Release HOL4 and HOLLight Importer.
--- a/src/HOL/Import/HOL/HOL4Base.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/HOL4Base.thy Mon Sep 26 16:10:19 2005 +0200
@@ -5,21 +5,21 @@
;setup_theory bool
constdefs
- ARB :: "'a::type"
+ ARB :: "'a"
"ARB == SOME x::'a::type. True"
lemma ARB_DEF: "ARB = (SOME x::'a::type. True)"
by (import bool ARB_DEF)
constdefs
- IN :: "'a::type => ('a::type => bool) => bool"
+ IN :: "'a => ('a => bool) => bool"
"IN == %(x::'a::type) f::'a::type => bool. f x"
lemma IN_DEF: "IN = (%(x::'a::type) f::'a::type => bool. f x)"
by (import bool IN_DEF)
constdefs
- RES_FORALL :: "('a::type => bool) => ('a::type => bool) => bool"
+ RES_FORALL :: "('a => bool) => ('a => bool) => bool"
"RES_FORALL ==
%(p::'a::type => bool) m::'a::type => bool. ALL x::'a::type. IN x p --> m x"
@@ -29,7 +29,7 @@
by (import bool RES_FORALL_DEF)
constdefs
- RES_EXISTS :: "('a::type => bool) => ('a::type => bool) => bool"
+ RES_EXISTS :: "('a => bool) => ('a => bool) => bool"
"RES_EXISTS ==
%(p::'a::type => bool) m::'a::type => bool. EX x::'a::type. IN x p & m x"
@@ -38,7 +38,7 @@
by (import bool RES_EXISTS_DEF)
constdefs
- RES_EXISTS_UNIQUE :: "('a::type => bool) => ('a::type => bool) => bool"
+ RES_EXISTS_UNIQUE :: "('a => bool) => ('a => bool) => bool"
"RES_EXISTS_UNIQUE ==
%(p::'a::type => bool) m::'a::type => bool.
RES_EXISTS p m &
@@ -53,7 +53,7 @@
by (import bool RES_EXISTS_UNIQUE_DEF)
constdefs
- RES_SELECT :: "('a::type => bool) => ('a::type => bool) => 'a::type"
+ RES_SELECT :: "('a => bool) => ('a => bool) => 'a"
"RES_SELECT ==
%(p::'a::type => bool) m::'a::type => bool. SOME x::'a::type. IN x p & m x"
@@ -240,7 +240,7 @@
by (import bool UEXISTS_SIMP)
consts
- RES_ABSTRACT :: "('a::type => bool) => ('a::type => 'b::type) => 'a::type => 'b::type"
+ RES_ABSTRACT :: "('a => bool) => ('a => 'b) => 'a => 'b"
specification (RES_ABSTRACT) RES_ABSTRACT_DEF: "(ALL (p::'a::type => bool) (m::'a::type => 'b::type) x::'a::type.
IN x p --> RES_ABSTRACT p m x = m x) &
@@ -265,15 +265,14 @@
;setup_theory combin
constdefs
- K :: "'a::type => 'b::type => 'a::type"
+ K :: "'a => 'b => 'a"
"K == %(x::'a::type) y::'b::type. x"
lemma K_DEF: "K = (%(x::'a::type) y::'b::type. x)"
by (import combin K_DEF)
constdefs
- S :: "('a::type => 'b::type => 'c::type)
-=> ('a::type => 'b::type) => 'a::type => 'c::type"
+ S :: "('a => 'b => 'c) => ('a => 'b) => 'a => 'c"
"S ==
%(f::'a::type => 'b::type => 'c::type) (g::'a::type => 'b::type)
x::'a::type. f x (g x)"
@@ -284,7 +283,7 @@
by (import combin S_DEF)
constdefs
- I :: "'a::type => 'a::type"
+ I :: "'a => 'a"
"(op ==::('a::type => 'a::type) => ('a::type => 'a::type) => prop)
(I::'a::type => 'a::type)
((S::('a::type => ('a::type => 'a::type) => 'a::type)
@@ -301,7 +300,7 @@
by (import combin I_DEF)
constdefs
- C :: "('a::type => 'b::type => 'c::type) => 'b::type => 'a::type => 'c::type"
+ C :: "('a => 'b => 'c) => 'b => 'a => 'c"
"C == %(f::'a::type => 'b::type => 'c::type) (x::'b::type) y::'a::type. f y x"
lemma C_DEF: "C =
@@ -309,7 +308,7 @@
by (import combin C_DEF)
constdefs
- W :: "('a::type => 'a::type => 'b::type) => 'a::type => 'b::type"
+ W :: "('a => 'a => 'b) => 'a => 'b"
"W == %(f::'a::type => 'a::type => 'b::type) x::'a::type. f x x"
lemma W_DEF: "W = (%(f::'a::type => 'a::type => 'b::type) x::'a::type. f x x)"
@@ -547,7 +546,7 @@
;setup_theory marker
consts
- stmarker :: "'a::type => 'a::type"
+ stmarker :: "'a => 'a"
defs
stmarker_primdef: "stmarker == %x::'a::type. x"
@@ -584,7 +583,7 @@
;setup_theory relation
constdefs
- TC :: "('a::type => 'a::type => bool) => 'a::type => 'a::type => bool"
+ TC :: "('a => 'a => bool) => 'a => 'a => bool"
"TC ==
%(R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
ALL P::'a::type => 'a::type => bool.
@@ -603,7 +602,7 @@
by (import relation TC_DEF)
constdefs
- RTC :: "('a::type => 'a::type => bool) => 'a::type => 'a::type => bool"
+ RTC :: "('a => 'a => bool) => 'a => 'a => bool"
"RTC ==
%(R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
ALL P::'a::type => 'a::type => bool.
@@ -622,7 +621,7 @@
by (import relation RTC_DEF)
consts
- RC :: "('a::type => 'a::type => bool) => 'a::type => 'a::type => bool"
+ RC :: "('a => 'a => bool) => 'a => 'a => bool"
defs
RC_primdef: "RC ==
@@ -633,7 +632,7 @@
by (import relation RC_def)
consts
- transitive :: "('a::type => 'a::type => bool) => bool"
+ transitive :: "('a => 'a => bool) => bool"
defs
transitive_primdef: "transitive ==
@@ -646,7 +645,7 @@
by (import relation transitive_def)
constdefs
- pred_reflexive :: "('a::type => 'a::type => bool) => bool"
+ pred_reflexive :: "('a => 'a => bool) => bool"
"pred_reflexive == %R::'a::type => 'a::type => bool. ALL x::'a::type. R x x"
lemma reflexive_def: "ALL R::'a::type => 'a::type => bool.
@@ -790,7 +789,7 @@
by (import relation RTC_MONOTONE)
constdefs
- WF :: "('a::type => 'a::type => bool) => bool"
+ WF :: "('a => 'a => bool) => bool"
"WF ==
%R::'a::type => 'a::type => bool.
ALL B::'a::type => bool.
