt x xa"
by (import hollight SQRT_MONO_LT_EQ)
lemma SQRT_MONO_LE_EQ: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
real_le (sqrt x) (sqrt xa) = real_le x xa"
by (import hollight SQRT_MONO_LE_EQ)
lemma SQRT_INJ: "ALL (x::hollight.real) xa::hollight.real.
real_le (real_of_num 0) x & real_le (real_of_num 0) xa -->
(sqrt x = sqrt xa) = (x = xa)"
by (import hollight SQRT_INJ)
lemma SQRT_EVEN_POW2: "ALL n::nat.
EVEN n -->
sqrt (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) n) =
real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight SQRT_EVEN_POW2)
lemma REAL_DIV_SQRT: "ALL x::hollight.real.
real_le (real_of_num 0) x --> real_div x (sqrt x) = sqrt x"
by (import hollight REAL_DIV_SQRT)
lemma POW_2_SQRT_ABS: "ALL x::hollight.real.
sqrt (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))) = real_abs x"
by (import hollight POW_2_SQRT_ABS)
lemma SQRT_EQ_0: "ALL x::hollight.real.
real_le (real_of_num 0) x -->
(sqrt x = real_of_num 0) = (x = real_of_num 0)"
by (import hollight SQRT_EQ_0)
lemma REAL_LE_LSQRT: "ALL (x::hollight.real) y::hollight.real.
real_le (real_of_num 0) x &
real_le (real_of_num 0) y &
real_le x (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 0))) -->
real_le (sqrt x) y"
by (import hollight REAL_LE_LSQRT)
lemma REAL_LE_POW_2: "ALL x::hollight.real.
real_le (real_of_num 0) (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight REAL_LE_POW_2)
lemma REAL_LE_RSQRT: "ALL (x::hollight.real) y::hollight.real.
real_le (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))) y -->
real_le x (sqrt y)"
by (import hollight REAL_LE_RSQRT)
lemma SIN_0: "sin (real_of_num 0) = real_of_num 0"
by (import hollight SIN_0)
lemma COS_0: "cos (real_of_num 0) = real_of_num (NUMERAL_BIT1 0)"
by (import hollight COS_0)
lemma SIN_CIRCLE: "ALL x::hollight.real.
real_add (real_pow (sin x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight SIN_CIRCLE)
lemma SIN_BOUND: "ALL x::hollight.real.
real_le (real_abs (sin x)) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight SIN_BOUND)
lemma SIN_BOUNDS: "ALL x::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) (sin x) &
real_le (sin x) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight SIN_BOUNDS)
lemma COS_BOUND: "ALL x::hollight.real.
real_le (real_abs (cos x)) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight COS_BOUND)
lemma COS_BOUNDS: "ALL x::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) (cos x) &
real_le (cos x) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight COS_BOUNDS)
lemma SIN_COS_ADD: "ALL (x::hollight.real) y::hollight.real.
real_add
(real_pow
(real_sub (sin (real_add x y))
(real_add (real_mul (sin x) (cos y)) (real_mul (cos x) (sin y))))
(NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_pow
(real_sub (cos (real_add x y))
(real_sub (real_mul (cos x) (cos y)) (real_mul (sin x) (sin y))))
(NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_of_num 0"
by (import hollight SIN_COS_ADD)
lemma SIN_COS_NEG: "ALL x::hollight.real.
real_add
(real_pow (real_add (sin (real_neg x)) (sin x))
(NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_pow (real_sub (cos (real_neg x)) (cos x))
(NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_of_num 0"
by (import hollight SIN_COS_NEG)
lemma SIN_ADD: "ALL (x::hollight.real) y::hollight.real.
sin (real_add x y) =
real_add (real_mul (sin x) (cos y)) (real_mul (cos x) (sin y))"
by (import hollight SIN_ADD)
lemma COS_ADD: "ALL (x::hollight.real) y::hollight.real.
cos (real_add x y) =
real_sub (real_mul (cos x) (cos y)) (real_mul (sin x) (sin y))"
by (import hollight COS_ADD)
lemma SIN_NEG: "ALL x::hollight.real. sin (real_neg x) = real_neg (sin x)"
by (import hollight SIN_NEG)
lemma COS_NEG: "ALL x::hollight.real. cos (real_neg x) = cos x"
by (import hollight COS_NEG)
lemma SIN_DOUBLE: "ALL x::hollight.real.
sin (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x) =
real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_mul (sin x) (cos x))"
by (import hollight SIN_DOUBLE)
lemma COS_DOUBLE: "ALL x::hollight.real.
cos (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x) =
real_sub (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_pow (sin x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight COS_DOUBLE)
lemma COS_ABS: "ALL x::hollight.real. cos (real_abs x) = cos x"
by (import hollight COS_ABS)
lemma SIN_PAIRED: "ALL x::hollight.real.
sums
(%n::nat.
real_mul
(real_div (real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) n)
(real_of_num
(FACT (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n + NUMERAL_BIT1 0))))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n + NUMERAL_BIT1 0)))
(sin x)"
by (import hollight SIN_PAIRED)
lemma SIN_POS: "ALL x::hollight.real.
real_lt (real_of_num 0) x &
real_lt x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) -->
real_lt (real_of_num 0) (sin x)"
by (import hollight SIN_POS)
lemma COS_PAIRED: "ALL x::hollight.real.
sums
(%n::nat.
real_mul
(real_div (real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) n)
(real_of_num (FACT (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n))))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n)))
(cos x)"
by (import hollight COS_PAIRED)
lemma COS_2: "real_lt (cos (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) (real_of_num 0)"
by (import hollight COS_2)
lemma COS_ISZERO: "EX! x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) &
cos x = real_of_num 0"
by (import hollight COS_ISZERO)
constdefs
pi :: "hollight.real"
"pi ==
real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(SOME x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) &
cos x = real_of_num 0)"
lemma DEF_pi: "pi =
real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(SOME x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) &
cos x = real_of_num 0)"
by (import hollight DEF_pi)
lemma PI2: "real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
(SOME x::hollight.real.