@@ -816,7 +815,7 @@
by (import relation WF_NOT_REFL)
constdefs
- EMPTY_REL :: "'a::type => 'a::type => bool"
+ EMPTY_REL :: "'a => 'a => bool"
"EMPTY_REL == %(x::'a::type) y::'a::type. False"
lemma EMPTY_REL_DEF: "ALL (x::'a::type) y::'a::type. EMPTY_REL x y = False"
@@ -833,8 +832,7 @@
by (import relation WF_TC)
consts
- inv_image :: "('b::type => 'b::type => bool)
-=> ('a::type => 'b::type) => 'a::type => 'a::type => bool"
+ inv_image :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool"
defs
inv_image_primdef: "relation.inv_image ==
@@ -850,8 +848,7 @@
by (import relation WF_inv_image)
constdefs
- RESTRICT :: "('a::type => 'b::type)
-=> ('a::type => 'a::type => bool) => 'a::type => 'a::type => 'b::type"
+ RESTRICT :: "('a => 'b) => ('a => 'a => bool) => 'a => 'a => 'b"
"RESTRICT ==
%(f::'a::type => 'b::type) (R::'a::type => 'a::type => bool) (x::'a::type)
y::'a::type. if R y x then f y else ARB"
@@ -865,9 +862,7 @@
by (import relation RESTRICT_LEMMA)
consts
- approx :: "('a::type => 'a::type => bool)
-=> (('a::type => 'b::type) => 'a::type => 'b::type)
- => 'a::type => ('a::type => 'b::type) => bool"
+ approx :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => ('a => 'b) => bool"
defs
approx_primdef: "approx ==
@@ -883,9 +878,7 @@
by (import relation approx_def)
consts
- the_fun :: "('a::type => 'a::type => bool)
-=> (('a::type => 'b::type) => 'a::type => 'b::type)
- => 'a::type => 'a::type => 'b::type"
+ the_fun :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => 'a => 'b"
defs
the_fun_primdef: "the_fun ==
@@ -899,8 +892,7 @@
by (import relation the_fun_def)
constdefs
- WFREC :: "('a::type => 'a::type => bool)
-=> (('a::type => 'b::type) => 'a::type => 'b::type) => 'a::type => 'b::type"
+ WFREC :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => 'b"
"WFREC ==
%(R::'a::type => 'a::type => bool)
(M::('a::type => 'b::type) => 'a::type => 'b::type) x::'a::type.
@@ -1061,9 +1053,7 @@
by (import pair pair_case_cong)
constdefs
- LEX :: "('a::type => 'a::type => bool)
-=> ('b::type => 'b::type => bool)
- => 'a::type * 'b::type => 'a::type * 'b::type => bool"
+ LEX :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool"
"LEX ==
%(R1::'a::type => 'a::type => bool) (R2::'b::type => 'b::type => bool)
(s::'a::type, t::'b::type) (u::'a::type, v::'b::type).
@@ -1080,9 +1070,7 @@
by (import pair WF_LEX)
constdefs
- RPROD :: "('a::type => 'a::type => bool)
-=> ('b::type => 'b::type => bool)
- => 'a::type * 'b::type => 'a::type * 'b::type => bool"
+ RPROD :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool"
"RPROD ==
%(R1::'a::type => 'a::type => bool) (R2::'b::type => 'b::type => bool)
(s::'a::type, t::'b::type) (u::'a::type, v::'b::type). R1 s u & R2 t v"
@@ -1104,7 +1092,7 @@
;setup_theory prim_rec
-lemma LESS_0_0: "(0::nat) < Suc (0::nat)"
+lemma LESS_0_0: "0 < Suc 0"
by (import prim_rec LESS_0_0)
lemma LESS_LEMMA1: "ALL (x::nat) xa::nat. x < Suc xa --> x = xa | x < xa"
@@ -1126,7 +1114,7 @@
by (import prim_rec NOT_LESS_EQ)
constdefs
- SIMP_REC_REL :: "(nat => 'a::type) => 'a::type => ('a::type => 'a::type) => nat => bool"
+ SIMP_REC_REL :: "(nat => 'a) => 'a => ('a => 'a) => nat => bool"
"(op ==::((nat => 'a::type)
=> 'a::type => ('a::type => 'a::type) => nat => bool)
=> ((nat => 'a::type)
@@ -1184,7 +1172,7 @@
by (import prim_rec SIMP_REC_REL_UNIQUE_RESULT)
consts
- SIMP_REC :: "'a::type => ('a::type => 'a::type) => nat => 'a::type"
+ SIMP_REC :: "'a => ('a => 'a) => nat => 'a"
specification (SIMP_REC) SIMP_REC: "ALL (x::'a::type) (f'::'a::type => 'a::type) n::nat.
EX g::nat => 'a::type.
@@ -1195,23 +1183,22 @@
by (import prim_rec LESS_SUC_SUC)
lemma SIMP_REC_THM: "ALL (x::'a::type) f::'a::type => 'a::type.
- SIMP_REC x f (0::nat) = x &
+ SIMP_REC x f 0 = x &
(ALL m::nat. SIMP_REC x f (Suc m) = f (SIMP_REC x f m))"
by (import prim_rec SIMP_REC_THM)
constdefs
PRE :: "nat => nat"
- "PRE == %m::nat. if m = (0::nat) then 0::nat else SOME n::nat. m = Suc n"
-
-lemma PRE_DEF: "ALL m::nat.
- PRE m = (if m = (0::nat) then 0::nat else SOME n::nat. m = Suc n)"
+ "PRE == %m::nat. if m = 0 then 0 else SOME n::nat. m = Suc n"
+
+lemma PRE_DEF: "ALL m::nat. PRE m = (if m = 0 then 0 else SOME n::nat. m = Suc n)"
by (import prim_rec PRE_DEF)
-lemma PRE: "PRE (0::nat) = (0::nat) & (ALL m::nat. PRE (Suc m) = m)"
+lemma PRE: "PRE 0 = 0 & (ALL m::nat. PRE (Suc m) = m)"
by (import prim_rec PRE)
constdefs
- PRIM_REC_FUN :: "'a::type => ('a::type => nat => 'a::type) => nat => nat => 'a::type"
+ PRIM_REC_FUN :: "'a => ('a => nat => 'a) => nat => nat => 'a"
"PRIM_REC_FUN ==
%(x::'a::type) f::'a::type => nat => 'a::type.
SIMP_REC (%n::nat. x) (%(fun::nat => 'a::type) n::nat. f (fun (PRE n)) n)"
@@ -1222,13 +1209,13 @@
by (import prim_rec PRIM_REC_FUN)
lemma PRIM_REC_EQN: "ALL (x::'a::type) f::'a::type => nat => 'a::type.
- (ALL n::nat. PRIM_REC_FUN x f (0::nat) n = x) &
+ (ALL n::nat. PRIM_REC_FUN x f 0 n = x) &
(ALL (m::nat) n::nat.
PRIM_REC_FUN x f (Suc m) n = f (PRIM_REC_FUN x f m (PRE n)) n)"
by (import prim_rec PRIM_REC_EQN)
constdefs
- PRIM_REC :: "'a::type => ('a::type => nat => 'a::type) => nat => 'a::type"
+ PRIM_REC :: "'a => ('a => nat => 'a) => nat => 'a"
"PRIM_REC ==
%(x::'a::type) (f::'a::type => nat => 'a::type) m::nat.
PRIM_REC_FUN x f m (PRE m)"
@@ -1238,28 +1225,27 @@
by (import prim_rec PRIM_REC)
lemma PRIM_REC_THM: "ALL (x::'a::type) f::'a::type => nat => 'a::type.
- PRIM_REC x f (0::nat) = x &
+ PRIM_REC x f 0 = x &
(ALL m::nat. PRIM_REC x f (Suc m) = f (PRIM_REC x f m) m)"
by (import prim_rec PRIM_REC_THM)
lemma DC: "ALL (P::'a::type => bool) (R::'a::type => 'a::type => bool) a::'a::type.
P a & (ALL x::'a::type. P x --> (EX y::'a::type. P y & R x y)) -->
(EX x::nat => 'a::type.
- x (0::nat) = a & (ALL n::nat. P (x n) & R (x n) (x (Suc n))))"
+ x 0 = a & (ALL n::nat. P (x n) & R (x n) (x (Suc n))))"
by (import prim_rec DC)
lemma num_Axiom_old: "ALL (e::'a::type) f::'a::type => nat => 'a::type.