real_le (real_of_num 0) x &
real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) &
cos x = real_of_num 0)"
by (import hollight PI2)
lemma COS_PI2: "cos (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) =
real_of_num 0"
by (import hollight COS_PI2)
lemma PI2_BOUNDS: "real_lt (real_of_num 0)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
real_lt (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))"
by (import hollight PI2_BOUNDS)
lemma PI_POS: "real_lt (real_of_num 0) pi"
by (import hollight PI_POS)
lemma SIN_PI2: "sin (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight SIN_PI2)
lemma COS_PI: "cos pi = real_neg (real_of_num (NUMERAL_BIT1 0))"
by (import hollight COS_PI)
lemma SIN_PI: "sin pi = real_of_num 0"
by (import hollight SIN_PI)
lemma SIN_COS: "ALL x::hollight.real.
sin x =
cos (real_sub (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x)"
by (import hollight SIN_COS)
lemma COS_SIN: "ALL x::hollight.real.
cos x =
sin (real_sub (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x)"
by (import hollight COS_SIN)
lemma SIN_PERIODIC_PI: "ALL x::hollight.real. sin (real_add x pi) = real_neg (sin x)"
by (import hollight SIN_PERIODIC_PI)
lemma COS_PERIODIC_PI: "ALL x::hollight.real. cos (real_add x pi) = real_neg (cos x)"
by (import hollight COS_PERIODIC_PI)
lemma SIN_PERIODIC: "ALL x::hollight.real.
sin (real_add x
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi)) =
sin x"
by (import hollight SIN_PERIODIC)
lemma COS_PERIODIC: "ALL x::hollight.real.
cos (real_add x
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi)) =
cos x"
by (import hollight COS_PERIODIC)
lemma COS_NPI: "ALL n::nat.
cos (real_mul (real_of_num n) pi) =
real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) n"
by (import hollight COS_NPI)
lemma SIN_NPI: "ALL n::nat. sin (real_mul (real_of_num n) pi) = real_of_num 0"
by (import hollight SIN_NPI)
lemma SIN_POS_PI2: "ALL x::hollight.real.
real_lt (real_of_num 0) x &
real_lt x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt (real_of_num 0) (sin x)"
by (import hollight SIN_POS_PI2)
lemma COS_POS_PI2: "ALL x::hollight.real.
real_lt (real_of_num 0) x &
real_lt x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt (real_of_num 0) (cos x)"
by (import hollight COS_POS_PI2)
lemma COS_POS_PI: "ALL x::hollight.real.
real_lt
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_lt x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt (real_of_num 0) (cos x)"
by (import hollight COS_POS_PI)
lemma SIN_POS_PI: "ALL x::hollight.real.
real_lt (real_of_num 0) x & real_lt x pi -->
real_lt (real_of_num 0) (sin x)"
by (import hollight SIN_POS_PI)
lemma SIN_POS_PI_LE: "ALL x::hollight.real.
real_le (real_of_num 0) x & real_le x pi -->
real_le (real_of_num 0) (sin x)"
by (import hollight SIN_POS_PI_LE)
lemma COS_TOTAL: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
(EX! x::hollight.real.
real_le (real_of_num 0) x & real_le x pi & cos x = y)"
by (import hollight COS_TOTAL)
lemma SIN_TOTAL: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
(EX! x::hollight.real.
real_le
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_le x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin x = y)"
by (import hollight SIN_TOTAL)
lemma COS_ZERO_LEMMA: "ALL x::hollight.real.
real_le (real_of_num 0) x & cos x = real_of_num 0 -->
(EX n::nat.
~ EVEN n &
x =
real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight COS_ZERO_LEMMA)
lemma SIN_ZERO_LEMMA: "ALL x::hollight.real.
real_le (real_of_num 0) x & sin x = real_of_num 0 -->
(EX n::nat.
EVEN n &
x =
real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight SIN_ZERO_LEMMA)
lemma COS_ZERO: "ALL x::hollight.real.
(cos x = real_of_num 0) =
((EX n::nat.
~ EVEN n &
x =
real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))) |
(EX n::nat.
~ EVEN n &
x =
real_neg
(real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))))"
by (import hollight COS_ZERO)
lemma SIN_ZERO: "ALL x::hollight.real.
(sin x = real_of_num 0) =
((EX n::nat.
EVEN n &
x =
real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))) |
(EX n::nat.
EVEN n &
x =
real_neg
(real_mul (real_of_num n)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))))"
by (import hollight SIN_ZERO)
lemma SIN_ZERO_PI: "ALL x::hollight.real.
(sin x = real_of_num 0) =
((EX n::nat. x = real_mul (real_of_num n) pi) |
(EX n::nat. x = real_neg (real_mul (real_of_num n) pi)))"
by (import hollight SIN_ZERO_PI)
lemma COS_ONE_2PI: "ALL x::hollight.real.
(cos x = real_of_num (NUMERAL_BIT1 0)) =
((EX n::nat.
x =
real_mul (real_of_num n)
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi)) |
(EX n::nat.
x =
real_neg
(real_mul (real_of_num n)
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi))))"
by (import hollight COS_ONE_2PI)
constdefs
tan :: "hollight.real => hollight.real"
"tan == %u::hollight.real. real_div (sin u) (cos u)"
lemma DEF_tan: "tan = (%u::hollight.real. real_div (sin u) (cos u))"
by (import hollight DEF_tan)
lemma TAN_0: "tan (real_of_num 0) = real_of_num 0"
by (import hollight TAN_0)
lemma TAN_PI: "tan pi = real_of_num 0"
by (import hollight TAN_PI)
lemma TAN_NPI: "ALL n::nat. tan (real_mul (real_of_num n) pi) = real_of_num 0"
by (import hollight TAN_NPI)
lemma TAN_NEG: "ALL x::hollight.real. tan (real_neg x) = real_neg (tan x)"
by (import hollight TAN_NEG)
lemma TAN_PERIODIC: "ALL x::hollight.real.
tan (real_add x
(real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi)) =
tan x"
by (import hollight TAN_PERIODIC)
lemma TAN_PERIODIC_PI: "ALL x::hollight.real. tan (real_add x pi) = tan x"
by (import hollight TAN_PERIODIC_PI)
lemma TAN_PERIODIC_NPI: "ALL (x::hollight.real) n::nat.
tan (real_add x (real_mul (real_of_num n) pi)) = tan x"
by (import hollight TAN_PERIODIC_NPI)
lemma TAN_ADD: "ALL (x::hollight.real) y::hollight.real.
cos x ~= real_of_num 0 &
cos y ~= real_of_num 0 & cos (real_add x y) ~= real_of_num 0 -->
tan (real_add x y) =
real_div (real_add (tan x) (tan y))
(real_sub (real_of_num (NUMERAL_BIT1 0)) (real_mul (tan x) (tan y)))"
by (import hollight TAN_ADD)
lemma TAN_DOUBLE: "ALL x::hollight.real.
cos x ~= real_of_num 0 &
cos (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x) ~=
real_of_num 0 -->
tan (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) x) =
real_div (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) (tan x))
(real_sub (real_of_num (NUMERAL_BIT1 0))
(real_pow (tan x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight TAN_DOUBLE)
lemma TAN_POS_PI2: "ALL x::hollight.real.
real_lt (real_of_num 0) x &
real_lt x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt (real_of_num 0) (tan x)"
by (import hollight TAN_POS_PI2)
lemma DIFF_TAN: "ALL x::hollight.real.
cos x ~= real_of_num 0 -->
diffl tan (real_inv (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) x"
by (import hollight DIFF_TAN)
lemma DIFF_TAN_COMPOSITE: "diffl (g::hollight.real => hollight.real) (m::hollight.real)
(x::hollight.real) &
cos (g x) ~= real_of_num 0 -->
diffl (%x::hollight.real. tan (g x))
(real_mul (real_inv (real_pow (cos (g x)) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
m)
x"
by (import hollight DIFF_TAN_COMPOSITE)
lemma TAN_TOTAL_LEMMA: "ALL y::hollight.real.
real_lt (real_of_num 0) y -->
(EX x::hollight.real.
real_lt (real_of_num 0) x &
real_lt x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
real_lt y (tan x))"
by (import hollight TAN_TOTAL_LEMMA)
lemma TAN_TOTAL_POS: "ALL y::hollight.real.
real_le (real_of_num 0) y -->
(EX x::hollight.real.
real_le (real_of_num 0) x &
real_lt x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = y)"
by (import hollight TAN_TOTAL_POS)
lemma TAN_TOTAL: "ALL y::hollight.real.
EX! x::hollight.real.
real_lt
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_lt x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = y"
by (import hollight TAN_TOTAL)
lemma PI2_PI4: "real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight PI2_PI4)
lemma TAN_PI4: "tan (real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0))))) =
real_of_num (NUMERAL_BIT1 0)"
by (import hollight TAN_PI4)
lemma TAN_COT: "ALL x::hollight.real.
tan (real_sub (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x) =
real_inv (tan x)"
by (import hollight TAN_COT)
lemma TAN_BOUND_PI2: "ALL x::hollight.real.
real_lt (real_abs x)
(real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0))))) -->
real_lt (real_abs (tan x)) (real_of_num (NUMERAL_BIT1 0))"
by (import hollight TAN_BOUND_PI2)
lemma TAN_ABS_GE_X: "ALL x::hollight.real.
real_lt (real_abs x)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_le (real_abs x) (real_abs (tan x))"
by (import hollight TAN_ABS_GE_X)
constdefs
asn :: "hollight.real => hollight.real"
"asn ==
%u::hollight.real.
SOME x::hollight.real.
real_le
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_le x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin x = u"
lemma DEF_asn: "asn =
(%u::hollight.real.
SOME x::hollight.real.
real_le
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_le x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin x = u)"
by (import hollight DEF_asn)
constdefs
acs :: "hollight.real => hollight.real"
"acs ==
%u::hollight.real.
SOME x::hollight.real.
real_le (real_of_num 0) x & real_le x pi & cos x = u"
lemma DEF_acs: "acs =
(%u::hollight.real.
SOME x::hollight.real.
real_le (real_of_num 0) x & real_le x pi & cos x = u)"
by (import hollight DEF_acs)
constdefs
atn :: "hollight.real => hollight.real"
"atn ==
%u::hollight.real.
SOME x::hollight.real.
real_lt
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_lt x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = u"
lemma DEF_atn: "atn =
(%u::hollight.real.
SOME x::hollight.real.
real_lt
(real_neg
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_lt x
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = u)"
by (import hollight DEF_atn)
lemma ASN: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
real_le
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
(asn y) &
real_le (asn y)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin (asn y) = y"
by (import hollight ASN)
lemma ASN_SIN: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
sin (asn y) = y"
by (import hollight ASN_SIN)
lemma ASN_BOUNDS: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
real_le
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
(asn y) &
real_le (asn y)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight ASN_BOUNDS)
lemma ASN_BOUNDS_LT: "ALL y::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_lt y (real_of_num (NUMERAL_BIT1 0)) -->
real_lt
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
(asn y) &
real_lt (asn y)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight ASN_BOUNDS_LT)
lemma SIN_ASN: "ALL x::hollight.real.
real_le
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_le x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
asn (sin x) = x"
by (import hollight SIN_ASN)
lemma ACS: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
real_le (real_of_num 0) (acs y) & real_le (acs y) pi & cos (acs y) = y"
by (import hollight ACS)
lemma ACS_COS: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
cos (acs y) = y"
by (import hollight ACS_COS)
lemma ACS_BOUNDS: "ALL y::hollight.real.