EX! fn1::nat => 'a::type.
- fn1 (0::nat) = e & (ALL n::nat. fn1 (Suc n) = f (fn1 n) n)"
+ fn1 0 = e & (ALL n::nat. fn1 (Suc n) = f (fn1 n) n)"
by (import prim_rec num_Axiom_old)
lemma num_Axiom: "ALL (e::'a::type) f::nat => 'a::type => 'a::type.
- EX x::nat => 'a::type.
- x (0::nat) = e & (ALL n::nat. x (Suc n) = f n (x n))"
+ EX x::nat => 'a::type. x 0 = e & (ALL n::nat. x (Suc n) = f n (x n))"
by (import prim_rec num_Axiom)
consts
- wellfounded :: "('a::type => 'a::type => bool) => bool"
+ wellfounded :: "('a => 'a => bool) => bool"
defs
wellfounded_primdef: "wellfounded ==
@@ -1281,7 +1267,7 @@
by (import prim_rec WF_LESS)
consts
- measure :: "('a::type => nat) => 'a::type => 'a::type => bool"
+ measure :: "('a => nat) => 'a => 'a => bool"
defs
measure_primdef: "prim_rec.measure == relation.inv_image op <"
@@ -1310,28 +1296,28 @@
consts
EVEN :: "nat => bool"
-specification (EVEN) EVEN: "EVEN (0::nat) = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
+specification (EVEN) EVEN: "EVEN 0 = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
by (import arithmetic EVEN)
consts
ODD :: "nat => bool"
-specification (ODD) ODD: "ODD (0::nat) = False & (ALL n::nat. ODD (Suc n) = (~ ODD n))"
+specification (ODD) ODD: "ODD 0 = False & (ALL n::nat. ODD (Suc n) = (~ ODD n))"
by (import arithmetic ODD)
-lemma TWO: "(2::nat) = Suc (1::nat)"
+lemma TWO: "2 = Suc 1"
by (import arithmetic TWO)
-lemma NORM_0: "(0::nat) = (0::nat)"
+lemma NORM_0: "(op =::nat => nat => bool) (0::nat) (0::nat)"
by (import arithmetic NORM_0)
lemma num_case_compute: "ALL n::nat.
nat_case (f::'a::type) (g::nat => 'a::type) n =
- (if n = (0::nat) then f else g (PRE n))"
+ (if n = 0 then f else g (PRE n))"
by (import arithmetic num_case_compute)
-lemma ADD_CLAUSES: "(0::nat) + (m::nat) = m &
-m + (0::nat) = m & Suc m + (n::nat) = Suc (m + n) & m + Suc n = Suc (m + n)"
+lemma ADD_CLAUSES: "0 + (m::nat) = m &
+m + 0 = m & Suc m + (n::nat) = Suc (m + n) & m + Suc n = Suc (m + n)"
by (import arithmetic ADD_CLAUSES)
lemma LESS_ADD: "ALL (m::nat) n::nat. n < m --> (EX p::nat. p + n = m)"
@@ -1350,7 +1336,7 @@
lemma LESS_NOT_SUC: "ALL (m::nat) n::nat. m < n & n ~= Suc m --> Suc m < n"
by (import arithmetic LESS_NOT_SUC)
-lemma LESS_0_CASES: "ALL m::nat. (0::nat) = m | (0::nat) < m"
+lemma LESS_0_CASES: "ALL m::nat. 0 = m | 0 < m"
by (import arithmetic LESS_0_CASES)
lemma LESS_CASES_IMP: "ALL (m::nat) n::nat. ~ m < n & m ~= n --> n < m"
@@ -1362,44 +1348,41 @@
lemma LESS_EQ_SUC_REFL: "ALL m::nat. m <= Suc m"
by (import arithmetic LESS_EQ_SUC_REFL)
-lemma LESS_ADD_NONZERO: "ALL (m::nat) n::nat. n ~= (0::nat) --> m < m + n"
+lemma LESS_ADD_NONZERO: "ALL (m::nat) n::nat. n ~= 0 --> m < m + n"
by (import arithmetic LESS_ADD_NONZERO)
lemma LESS_EQ_ANTISYM: "ALL (x::nat) xa::nat. ~ (x < xa & xa <= x)"
by (import arithmetic LESS_EQ_ANTISYM)
-lemma SUB_0: "ALL m::nat. (0::nat) - m = (0::nat) & m - (0::nat) = m"
+lemma SUB_0: "ALL m::nat. 0 - m = 0 & m - 0 = m"
by (import arithmetic SUB_0)
-lemma SUC_SUB1: "ALL m::nat. Suc m - (1::nat) = m"
+lemma SUC_SUB1: "ALL m::nat. Suc m - 1 = m"
by (import arithmetic SUC_SUB1)
-lemma PRE_SUB1: "ALL m::nat. PRE m = m - (1::nat)"
+lemma PRE_SUB1: "ALL m::nat. PRE m = m - 1"
by (import arithmetic PRE_SUB1)
lemma MULT_CLAUSES: "ALL (x::nat) xa::nat.
- (0::nat) * x = (0::nat) &
- x * (0::nat) = (0::nat) &
- (1::nat) * x = x &
- x * (1::nat) = x & Suc x * xa = x * xa + xa & x * Suc xa = x + x * xa"
+ 0 * x = 0 &
+ x * 0 = 0 &
+ 1 * x = x &
+ x * 1 = x & Suc x * xa = x * xa + xa & x * Suc xa = x + x * xa"
by (import arithmetic MULT_CLAUSES)
lemma PRE_SUB: "ALL (m::nat) n::nat. PRE (m - n) = PRE m - n"
by (import arithmetic PRE_SUB)
-lemma ADD_EQ_1: "ALL (m::nat) n::nat.
- (m + n = (1::nat)) =
- (m = (1::nat) & n = (0::nat) | m = (0::nat) & n = (1::nat))"
+lemma ADD_EQ_1: "ALL (m::nat) n::nat. (m + n = 1) = (m = 1 & n = 0 | m = 0 & n = 1)"
by (import arithmetic ADD_EQ_1)
-lemma ADD_INV_0_EQ: "ALL (m::nat) n::nat. (m + n = m) = (n = (0::nat))"
+lemma ADD_INV_0_EQ: "ALL (m::nat) n::nat. (m + n = m) = (n = 0)"
by (import arithmetic ADD_INV_0_EQ)
-lemma PRE_SUC_EQ: "ALL (m::nat) n::nat. (0::nat) < n --> (m = PRE n) = (Suc m = n)"
+lemma PRE_SUC_EQ: "ALL (m::nat) n::nat. 0 < n --> (m = PRE n) = (Suc m = n)"
by (import arithmetic PRE_SUC_EQ)
-lemma INV_PRE_EQ: "ALL (m::nat) n::nat.
- (0::nat) < m & (0::nat) < n --> (PRE m = PRE n) = (m = n)"
+lemma INV_PRE_EQ: "ALL (m::nat) n::nat. 0 < m & 0 < n --> (PRE m = PRE n) = (m = n)"
by (import arithmetic INV_PRE_EQ)
lemma LESS_SUC_NOT: "ALL (m::nat) n::nat. m < n --> ~ n < Suc m"
@@ -1408,7 +1391,7 @@
lemma ADD_EQ_SUB: "ALL (m::nat) (n::nat) p::nat. n <= p --> (m + n = p) = (m = p - n)"
by (import arithmetic ADD_EQ_SUB)
-lemma LESS_ADD_1: "ALL (x::nat) xa::nat. xa < x --> (EX xb::nat. x = xa + (xb + (1::nat)))"
+lemma LESS_ADD_1: "ALL (x::nat) xa::nat. xa < x --> (EX xb::nat. x = xa + (xb + 1))"
by (import arithmetic LESS_ADD_1)
lemma NOT_ODD_EQ_EVEN: "ALL (n::nat) m::nat. Suc (n + n) ~= m + m"
@@ -1440,16 +1423,16 @@
((Not::bool => bool) (P m)))))))"
by (import arithmetic WOP)
-lemma INV_PRE_LESS: "ALL m>0::nat. ALL n::nat. (PRE m < PRE n) = (m < n)"
+lemma INV_PRE_LESS: "ALL m>0. ALL n::nat. (PRE m < PRE n) = (m < n)"
by (import arithmetic INV_PRE_LESS)
-lemma INV_PRE_LESS_EQ: "ALL n>0::nat. ALL m::nat. (PRE m <= PRE n) = (m <= n)"
+lemma INV_PRE_LESS_EQ: "ALL n>0. ALL m::nat. (PRE m <= PRE n) = (m <= n)"
by (import arithmetic INV_PRE_LESS_EQ)
-lemma SUB_EQ_EQ_0: "ALL (m::nat) n::nat. (m - n = m) = (m = (0::nat) | n = (0::nat))"
+lemma SUB_EQ_EQ_0: "ALL (m::nat) n::nat. (m - n = m) = (m = 0 | n = 0)"
by (import arithmetic SUB_EQ_EQ_0)
-lemma SUB_LESS_OR: "ALL (m::nat) n::nat. n < m --> n <= m - (1::nat)"
+lemma SUB_LESS_OR: "ALL (m::nat) n::nat. n < m --> n <= m - 1"
by (import arithmetic SUB_LESS_OR)
lemma LESS_SUB_ADD_LESS: "ALL (n::nat) (m::nat) i::nat. i < n - m --> i + m < n"
@@ -1468,15 +1451,13 @@
xa <= x & xb <= x --> (x - xa = x - xb) = (xa = xb)"
by (import arithmetic SUB_CANCEL)
-lemma NOT_EXP_0: "ALL (m::nat) n::nat. Suc n ^ m ~= (0::nat)"
+lemma NOT_EXP_0: "ALL (m::nat) n::nat. Suc n ^ m ~= 0"
by (import arithmetic NOT_EXP_0)
-lemma ZERO_LESS_EXP: "ALL (m::nat) n::nat. (0::nat) < Suc n ^ m"
+lemma ZERO_LESS_EXP: "ALL (m::nat) n::nat. 0 < Suc n ^ m"
by (import arithmetic ZERO_LESS_EXP)
-lemma ODD_OR_EVEN: "ALL x::nat.
- EX xa::nat.
- x = Suc (Suc (0::nat)) * xa | x = Suc (Suc (0::nat)) * xa + (1::nat)"
+lemma ODD_OR_EVEN: "ALL x::nat. EX xa::nat. x = Suc (Suc 0) * xa | x = Suc (Suc 0) * xa + 1"
by (import arithmetic ODD_OR_EVEN)
lemma LESS_EXP_SUC_MONO: "ALL (n::nat) m::nat. Suc (Suc m) ^ n < Suc (Suc m) ^ Suc n"
@@ -1491,16 +1472,16 @@
lemma LESS_EQ_EXISTS: "ALL (m::nat) n::nat. (m <= n) = (EX p::nat. n = m + p)"
by (import arithmetic LESS_EQ_EXISTS)
-lemma MULT_EQ_1: "ALL (x::nat) y::nat. (x * y = (1::nat)) = (x = (1::nat) & y = (1::nat))"
+lemma MULT_EQ_1: "ALL (x::nat) y::nat. (x * y = 1) = (x = 1 & y = 1)"
by (import arithmetic MULT_EQ_1)
consts
FACT :: "nat => nat"
-specification (FACT) FACT: "FACT (0::nat) = (1::nat) & (ALL n::nat. FACT (Suc n) = Suc n * FACT n)"
+specification (FACT) FACT: "FACT 0 = 1 & (ALL n::nat. FACT (Suc n) = Suc n * FACT n)"
by (import arithmetic FACT)
-lemma FACT_LESS: "ALL n::nat. (0::nat) < FACT n"
+lemma FACT_LESS: "ALL n::nat. 0 < FACT n"
by (import arithmetic FACT_LESS)
lemma EVEN_ODD: "ALL n::nat. EVEN n = (~ ODD n)"
@@ -1527,24 +1508,24 @@
lemma ODD_MULT: "ALL (m::nat) n::nat. ODD (m * n) = (ODD m & ODD n)"
by (import arithmetic ODD_MULT)
-lemma EVEN_DOUBLE: "ALL n::nat. EVEN ((2::nat) * n)"
+lemma EVEN_DOUBLE: "ALL n::nat. EVEN (2 * n)"
by (import arithmetic EVEN_DOUBLE)
-lemma ODD_DOUBLE: "ALL x::nat. ODD (Suc ((2::nat) * x))"
+lemma ODD_DOUBLE: "ALL x::nat. ODD (Suc (2 * x))"
by (import arithmetic ODD_DOUBLE)
lemma EVEN_ODD_EXISTS: "ALL x::nat.
- (EVEN x --> (EX m::nat. x = (2::nat) * m)) &
- (ODD x --> (EX m::nat. x = Suc ((2::nat) * m)))"
+ (EVEN x --> (EX m::nat. x = 2 * m)) &
+ (ODD x --> (EX m::nat. x = Suc (2 * m)))"
by (import arithmetic EVEN_ODD_EXISTS)
-lemma EVEN_EXISTS: "ALL n::nat. EVEN n = (EX m::nat. n = (2::nat) * m)"
+lemma EVEN_EXISTS: "ALL n::nat. EVEN n = (EX m::nat. n = 2 * m)"
by (import arithmetic EVEN_EXISTS)
-lemma ODD_EXISTS: "ALL n::nat. ODD n = (EX m::nat. n = Suc ((2::nat) * m))"
+lemma ODD_EXISTS: "ALL n::nat. ODD n = (EX m::nat. n = Suc (2 * m))"
by (import arithmetic ODD_EXISTS)
-lemma NOT_SUC_LESS_EQ_0: "ALL x::nat. ~ Suc x <= (0::nat)"
+lemma NOT_SUC_LESS_EQ_0: "ALL x::nat. ~ Suc x <= 0"
by (import arithmetic NOT_SUC_LESS_EQ_0)
lemma NOT_LEQ: "ALL (x::nat) xa::nat. (~ x <= xa) = (Suc xa <= x)"
@@ -1573,45 +1554,41 @@
m - (n - p) = (if n <= p then m else m + p - n)"
by (import arithmetic SUB_LEFT_SUB)
-lemma SUB_LEFT_SUC: "ALL (m::nat) n::nat.