real_le (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_le y (real_of_num (NUMERAL_BIT1 0)) -->
real_le (real_of_num 0) (acs y) & real_le (acs y) pi"
by (import hollight ACS_BOUNDS)
lemma ACS_BOUNDS_LT: "ALL y::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) y &
real_lt y (real_of_num (NUMERAL_BIT1 0)) -->
real_lt (real_of_num 0) (acs y) & real_lt (acs y) pi"
by (import hollight ACS_BOUNDS_LT)
lemma COS_ACS: "ALL x::hollight.real.
real_le (real_of_num 0) x & real_le x pi --> acs (cos x) = x"
by (import hollight COS_ACS)
lemma ATN: "ALL y::hollight.real.
real_lt
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
(atn y) &
real_lt (atn y)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan (atn y) = y"
by (import hollight ATN)
lemma ATN_TAN: "ALL x::hollight.real. tan (atn x) = x"
by (import hollight ATN_TAN)
lemma ATN_BOUNDS: "ALL x::hollight.real.
real_lt
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
(atn x) &
real_lt (atn x)
(real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight ATN_BOUNDS)
lemma TAN_ATN: "ALL x::hollight.real.
real_lt
(real_neg (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x &
real_lt x (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
atn (tan x) = x"
by (import hollight TAN_ATN)
lemma ATN_0: "atn (real_of_num 0) = real_of_num 0"
by (import hollight ATN_0)
lemma ATN_1: "atn (real_of_num (NUMERAL_BIT1 0)) =
real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight ATN_1)
lemma ATN_NEG: "ALL x::hollight.real. atn (real_neg x) = real_neg (atn x)"
by (import hollight ATN_NEG)
lemma COS_ATN_NZ: "ALL x::hollight.real. cos (atn x) ~= real_of_num 0"
by (import hollight COS_ATN_NZ)
lemma TAN_SEC: "ALL x::hollight.real.
cos x ~= real_of_num 0 -->
real_add (real_of_num (NUMERAL_BIT1 0))
(real_pow (tan x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
real_pow (real_inv (cos x)) (NUMERAL_BIT0 (NUMERAL_BIT1 0))"
by (import hollight TAN_SEC)
lemma DIFF_ATN: "ALL x::hollight.real.
diffl atn
(real_inv
(real_add (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x"
by (import hollight DIFF_ATN)
lemma DIFF_ATN_COMPOSITE: "diffl (g::hollight.real => hollight.real) (m::hollight.real)
(x::hollight.real) -->
diffl (%x::hollight.real. atn (g x))
(real_mul
(real_inv
(real_add (real_of_num (NUMERAL_BIT1 0))
(real_pow (g x) (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
m)
x"
by (import hollight DIFF_ATN_COMPOSITE)
lemma ATN_MONO_LT: "ALL (x::hollight.real) y::hollight.real.
real_lt x y --> real_lt (atn x) (atn y)"
by (import hollight ATN_MONO_LT)
lemma ATN_MONO_LT_EQ: "ALL (x::hollight.real) y::hollight.real.
real_lt (atn x) (atn y) = real_lt x y"
by (import hollight ATN_MONO_LT_EQ)
lemma ATN_MONO_LE_EQ: "ALL (x::hollight.real) xa::hollight.real.
real_le (atn x) (atn xa) = real_le x xa"
by (import hollight ATN_MONO_LE_EQ)
lemma ATN_INJ: "ALL (x::hollight.real) xa::hollight.real. (atn x = atn xa) = (x = xa)"
by (import hollight ATN_INJ)
lemma ATN_POS_LT: "real_lt (real_of_num 0) (atn (x::hollight.real)) = real_lt (real_of_num 0) x"
by (import hollight ATN_POS_LT)
lemma ATN_POS_LE: "real_le (real_of_num 0) (atn (x::hollight.real)) = real_le (real_of_num 0) x"
by (import hollight ATN_POS_LE)
lemma ATN_LT_PI4_POS: "ALL x::hollight.real.
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
real_lt (atn x)
(real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight ATN_LT_PI4_POS)
lemma ATN_LT_PI4_NEG: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x -->
real_lt
(real_neg
(real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0))))))
(atn x)"
by (import hollight ATN_LT_PI4_NEG)
lemma ATN_LT_PI4: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
real_lt (real_abs (atn x))
(real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight ATN_LT_PI4)
lemma ATN_LE_PI4: "ALL x::hollight.real.
real_le (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
real_le (real_abs (atn x))
(real_div pi
(real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight ATN_LE_PI4)
lemma COS_SIN_SQRT: "ALL x::hollight.real.
real_le (real_of_num 0) (cos x) -->
cos x =
sqrt
(real_sub (real_of_num (NUMERAL_BIT1 0))
(real_pow (sin x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight COS_SIN_SQRT)
lemma COS_ASN_NZ: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
cos (asn x) ~= real_of_num 0"
by (import hollight COS_ASN_NZ)
lemma DIFF_ASN_COS: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
diffl asn (real_inv (cos (asn x))) x"
by (import hollight DIFF_ASN_COS)
lemma DIFF_ASN: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
diffl asn
(real_inv
(sqrt
(real_sub (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0))))))
x"
by (import hollight DIFF_ASN)
lemma SIN_COS_SQRT: "ALL x::hollight.real.
real_le (real_of_num 0) (sin x) -->
sin x =
sqrt
(real_sub (real_of_num (NUMERAL_BIT1 0))
(real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))"
by (import hollight SIN_COS_SQRT)
lemma SIN_ACS_NZ: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
sin (acs x) ~= real_of_num 0"
by (import hollight SIN_ACS_NZ)
lemma DIFF_ACS_SIN: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
diffl acs (real_inv (real_neg (sin (acs x)))) x"
by (import hollight DIFF_ACS_SIN)
lemma DIFF_ACS: "ALL x::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) -->
diffl acs
(real_neg
(real_inv
(sqrt
(real_sub (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))))
x"
by (import hollight DIFF_ACS)
lemma CIRCLE_SINCOS: "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 (NUMERAL_BIT1 0) -->
(EX t::hollight.real. x = cos t & y = sin t)"
by (import hollight CIRCLE_SINCOS)
lemma ACS_MONO_LT: "ALL (x::hollight.real) y::hollight.real.
real_lt (real_neg (real_of_num (NUMERAL_BIT1 0))) x &
real_lt x y & real_lt y (real_of_num (NUMERAL_BIT1 0)) -->
real_lt (acs y) (acs x)"
by (import hollight ACS_MONO_LT)
lemma LESS_SUC_EQ: "ALL (m::nat) n::nat. < m (Suc n) = <= m n"
by (import hollight LESS_SUC_EQ)
lemma LESS_1: "ALL x::nat. < x (NUMERAL_BIT1 0) = (x = 0)"
by (import hollight LESS_1)
constdefs
division :: "hollight.real * hollight.real => (nat => hollight.real) => bool"
"division ==
%(u::hollight.real * hollight.real) ua::nat => hollight.real.
ua 0 = fst u &
(EX N::nat.
(ALL n::nat. < n N --> real_lt (ua n) (ua (Suc n))) &
(ALL n::nat. >= n N --> ua n = snd u))"
lemma DEF_division: "division =
(%(u::hollight.real * hollight.real) ua::nat => hollight.real.
ua 0 = fst u &
(EX N::nat.
(ALL n::nat. < n N --> real_lt (ua n) (ua (Suc n))) &
(ALL n::nat. >= n N --> ua n = snd u)))"
by (import hollight DEF_division)
constdefs
dsize :: "(nat => hollight.real) => nat"
"dsize ==
%u::nat => hollight.real.
SOME N::nat.
(ALL n::nat. < n N --> real_lt (u n) (u (Suc n))) &
(ALL n::nat. >= n N --> u n = u N)"
lemma DEF_dsize: "dsize =
(%u::nat => hollight.real.
SOME N::nat.
(ALL n::nat. < n N --> real_lt (u n) (u (Suc n))) &
(ALL n::nat. >= n N --> u n = u N))"
by (import hollight DEF_dsize)
constdefs
tdiv :: "hollight.real * hollight.real
=> (nat => hollight.real) * (nat => hollight.real) => bool"
"tdiv ==
%(u::hollight.real * hollight.real)
ua::(nat => hollight.real) * (nat => hollight.real).
division (fst u, snd u) (fst ua) &
(ALL n::nat.
real_le (fst ua n) (snd ua n) & real_le (snd ua n) (fst ua (Suc n)))"
lemma DEF_tdiv: "tdiv =
(%(u::hollight.real * hollight.real)
ua::(nat => hollight.real) * (nat => hollight.real).
division (fst u, snd u) (fst ua) &
(ALL n::nat.
real_le (fst ua n) (snd ua n) &
real_le (snd ua n) (fst ua (Suc n))))"
by (import hollight DEF_tdiv)
constdefs
gauge :: "(hollight.real => bool) => (hollight.real => hollight.real) => bool"
"gauge ==
%(u::hollight.real => bool) ua::hollight.real => hollight.real.
ALL x::hollight.real. u x --> real_lt (real_of_num 0) (ua x)"
lemma DEF_gauge: "gauge =
(%(u::hollight.real => bool) ua::hollight.real => hollight.real.
ALL x::hollight.real. u x --> real_lt (real_of_num 0) (ua x))"
by (import hollight DEF_gauge)
constdefs
fine :: "(hollight.real => hollight.real)
=> (nat => hollight.real) * (nat => hollight.real) => bool"
"fine ==
%(u::hollight.real => hollight.real)
ua::(nat => hollight.real) * (nat => hollight.real).
ALL n::nat.
< n (dsize (fst ua)) -->
real_lt (real_sub (fst ua (Suc n)) (fst ua n)) (u (snd ua n))"
lemma DEF_fine: "fine =
(%(u::hollight.real => hollight.real)
ua::(nat => hollight.real) * (nat => hollight.real).
ALL n::nat.
< n (dsize (fst ua)) -->
real_lt (real_sub (fst ua (Suc n)) (fst ua n)) (u (snd ua n)))"
by (import hollight DEF_fine)
constdefs
rsum :: "(nat => hollight.real) * (nat => hollight.real)
=> (hollight.real => hollight.real) => hollight.real"
"rsum ==
%(u::(nat => hollight.real) * (nat => hollight.real))
ua::hollight.real => hollight.real.
psum (0, dsize (fst u))
(%n::nat. real_mul (ua (snd u n)) (real_sub (fst u (Suc n)) (fst u n)))"
lemma DEF_rsum: "rsum =
(%(u::(nat => hollight.real) * (nat => hollight.real))
ua::hollight.real => hollight.real.
psum (0, dsize (fst u))
(%n::nat.
real_mul (ua (snd u n)) (real_sub (fst u (Suc n)) (fst u n))))"
by (import hollight DEF_rsum)
constdefs
defint :: "hollight.real * hollight.real
=> (hollight.real => hollight.real) => hollight.real => bool"
"defint ==
%(u::hollight.real * hollight.real) (ua::hollight.real => hollight.real)
ub::hollight.real.
ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX g::hollight.real => hollight.real.
gauge (%x::hollight.real. real_le (fst u) x & real_le x (snd u))
g &
(ALL (D::nat => hollight.real) p::nat => hollight.real.
tdiv (fst u, snd u) (D, p) & fine g (D, p) -->
real_lt (real_abs (real_sub (rsum (D, p) ua) ub)) e))"
lemma DEF_defint: "defint =
(%(u::hollight.real * hollight.real) (ua::hollight.real => hollight.real)
ub::hollight.real.