- Suc (m - n) = (if m <= n then Suc (0::nat) else Suc m - n)"
+lemma SUB_LEFT_SUC: "ALL (m::nat) n::nat. Suc (m - n) = (if m <= n then Suc 0 else Suc m - n)"
by (import arithmetic SUB_LEFT_SUC)
-lemma SUB_LEFT_LESS_EQ: "ALL (m::nat) (n::nat) p::nat. (m <= n - p) = (m + p <= n | m <= (0::nat))"
+lemma SUB_LEFT_LESS_EQ: "ALL (m::nat) (n::nat) p::nat. (m <= n - p) = (m + p <= n | m <= 0)"
by (import arithmetic SUB_LEFT_LESS_EQ)
lemma SUB_RIGHT_LESS_EQ: "ALL (m::nat) (n::nat) p::nat. (m - n <= p) = (m <= n + p)"
by (import arithmetic SUB_RIGHT_LESS_EQ)
-lemma SUB_RIGHT_LESS: "ALL (m::nat) (n::nat) p::nat. (m - n < p) = (m < n + p & (0::nat) < p)"
+lemma SUB_RIGHT_LESS: "ALL (m::nat) (n::nat) p::nat. (m - n < p) = (m < n + p & 0 < p)"
by (import arithmetic SUB_RIGHT_LESS)
-lemma SUB_RIGHT_GREATER_EQ: "ALL (m::nat) (n::nat) p::nat. (p <= m - n) = (n + p <= m | p <= (0::nat))"
+lemma SUB_RIGHT_GREATER_EQ: "ALL (m::nat) (n::nat) p::nat. (p <= m - n) = (n + p <= m | p <= 0)"
by (import arithmetic SUB_RIGHT_GREATER_EQ)
-lemma SUB_LEFT_GREATER: "ALL (m::nat) (n::nat) p::nat. (n - p < m) = (n < m + p & (0::nat) < m)"
+lemma SUB_LEFT_GREATER: "ALL (m::nat) (n::nat) p::nat. (n - p < m) = (n < m + p & 0 < m)"
by (import arithmetic SUB_LEFT_GREATER)
lemma SUB_RIGHT_GREATER: "ALL (m::nat) (n::nat) p::nat. (p < m - n) = (n + p < m)"
by (import arithmetic SUB_RIGHT_GREATER)
-lemma SUB_LEFT_EQ: "ALL (m::nat) (n::nat) p::nat.
- (m = n - p) = (m + p = n | m <= (0::nat) & n <= p)"
+lemma SUB_LEFT_EQ: "ALL (m::nat) (n::nat) p::nat. (m = n - p) = (m + p = n | m <= 0 & n <= p)"
by (import arithmetic SUB_LEFT_EQ)
-lemma SUB_RIGHT_EQ: "ALL (m::nat) (n::nat) p::nat.
- (m - n = p) = (m = n + p | m <= n & p <= (0::nat))"
+lemma SUB_RIGHT_EQ: "ALL (m::nat) (n::nat) p::nat. (m - n = p) = (m = n + p | m <= n & p <= 0)"
by (import arithmetic SUB_RIGHT_EQ)
-lemma LE: "(ALL n::nat. (n <= (0::nat)) = (n = (0::nat))) &
+lemma LE: "(ALL n::nat. (n <= 0) = (n = 0)) &
(ALL (m::nat) n::nat. (m <= Suc n) = (m = Suc n | m <= n))"
by (import arithmetic LE)
-lemma DA: "ALL (k::nat) n::nat.
- (0::nat) < n --> (EX (x::nat) q::nat. k = q * n + x & x < n)"
+lemma DA: "ALL (k::nat) n::nat. 0 < n --> (EX (x::nat) q::nat. k = q * n + x & x < n)"
by (import arithmetic DA)
-lemma DIV_LESS_EQ: "ALL n>0::nat. ALL k::nat. k div n <= k"
+lemma DIV_LESS_EQ: "ALL n>0. ALL k::nat. k div n <= k"
by (import arithmetic DIV_LESS_EQ)
lemma DIV_UNIQUE: "ALL (n::nat) (k::nat) q::nat.
@@ -1625,92 +1602,108 @@
lemma DIV_MULT: "ALL (n::nat) r::nat. r < n --> (ALL q::nat. (q * n + r) div n = q)"
by (import arithmetic DIV_MULT)
-lemma MOD_EQ_0: "ALL n>0::nat. ALL k::nat. k * n mod n = (0::nat)"
+lemma MOD_EQ_0: "ALL n>0. ALL k::nat. k * n mod n = 0"
by (import arithmetic MOD_EQ_0)
-lemma ZERO_MOD: "ALL n>0::nat. (0::nat) mod n = (0::nat)"
+lemma ZERO_MOD: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
+ ((op =::nat => nat => bool) ((op mod::nat => nat => nat) (0::nat) n)
+ (0::nat)))"
by (import arithmetic ZERO_MOD)
-lemma ZERO_DIV: "ALL n>0::nat. (0::nat) div n = (0::nat)"
+lemma ZERO_DIV: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
+ ((op =::nat => nat => bool) ((op div::nat => nat => nat) (0::nat) n)
+ (0::nat)))"
by (import arithmetic ZERO_DIV)
lemma MOD_MULT: "ALL (n::nat) r::nat. r < n --> (ALL q::nat. (q * n + r) mod n = r)"
by (import arithmetic MOD_MULT)
-lemma MOD_TIMES: "ALL n>0::nat. ALL (q::nat) r::nat. (q * n + r) mod n = r mod n"
+lemma MOD_TIMES: "ALL n>0. ALL (q::nat) r::nat. (q * n + r) mod n = r mod n"
by (import arithmetic MOD_TIMES)
-lemma MOD_PLUS: "ALL n>0::nat. ALL (j::nat) k::nat. (j mod n + k mod n) mod n = (j + k) mod n"
+lemma MOD_PLUS: "ALL n>0. ALL (j::nat) k::nat. (j mod n + k mod n) mod n = (j + k) mod n"
by (import arithmetic MOD_PLUS)
-lemma MOD_MOD: "ALL n>0::nat. ALL k::nat. k mod n mod n = k mod n"
+lemma MOD_MOD: "ALL n>0. ALL k::nat. k mod n mod n = k mod n"
by (import arithmetic MOD_MOD)
-lemma ADD_DIV_ADD_DIV: "ALL x>0::nat. ALL (xa::nat) r::nat. (xa * x + r) div x = xa + r div x"
+lemma ADD_DIV_ADD_DIV: "ALL x>0. ALL (xa::nat) r::nat. (xa * x + r) div x = xa + r div x"
by (import arithmetic ADD_DIV_ADD_DIV)
lemma MOD_MULT_MOD: "ALL (m::nat) n::nat.
- (0::nat) < n & (0::nat) < m -->
- (ALL x::nat. x mod (n * m) mod n = x mod n)"
+ 0 < n & 0 < m --> (ALL x::nat. x mod (n * m) mod n = x mod n)"
by (import arithmetic MOD_MULT_MOD)
-lemma DIVMOD_ID: "ALL n>0::nat. n div n = (1::nat) & n mod n = (0::nat)"
+lemma DIVMOD_ID: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
+ ((op &::bool => bool => bool)
+ ((op =::nat => nat => bool) ((op div::nat => nat => nat) n n)
+ (1::nat))
+ ((op =::nat => nat => bool) ((op mod::nat => nat => nat) n n)
+ (0::nat))))"
by (import arithmetic DIVMOD_ID)
lemma DIV_DIV_DIV_MULT: "ALL (x::nat) xa::nat.
- (0::nat) < x & (0::nat) < xa -->
- (ALL xb::nat. xb div x div xa = xb div (x * xa))"
+ 0 < x & 0 < xa --> (ALL xb::nat. xb div x div xa = xb div (x * xa))"
by (import arithmetic DIV_DIV_DIV_MULT)
lemma DIV_P: "ALL (P::nat => bool) (p::nat) q::nat.
- (0::nat) < q -->
- P (p div q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P k)"
+ 0 < q --> P (p div q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P k)"
by (import arithmetic DIV_P)
lemma MOD_P: "ALL (P::nat => bool) (p::nat) q::nat.
- (0::nat) < q -->
- P (p mod q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P r)"
+ 0 < q --> P (p mod q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P r)"
by (import arithmetic MOD_P)
-lemma MOD_TIMES2: "ALL n>0::nat. ALL (j::nat) k::nat. j mod n * (k mod n) mod n = j * k mod n"
+lemma MOD_TIMES2: "ALL n>0. ALL (j::nat) k::nat. j mod n * (k mod n) mod n = j * k mod n"
by (import arithmetic MOD_TIMES2)
lemma MOD_COMMON_FACTOR: "ALL (n::nat) (p::nat) q::nat.
- (0::nat) < n & (0::nat) < q --> n * (p mod q) = n * p mod (n * q)"
+ 0 < n & 0 < q --> n * (p mod q) = n * p mod (n * q)"
by (import arithmetic MOD_COMMON_FACTOR)
lemma num_case_cong: "ALL (M::nat) (M'::nat) (b::'a::type) f::nat => 'a::type.
M = M' &
- (M' = (0::nat) --> b = (b'::'a::type)) &
+ (M' = 0 --> b = (b'::'a::type)) &
(ALL n::nat. M' = Suc n --> f n = (f'::nat => 'a::type) n) -->
nat_case b f M = nat_case b' f' M'"
by (import arithmetic num_case_cong)
lemma SUC_ELIM_THM: "ALL P::nat => nat => bool.
- (ALL n::nat. P (Suc n) n) = (ALL n>0::nat. P n (n - (1::nat)))"
+ (ALL n::nat. P (Suc n) n) = (ALL n>0. P n (n - 1))"
by (import arithmetic SUC_ELIM_THM)
lemma SUB_ELIM_THM: "(P::nat => bool) ((a::nat) - (b::nat)) =
-(ALL x::nat. (b = a + x --> P (0::nat)) & (a = b + x --> P x))"
+(ALL x::nat. (b = a + x --> P 0) & (a = b + x --> P x))"
by (import arithmetic SUB_ELIM_THM)
lemma PRE_ELIM_THM: "(P::nat => bool) (PRE (n::nat)) =
-(ALL m::nat. (n = (0::nat) --> P (0::nat)) & (n = Suc m --> P m))"
+(ALL m::nat. (n = 0 --> P 0) & (n = Suc m --> P m))"
by (import arithmetic PRE_ELIM_THM)
-lemma MULT_INCREASES: "ALL (m::nat) n::nat. (1::nat) < m & (0::nat) < n --> Suc n <= m * n"
+lemma MULT_INCREASES: "ALL (m::nat) n::nat. 1 < m & 0 < n --> Suc n <= m * n"
by (import arithmetic MULT_INCREASES)
-lemma EXP_ALWAYS_BIG_ENOUGH: "ALL b>1::nat. ALL n::nat. EX m::nat. n <= b ^ m"
+lemma EXP_ALWAYS_BIG_ENOUGH: "ALL b>1. ALL n::nat. EX m::nat. n <= b ^ m"
by (import arithmetic EXP_ALWAYS_BIG_ENOUGH)
-lemma EXP_EQ_0: "ALL (n::nat) m::nat. (n ^ m = (0::nat)) = (n = (0::nat) & (0::nat) < m)"
+lemma EXP_EQ_0: "ALL (n::nat) m::nat. (n ^ m = 0) = (n = 0 & 0 < m)"
by (import arithmetic EXP_EQ_0)
-lemma EXP_1: "ALL x::nat. (1::nat) ^ x = (1::nat) & x ^ (1::nat) = x"
+lemma EXP_1: "(All::(nat => bool) => bool)
+ (%x::nat.
+ (op &::bool => bool => bool)
+ ((op =::nat => nat => bool) ((op ^::nat => nat => nat) (1::nat) x)
+ (1::nat))
+ ((op =::nat => nat => bool) ((op ^::nat => nat => nat) x (1::nat)) x))"
by (import arithmetic EXP_1)
-lemma EXP_EQ_1: "ALL (n::nat) m::nat. (n ^ m = (1::nat)) = (n = (1::nat) | m = (0::nat))"
+lemma EXP_EQ_1: "ALL (n::nat) m::nat. (n ^ m = 1) = (n = 1 | m = 0)"
by (import arithmetic EXP_EQ_1)
lemma MIN_MAX_EQ: "ALL (x::nat) xa::nat. (min x xa = max x xa) = (x = xa)"
@@ -1741,10 +1734,10 @@
lemma MAX_LE: "ALL (x::nat) xa::nat. xa <= max xa x & x <= max xa x"
by (import arithmetic MAX_LE)
-lemma MIN_0: "ALL x::nat. min x (0::nat) = (0::nat) & min (0::nat) x = (0::nat)"
+lemma MIN_0: "ALL x::nat. min x 0 = 0 & min 0 x = 0"
by (import arithmetic MIN_0)
-lemma MAX_0: "ALL x::nat. max x (0::nat) = x & max (0::nat) x = x"
+lemma MAX_0: "ALL x::nat. max x 0 = x & max 0 x = x"
by (import arithmetic MAX_0)
lemma EXISTS_GREATEST: "ALL P::nat => bool.
@@ -1758,9 +1751,9 @@
constdefs
trat_1 :: "nat * nat"
- "trat_1 == (0::nat, 0::nat)"
-
-lemma trat_1: "trat_1 = (0::nat, 0::nat)"
+ "trat_1 == (0, 0)"
+
+lemma trat_1: "trat_1 = (0, 0)"
by (import hrat trat_1)
constdefs
@@ -1794,7 +1787,7 @@
consts
trat_sucint :: "nat => nat * nat"
-specification (trat_sucint) trat_sucint: "trat_sucint (0::nat) = trat_1 &
+specification (trat_sucint) trat_sucint: "trat_sucint 0 = trat_1 &
(ALL n::nat. trat_sucint (Suc n) = trat_add (trat_sucint n) trat_1)"
by (import hrat trat_sucint)
@@ -1882,14 +1875,14 @@
(EX d::nat * nat. trat_eq i (trat_add h d))"
by (import hrat TRAT_ADD_TOTAL)
-lemma TRAT_SUCINT_0: "ALL n::nat. trat_eq (trat_sucint n) (n, 0::nat)"
+lemma TRAT_SUCINT_0: "ALL n::nat. trat_eq (trat_sucint n) (n, 0)"
by (import hrat TRAT_SUCINT_0)
lemma TRAT_ARCH: "ALL h::nat * nat.
EX (n::nat) d::nat * nat. trat_eq (trat_sucint n) (trat_add h d)"
by (import hrat TRAT_ARCH)
-lemma TRAT_SUCINT: "trat_eq (trat_sucint (0::nat)) trat_1 &
+lemma TRAT_SUCINT: "trat_eq (trat_sucint 0) trat_1 &
(ALL n::nat.