ALL e::hollight.real.
real_lt (real_of_num 0) e -->
(EX g::hollight.real => hollight.real.
gauge (%x::hollight.real. real_le (fst u) x & real_le x (snd u))
g &
(ALL (D::nat => hollight.real) p::nat => hollight.real.
tdiv (fst u, snd u) (D, p) & fine g (D, p) -->
real_lt (real_abs (real_sub (rsum (D, p) ua) ub)) e)))"
by (import hollight DEF_defint)
lemma DIVISION_0: "ALL (a::hollight.real) b::hollight.real.
a = b --> dsize (%n::nat. COND (n = 0) a b) = 0"
by (import hollight DIVISION_0)
lemma DIVISION_1: "ALL (a::hollight.real) b::hollight.real.
real_lt a b --> dsize (%n::nat. COND (n = 0) a b) = NUMERAL_BIT1 0"
by (import hollight DIVISION_1)
lemma DIVISION_SINGLE: "ALL (a::hollight.real) b::hollight.real.
real_le a b --> division (a, b) (%n::nat. COND (n = 0) a b)"
by (import hollight DIVISION_SINGLE)
lemma DIVISION_LHS: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D --> D 0 = a"
by (import hollight DIVISION_LHS)
lemma DIVISION_THM: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D =
(D 0 = a &
(ALL n::nat. < n (dsize D) --> real_lt (D n) (D (Suc n))) &
(ALL n::nat. >= n (dsize D) --> D n = b))"
by (import hollight DIVISION_THM)
lemma DIVISION_RHS: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D --> D (dsize D) = b"
by (import hollight DIVISION_RHS)
lemma DIVISION_LT_GEN: "ALL (D::nat => hollight.real) (a::hollight.real) (b::hollight.real) (m::nat)
n::nat.
division (a, b) D & < m n & <= n (dsize D) --> real_lt (D m) (D n)"
by (import hollight DIVISION_LT_GEN)
lemma DIVISION_LT: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D -->
(ALL n::nat. < n (dsize D) --> real_lt (D 0) (D (Suc n)))"
by (import hollight DIVISION_LT)
lemma DIVISION_LE: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D --> real_le a b"
by (import hollight DIVISION_LE)
lemma DIVISION_GT: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D -->
(ALL n::nat. < n (dsize D) --> real_lt (D n) (D (dsize D)))"
by (import hollight DIVISION_GT)
lemma DIVISION_EQ: "ALL (D::nat => hollight.real) (a::hollight.real) b::hollight.real.
division (a, b) D --> (a = b) = (dsize D = 0)"
by (import hollight DIVISION_EQ)
lemma DIVISION_LBOUND: "ALL (x::nat => hollight.real) (xa::hollight.real) (xb::hollight.real)
xc::nat. division (xa, xb) x --> real_le xa (x xc)"
by (import hollight DIVISION_LBOUND)
lemma DIVISION_LBOUND_LT: "ALL (x::nat => hollight.real) (xa::hollight.real) (xb::hollight.real)
xc::nat. division (xa, xb) x & dsize x ~= 0 --> real_lt xa (x (Suc xc))"
by (import hollight DIVISION_LBOUND_LT)
lemma DIVISION_UBOUND: "ALL (x::nat => hollight.real) (xa::hollight.real) (xb::hollight.real)
xc::nat. division (xa, xb) x --> real_le (x xc) xb"
by (import hollight DIVISION_UBOUND)
lemma DIVISION_UBOUND_LT: "ALL (D::nat => hollight.real) (a::hollight.real) (b::hollight.real) n::nat.
division (a, b) D & < n (dsize D) --> real_lt (D n) b"
by (import hollight DIVISION_UBOUND_LT)
lemma DIVISION_APPEND_LEMMA1: "ALL (a::hollight.real) (b::hollight.real) (c::hollight.real)
(D1::nat => hollight.real) D2::nat => hollight.real.
division (a, b) D1 & division (b, c) D2 -->
(ALL n::nat.
< n (dsize D1 + dsize D2) -->
real_lt (COND (< n (dsize D1)) (D1 n) (D2 (n - dsize D1)))
(COND (< (Suc n) (dsize D1)) (D1 (Suc n))
(D2 (Suc n - dsize D1)))) &
(ALL n::nat.
>= n (dsize D1 + dsize D2) -->
COND (< n (dsize D1)) (D1 n) (D2 (n - dsize D1)) =
COND (< (dsize D1 + dsize D2) (dsize D1)) (D1 (dsize D1 + dsize D2))
(D2 (dsize D1 + dsize D2 - dsize D1)))"
by (import hollight DIVISION_APPEND_LEMMA1)
lemma DIVISION_APPEND_LEMMA2: "ALL (a::hollight.real) (b::hollight.real) (c::hollight.real)
(D1::nat => hollight.real) D2::nat => hollight.real.
division (a, b) D1 & division (b, c) D2 -->
dsize (%n::nat. COND (< n (dsize D1)) (D1 n) (D2 (n - dsize D1))) =
dsize D1 + dsize D2"
by (import hollight DIVISION_APPEND_LEMMA2)
lemma DIVISION_APPEND: "ALL (a::hollight.real) (b::hollight.real) c::hollight.real.
(EX (D1::nat => hollight.real) p1::nat => hollight.real.
tdiv (a, b) (D1, p1) &
fine (g::hollight.real => hollight.real) (D1, p1)) &
(EX (D2::nat => hollight.real) p2::nat => hollight.real.
tdiv (b, c) (D2, p2) & fine g (D2, p2)) -->
(EX (x::nat => hollight.real) p::nat => hollight.real.
tdiv (a, c) (x, p) & fine g (x, p))"
by (import hollight DIVISION_APPEND)
lemma DIVISION_EXISTS: "ALL (a::hollight.real) (b::hollight.real) g::hollight.real => hollight.real.
real_le a b & gauge (%x::hollight.real. real_le a x & real_le x b) g -->
(EX (D::nat => hollight.real) p::nat => hollight.real.
tdiv (a, b) (D, p) & fine g (D, p))"
by (import hollight DIVISION_EXISTS)
lemma GAUGE_MIN: "ALL (E::hollight.real => bool) (g1::hollight.real => hollight.real)
g2::hollight.real => hollight.real.
gauge E g1 & gauge E g2 -->
gauge E (%x::hollight.real. COND (real_lt (g1 x) (g2 x)) (g1 x) (g2 x))"
by (import hollight GAUGE_MIN)
lemma FINE_MIN: "ALL (g1::hollight.real => hollight.real)
(g2::hollight.real => hollight.real) (D::nat => hollight.real)
p::nat => hollight.real.