trat_eq (trat_sucint (Suc n)) (trat_add (trat_sucint n) trat_1))"
by (import hrat TRAT_SUCINT)
@@ -1989,7 +1982,7 @@
lemma HRAT_ARCH: "ALL h::hrat. EX (x::nat) xa::hrat. hrat_sucint x = hrat_add h xa"
by (import hrat HRAT_ARCH)
-lemma HRAT_SUCINT: "hrat_sucint (0::nat) = hrat_1 &
+lemma HRAT_SUCINT: "hrat_sucint 0 = hrat_1 &
(ALL x::nat. hrat_sucint (Suc x) = hrat_add (hrat_sucint x) hrat_1)"
by (import hrat HRAT_SUCINT)
@@ -2382,40 +2375,243 @@
lemma iiSUC: "ALL n::nat. iiSUC n = Suc (Suc n)"
by (import numeral iiSUC)
-lemma numeral_distrib: "(ALL x::nat. (0::nat) + x = x) &
-(ALL x::nat. x + (0::nat) = x) &
-(ALL (x::nat) xa::nat. NUMERAL x + NUMERAL xa = NUMERAL (iZ (x + xa))) &
-(ALL x::nat. (0::nat) * x = (0::nat)) &
-(ALL x::nat. x * (0::nat) = (0::nat)) &
-(ALL (x::nat) xa::nat. NUMERAL x * NUMERAL xa = NUMERAL (x * xa)) &
-(ALL x::nat. (0::nat) - x = (0::nat)) &
-(ALL x::nat. x - (0::nat) = x) &
-(ALL (x::nat) xa::nat. NUMERAL x - NUMERAL xa = NUMERAL (x - xa)) &
-(ALL x::nat. (0::nat) ^ NUMERAL (NUMERAL_BIT1 x) = (0::nat)) &
-(ALL x::nat. (0::nat) ^ NUMERAL (NUMERAL_BIT2 x) = (0::nat)) &
-(ALL x::nat. x ^ (0::nat) = (1::nat)) &
-(ALL (x::nat) xa::nat. NUMERAL x ^ NUMERAL xa = NUMERAL (x ^ xa)) &
-Suc (0::nat) = (1::nat) &
-(ALL x::nat. Suc (NUMERAL x) = NUMERAL (Suc x)) &
-PRE (0::nat) = (0::nat) &
-(ALL x::nat. PRE (NUMERAL x) = NUMERAL (PRE x)) &
-(ALL x::nat. (NUMERAL x = (0::nat)) = (x = ALT_ZERO)) &
-(ALL x::nat. ((0::nat) = NUMERAL x) = (x = ALT_ZERO)) &
-(ALL (x::nat) xa::nat. (NUMERAL x = NUMERAL xa) = (x = xa)) &
-(ALL x::nat. (x < (0::nat)) = False) &
-(ALL x::nat. ((0::nat) < NUMERAL x) = (ALT_ZERO < x)) &
-(ALL (x::nat) xa::nat. (NUMERAL x < NUMERAL xa) = (x < xa)) &
-(ALL x::nat. (x < (0::nat)) = False) &
-(ALL x::nat. ((0::nat) < NUMERAL x) = (ALT_ZERO < x)) &
-(ALL (x::nat) xa::nat. (NUMERAL xa < NUMERAL x) = (xa < x)) &
-(ALL x::nat. ((0::nat) <= x) = True) &
-(ALL x::nat. (NUMERAL x <= (0::nat)) = (x <= ALT_ZERO)) &
-(ALL (x::nat) xa::nat. (NUMERAL x <= NUMERAL xa) = (x <= xa)) &
-(ALL x::nat. ((0::nat) <= x) = True) &
-(ALL x::nat. (x <= (0::nat)) = (x = (0::nat))) &
-(ALL (x::nat) xa::nat. (NUMERAL xa <= NUMERAL x) = (xa <= x)) &
-(ALL x::nat. ODD (NUMERAL x) = ODD x) &
-(ALL x::nat. EVEN (NUMERAL x) = EVEN x) & ~ ODD (0::nat) & EVEN (0::nat)"
+lemma numeral_distrib: "(op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool) ((op +::nat => nat => nat) (0::nat) x) x))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool) ((op +::nat => nat => nat) x (0::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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((NUMERAL::nat => nat)
+ ((iZ::nat => nat) ((op +::nat => nat => nat) x xa))))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op *::nat => nat => nat) (0::nat) x) (0::nat)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op *::nat => 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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((NUMERAL::nat => nat)
+ ((op *::nat => nat => nat) x xa)))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op -::nat => nat => nat) (0::nat) x) (0::nat)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op -::nat => nat => nat) x (0::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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((NUMERAL::nat => nat)
+ ((op -::nat => nat => nat) x xa)))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op ^::nat => nat => nat) (0::nat)
+ ((NUMERAL::nat => 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) (0::nat)
+ ((NUMERAL::nat => nat)
+ ((NUMERAL_BIT2::nat => nat) x)))
+ (0::nat)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op ^::nat => nat => nat) x (0::nat))
+ (1::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::nat => nat) x) ((NUMERAL::nat => nat) xa))
+ ((NUMERAL::nat => nat)
+ ((op ^::nat => nat => nat) x xa)))))
+ ((op &::bool => bool => bool)
+ ((op =::nat => nat => bool)
+ ((Suc::nat => nat) (0::nat)) (1::nat))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((Suc::nat => nat)
+((NUMERAL::nat => nat) x))
+ ((NUMERAL::nat => nat)
+((Suc::nat => nat) x))))
+ ((op &::bool => bool => bool)
+ ((op =::nat => nat => bool)
+ ((PRE::nat => nat) (0::nat)) (0::nat))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool) ((PRE::nat => nat) ((NUMERAL::nat => nat) x))
+ ((NUMERAL::nat => nat) ((PRE::nat => nat) x))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op =::nat => nat => bool) ((NUMERAL::nat => nat) x) (0::nat))
+ ((op =::nat => nat => bool) x (ALT_ZERO::nat))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op =::nat => nat => bool) (0::nat) ((NUMERAL::nat => nat) x))
+ ((op =::nat => nat => bool) x (ALT_ZERO::nat))))
+ ((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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((op =::nat => nat => bool) x xa))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <::nat => nat => bool) 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::nat => nat) x))
+ ((op <::nat => nat => bool) (ALT_ZERO::nat) x)))
+ ((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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((op <::nat => nat => bool) x xa))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <::nat => nat => bool) 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::nat => nat) x))
+ ((op <::nat => nat => bool) (ALT_ZERO::nat) x)))
+ ((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::nat => nat) xa)
+ ((NUMERAL::nat => nat) x))
+ ((op <::nat => nat => bool) xa x))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <=::nat => nat => bool) (0::nat) x)
+ (True::bool)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <=::nat => nat => bool)
+ ((NUMERAL::nat => nat) x) (0::nat))
+ ((op <=::nat => nat => bool) x (ALT_ZERO::nat))))
+ ((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::nat => nat) x)
+ ((NUMERAL::nat => nat) xa))
+ ((op <=::nat => nat => bool) x xa))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <=::nat => nat => bool) (0::nat) x)
+ (True::bool)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((op <=::nat => nat => bool) x (0::nat))
+ ((op =::nat => nat => bool) x (0::nat))))
+ ((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::nat => nat) xa) ((NUMERAL::nat => nat) x))
+ ((op <=::nat => nat => bool) xa x))))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((ODD::nat => bool)
+ ((NUMERAL::nat => nat) x))
+ ((ODD::nat => bool) x)))
+ ((op &::bool => bool => bool)
+ ((All::(nat => bool) => bool)
+ (%x::nat.
+ (op =::bool => bool => bool)
+ ((EVEN::nat => bool)
+((NUMERAL::nat => nat) x))
+ ((EVEN::nat => bool) x)))
+ ((op &::bool => bool => bool)
+ ((Not::bool => bool)
+ ((ODD::nat => bool) (0::nat)))
+ ((EVEN::nat => bool)
+ (0::nat))))))))))))))))))))))))))))))))))))"
by (import numeral numeral_distrib)
lemma numeral_iisuc: "iiSUC ALT_ZERO = NUMERAL_BIT2 ALT_ZERO &
@@ -2496,7 +2692,7 @@
by (import numeral bit_initiality)
consts
- iBIT_cases :: "nat => 'a::type => (nat => 'a::type) => (nat => 'a::type) => 'a::type"
+ iBIT_cases :: "nat => 'a => (nat => 'a) => (nat => 'a) => 'a"
specification (iBIT_cases) iBIT_cases: "(ALL (zf::'a::type) (bf1::nat => 'a::type) bf2::nat => 'a::type.
iBIT_cases ALT_ZERO zf bf1 bf2 = zf) &
@@ -2561,17 +2757,8 @@
NUMERAL_BIT1 (iSUB False xb xc)"
by (import numeral iSUB_THM)
-lemma numeral_sub: "(All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((NUMERAL::nat => nat) ((op -::nat => nat => nat) x xa))
- ((If::bool => nat => nat => nat)
- ((op <::nat => nat => bool) xa x)
- ((NUMERAL::nat => nat)
- ((iSUB::bool => nat => nat => nat) (True::bool) x xa))
- (0::nat))))"
+lemma numeral_sub: "ALL (x::nat) xa::nat.