fine (%x::hollight.real. COND (real_lt (g1 x) (g2 x)) (g1 x) (g2 x))
(D, p) -->
fine g1 (D, p) & fine g2 (D, p)"
by (import hollight FINE_MIN)
lemma DINT_UNIQ: "ALL (a::hollight.real) (b::hollight.real)
(f::hollight.real => hollight.real) (k1::hollight.real)
k2::hollight.real.
real_le a b & defint (a, b) f k1 & defint (a, b) f k2 --> k1 = k2"
by (import hollight DINT_UNIQ)
lemma INTEGRAL_NULL: "ALL (f::hollight.real => hollight.real) a::hollight.real.
defint (a, a) f (real_of_num 0)"
by (import hollight INTEGRAL_NULL)
lemma STRADDLE_LEMMA: "ALL (f::hollight.real => hollight.real) (f'::hollight.real => hollight.real)
(a::hollight.real) (b::hollight.real) e::hollight.real.
(ALL x::hollight.real. real_le a x & real_le x b --> diffl f (f' x) x) &
real_lt (real_of_num 0) e -->
(EX x::hollight.real => hollight.real.
gauge (%x::hollight.real. real_le a x & real_le x b) x &
(ALL (xa::hollight.real) (u::hollight.real) v::hollight.real.
real_le a u &
real_le u xa &
real_le xa v & real_le v b & real_lt (real_sub v u) (x xa) -->
real_le
(real_abs
(real_sub (real_sub (f v) (f u))
(real_mul (f' xa) (real_sub v u))))
(real_mul e (real_sub v u))))"
by (import hollight STRADDLE_LEMMA)
lemma FTC1: "ALL (f::hollight.real => hollight.real) (f'::hollight.real => hollight.real)
(a::hollight.real) b::hollight.real.
real_le a b &
(ALL x::hollight.real.
real_le a x & real_le x b --> diffl f (f' x) x) -->
defint (a, b) f' (real_sub (f b) (f a))"
by (import hollight FTC1)
lemma MCLAURIN: "ALL (f::hollight.real => hollight.real)
(diff::nat => hollight.real => hollight.real) (h::hollight.real) n::nat.
real_lt (real_of_num 0) h &
< 0 n &
diff 0 = f &
(ALL (m::nat) t::hollight.real.
< m n & real_le (real_of_num 0) t & real_le t h -->
diffl (diff m) (diff (Suc m) t) t) -->
(EX t::hollight.real.
real_lt (real_of_num 0) t &
real_lt t h &
f h =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow h m)))
(real_mul (real_div (diff n t) (real_of_num (FACT n)))
(real_pow h n)))"
by (import hollight MCLAURIN)
lemma MCLAURIN_NEG: "ALL (f::hollight.real => hollight.real)
(diff::nat => hollight.real => hollight.real) (h::hollight.real) n::nat.
real_lt h (real_of_num 0) &
< 0 n &
diff 0 = f &
(ALL (m::nat) t::hollight.real.
< m n & real_le h t & real_le t (real_of_num 0) -->
diffl (diff m) (diff (Suc m) t) t) -->
(EX t::hollight.real.
real_lt h t &
real_lt t (real_of_num 0) &
f h =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow h m)))
(real_mul (real_div (diff n t) (real_of_num (FACT n)))
(real_pow h n)))"
by (import hollight MCLAURIN_NEG)
lemma MCLAURIN_BI_LE: "ALL (f::hollight.real => hollight.real)
(diff::nat => hollight.real => hollight.real) (x::hollight.real) n::nat.
diff 0 = f &
(ALL (m::nat) t::hollight.real.
< m n & real_le (real_abs t) (real_abs x) -->
diffl (diff m) (diff (Suc m) t) t) -->
(EX xa::hollight.real.
real_le (real_abs xa) (real_abs x) &
f x =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow x m)))
(real_mul (real_div (diff n xa) (real_of_num (FACT n)))
(real_pow x n)))"
by (import hollight MCLAURIN_BI_LE)
lemma MCLAURIN_ALL_LT: "ALL (f::hollight.real => hollight.real)
diff::nat => hollight.real => hollight.real.
diff 0 = f &
(ALL (m::nat) x::hollight.real. diffl (diff m) (diff (Suc m) x) x) -->
(ALL (x::hollight.real) n::nat.
x ~= real_of_num 0 & < 0 n -->
(EX t::hollight.real.
real_lt (real_of_num 0) (real_abs t) &
real_lt (real_abs t) (real_abs x) &
f x =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_div (diff m (real_of_num 0))
(real_of_num (FACT m)))
(real_pow x m)))
(real_mul (real_div (diff n t) (real_of_num (FACT n)))
(real_pow x n))))"
by (import hollight MCLAURIN_ALL_LT)
lemma MCLAURIN_ZERO: "ALL (diff::nat => hollight.real => hollight.real) (n::nat) x::hollight.real.
x = real_of_num 0 & < 0 n -->
psum (0, n)
(%m::nat.
real_mul (real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow x m)) =
diff 0 (real_of_num 0)"
by (import hollight MCLAURIN_ZERO)
lemma MCLAURIN_ALL_LE: "ALL (f::hollight.real => hollight.real)
diff::nat => hollight.real => hollight.real.
diff 0 = f &
(ALL (m::nat) x::hollight.real. diffl (diff m) (diff (Suc m) x) x) -->
(ALL (x::hollight.real) n::nat.
EX t::hollight.real.
real_le (real_abs t) (real_abs x) &
f x =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow x m)))
(real_mul (real_div (diff n t) (real_of_num (FACT n)))
(real_pow x n)))"
by (import hollight MCLAURIN_ALL_LE)
lemma MCLAURIN_EXP_LEMMA: "exp = exp & (ALL (x::nat) xa::hollight.real. diffl exp (exp xa) xa)"
by (import hollight MCLAURIN_EXP_LEMMA)
lemma MCLAURIN_EXP_LT: "ALL (x::hollight.real) n::nat.
x ~= real_of_num 0 & < 0 n -->
(EX t::hollight.real.
real_lt (real_of_num 0) (real_abs t) &
real_lt (real_abs t) (real_abs x) &
exp x =
real_add
(psum (0, n)
(%m::nat. real_div (real_pow x m) (real_of_num (FACT m))))
(real_mul (real_div (exp t) (real_of_num (FACT n))) (real_pow x n)))"
by (import hollight MCLAURIN_EXP_LT)
lemma MCLAURIN_EXP_LE: "ALL (x::hollight.real) n::nat.
EX t::hollight.real.
real_le (real_abs t) (real_abs x) &
exp x =
real_add
(psum (0, n)
(%m::nat. real_div (real_pow x m) (real_of_num (FACT m))))
(real_mul (real_div (exp t) (real_of_num (FACT n))) (real_pow x n))"
by (import hollight MCLAURIN_EXP_LE)
lemma DIFF_LN_COMPOSITE: "ALL (g::hollight.real => hollight.real) (m::hollight.real) x::hollight.real.
diffl g m x & real_lt (real_of_num 0) (g x) -->
diffl (%x::hollight.real. ln (g x)) (real_mul (real_inv (g x)) m) x"
by (import hollight DIFF_LN_COMPOSITE)
lemma MCLAURIN_LN_POS: "ALL (x::hollight.real) n::nat.
real_lt (real_of_num 0) x & < 0 n -->
(EX t::hollight.real.
real_lt (real_of_num 0) t &
real_lt t x &
ln (real_add (real_of_num (NUMERAL_BIT1 0)) x) =
real_add
(psum (0, n)
(%m::nat.
real_mul
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) (Suc m))
(real_div (real_pow x m) (real_of_num m))))
(real_mul
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) (Suc n))
(real_div (real_pow x n)
(real_mul (real_of_num n)
(real_pow (real_add (real_of_num (NUMERAL_BIT1 0)) t) n)))))"
by (import hollight MCLAURIN_LN_POS)
lemma MCLAURIN_LN_NEG: "ALL (x::hollight.real) n::nat.
real_lt (real_of_num 0) x &
real_lt x (real_of_num (NUMERAL_BIT1 0)) & < 0 n -->
(EX t::hollight.real.
real_lt (real_of_num 0) t &
real_lt t x &
real_neg (ln (real_sub (real_of_num (NUMERAL_BIT1 0)) x)) =
real_add
(psum (0, n) (%m::nat. real_div (real_pow x m) (real_of_num m)))
(real_div (real_pow x n)
(real_mul (real_of_num n)
(real_pow (real_sub (real_of_num (NUMERAL_BIT1 0)) t) n))))"
by (import hollight MCLAURIN_LN_NEG)
lemma MCLAURIN_SIN: "ALL (x::hollight.real) n::nat.
real_le
(real_abs
(real_sub (sin x)
(psum (0, n)
(%m::nat.
real_mul
(COND (EVEN m) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (m - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT m))))
(real_pow x m)))))
(real_mul (real_inv (real_of_num (FACT n))) (real_pow (real_abs x) n))"
by (import hollight MCLAURIN_SIN)
lemma MCLAURIN_COS: "ALL (x::hollight.real) n::nat.
real_le
(real_abs
(real_sub (cos x)
(psum (0, n)
(%m::nat.
real_mul
(COND (EVEN m)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV m (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num (FACT m)))
(real_of_num 0))
(real_pow x m)))))
(real_mul (real_inv (real_of_num (FACT n))) (real_pow (real_abs x) n))"
by (import hollight MCLAURIN_COS)
lemma REAL_ATN_POWSER_SUMMABLE: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
summable
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n)))
(real_pow x n))"
by (import hollight REAL_ATN_POWSER_SUMMABLE)
lemma REAL_ATN_POWSER_DIFFS_SUMMABLE: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
summable
(%xa::nat.
real_mul
(diffs
(%n::nat.
COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n)))
xa)
(real_pow x xa))"
by (import hollight REAL_ATN_POWSER_DIFFS_SUMMABLE)
lemma REAL_ATN_POWSER_DIFFS_SUM: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
sums
(%n::nat.
real_mul
(diffs
(%n::nat.
COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n)))
n)
(real_pow x n))
(real_inv
(real_add (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))"
by (import hollight REAL_ATN_POWSER_DIFFS_SUM)
lemma REAL_ATN_POWSER_DIFFS_DIFFS_SUMMABLE: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
summable
(%xa::nat.
real_mul
(diffs
(diffs
(%n::nat.
COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n))))
xa)
(real_pow x xa))"
by (import hollight REAL_ATN_POWSER_DIFFS_DIFFS_SUMMABLE)
lemma REAL_ATN_POWSER_DIFFL: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
diffl
(%x::hollight.real.
suminf
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n)))
(real_pow x n)))
(real_inv
(real_add (real_of_num (NUMERAL_BIT1 0))
(real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x"
by (import hollight REAL_ATN_POWSER_DIFFL)
lemma REAL_ATN_POWSER: "ALL x::hollight.real.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
sums
(%n::nat.
real_mul
(COND (EVEN n) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (n - NUMERAL_BIT1 0) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num n)))
(real_pow x n))
(atn x)"
by (import hollight REAL_ATN_POWSER)
lemma MCLAURIN_ATN: "ALL (x::hollight.real) n::nat.
real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
real_le
(real_abs
(real_sub (atn x)
(psum (0, n)
(%m::nat.
real_mul
(COND (EVEN m) (real_of_num 0)
(real_div
(real_pow (real_neg (real_of_num (NUMERAL_BIT1 0)))
(DIV (m - NUMERAL_BIT1 0)
(NUMERAL_BIT0 (NUMERAL_BIT1 0))))
(real_of_num m)))
(real_pow x m)))))
(real_div (real_pow (real_abs x) n)
(real_sub (real_of_num (NUMERAL_BIT1 0)) (real_abs x)))"
by (import hollight MCLAURIN_ATN)
;end_setup
end