+ NUMERAL (x - xa) = (if xa < x then NUMERAL (iSUB True x xa) else 0)"
by (import numeral numeral_sub)
lemma iDUB_removal: "ALL x::nat.
@@ -2606,12 +2793,12 @@
~ ODD ALT_ZERO & ~ ODD (NUMERAL_BIT2 x) & ODD (NUMERAL_BIT1 x)"
by (import numeral numeral_evenodd)
-lemma numeral_fact: "ALL n::nat. FACT n = (if n = (0::nat) then 1::nat else n * FACT (PRE n))"
+lemma numeral_fact: "ALL n::nat. FACT n = (if n = 0 then 1 else n * FACT (PRE n))"
by (import numeral numeral_fact)
lemma numeral_funpow: "ALL n::nat.
((f::'a::type => 'a::type) ^ n) (x::'a::type) =
- (if n = (0::nat) then x else (f ^ (n - (1::nat))) (f x))"
+ (if n = 0 then x else (f ^ (n - 1)) (f x))"
by (import numeral numeral_funpow)
;end_setup
@@ -2627,9 +2814,9 @@
constdefs
NUMPAIR :: "nat => nat => nat"
- "NUMPAIR == %(x::nat) y::nat. (2::nat) ^ x * ((2::nat) * y + (1::nat))"
-
-lemma NUMPAIR: "ALL (x::nat) y::nat. NUMPAIR x y = (2::nat) ^ x * ((2::nat) * y + (1::nat))"
+ "NUMPAIR == %(x::nat) y::nat. 2 ^ x * (2 * y + 1)"
+
+lemma NUMPAIR: "ALL (x::nat) y::nat. NUMPAIR x y = 2 ^ x * (2 * y + 1)"
by (import ind_type NUMPAIR)
lemma NUMPAIR_INJ_LEMMA: "ALL (x::nat) (xa::nat) (xb::nat) xc::nat.
@@ -2649,10 +2836,9 @@
constdefs
NUMSUM :: "bool => nat => nat"
- "NUMSUM == %(b::bool) x::nat. if b then Suc ((2::nat) * x) else (2::nat) * x"
-
-lemma NUMSUM: "ALL (b::bool) x::nat.
- NUMSUM b x = (if b then Suc ((2::nat) * x) else (2::nat) * x)"
+ "NUMSUM == %(b::bool) x::nat. if b then Suc (2 * x) else 2 * x"
+
+lemma NUMSUM: "ALL (b::bool) x::nat. NUMSUM b x = (if b then Suc (2 * x) else 2 * x)"
by (import ind_type NUMSUM)
lemma NUMSUM_INJ: "ALL (b1::bool) (x1::nat) (b2::bool) x2::nat.
@@ -2667,7 +2853,7 @@
by (import ind_type NUMSUM_DEST)
constdefs
- INJN :: "nat => nat => 'a::type => bool"
+ INJN :: "nat => nat => 'a => bool"
"INJN == %(m::nat) (n::nat) a::'a::type. n = m"
lemma INJN: "ALL m::nat. INJN m = (%(n::nat) a::'a::type. n = m)"
@@ -2677,7 +2863,7 @@
by (import ind_type INJN_INJ)
constdefs
- INJA :: "'a::type => nat => 'a::type => bool"
+ INJA :: "'a => nat => 'a => bool"
"INJA == %(a::'a::type) (n::nat) b::'a::type. b = a"
lemma INJA: "ALL a::'a::type. INJA a = (%(n::nat) b::'a::type. b = a)"
@@ -2687,7 +2873,7 @@
by (import ind_type INJA_INJ)
constdefs
- INJF :: "(nat => nat => 'a::type => bool) => nat => 'a::type => bool"
+ INJF :: "(nat => nat => 'a => bool) => nat => 'a => bool"
"INJF == %(f::nat => nat => 'a::type => bool) n::nat. f (NUMFST n) (NUMSND n)"
lemma INJF: "ALL f::nat => nat => 'a::type => bool.
@@ -2699,8 +2885,7 @@
by (import ind_type INJF_INJ)
constdefs
- INJP :: "(nat => 'a::type => bool)
-=> (nat => 'a::type => bool) => nat => 'a::type => bool"
+ INJP :: "(nat => 'a => bool) => (nat => 'a => bool) => nat => 'a => bool"
"INJP ==
%(f1::nat => 'a::type => bool) (f2::nat => 'a::type => bool) (n::nat)
a::'a::type. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a"
@@ -2717,8 +2902,7 @@
by (import ind_type INJP_INJ)
constdefs
- ZCONSTR :: "nat
-=> 'a::type => (nat => nat => 'a::type => bool) => nat => 'a::type => bool"
+ ZCONSTR :: "nat => 'a => (nat => nat => 'a => bool) => nat => 'a => bool"
"ZCONSTR ==
%(c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
INJP (INJN (Suc c)) (INJP (INJA i) (INJF r))"
@@ -2728,10 +2912,10 @@
by (import ind_type ZCONSTR)
constdefs
- ZBOT :: "nat => 'a::type => bool"
- "ZBOT == INJP (INJN (0::nat)) (SOME z::nat => 'a::type => bool. True)"
-
-lemma ZBOT: "ZBOT = INJP (INJN (0::nat)) (SOME z::nat => 'a::type => bool. True)"
+ ZBOT :: "nat => 'a => bool"
+ "ZBOT == INJP (INJN 0) (SOME z::nat => 'a::type => bool. True)"
+
+lemma ZBOT: "ZBOT = INJP (INJN 0) (SOME z::nat => 'a::type => bool. True)"
by (import ind_type ZBOT)
lemma ZCONSTR_ZBOT: "ALL (x::nat) (xa::'a::type) xb::nat => nat => 'a::type => bool.
@@ -2739,7 +2923,7 @@
by (import ind_type ZCONSTR_ZBOT)
constdefs
- ZRECSPACE :: "(nat => 'a::type => bool) => bool"
+ ZRECSPACE :: "(nat => 'a => bool) => bool"
"ZRECSPACE ==
%a0::nat => 'a::type => bool.
ALL ZRECSPACE'::(nat => 'a::type => bool) => bool.
@@ -2805,22 +2989,22 @@
lemmas recspace_TY_DEF = typedef_hol2hol4 [OF type_definition_recspace]
consts
- mk_rec :: "(nat => 'a::type => bool) => 'a::type recspace"
- dest_rec :: "'a::type recspace => nat => 'a::type => bool"
+ mk_rec :: "(nat => 'a => bool) => 'a recspace"
+ dest_rec :: "'a recspace => nat => 'a => bool"
specification (dest_rec mk_rec) recspace_repfns: "(ALL a::'a::type recspace. mk_rec (dest_rec a) = a) &
(ALL r::nat => 'a::type => bool. ZRECSPACE r = (dest_rec (mk_rec r) = r))"
by (import ind_type recspace_repfns)
constdefs
- BOTTOM :: "'a::type recspace"
+ BOTTOM :: "'a recspace"
"BOTTOM == mk_rec ZBOT"
lemma BOTTOM: "BOTTOM = mk_rec ZBOT"
by (import ind_type BOTTOM)
constdefs
- CONSTR :: "nat => 'a::type => (nat => 'a::type recspace) => 'a::type recspace"
+ CONSTR :: "nat => 'a => (nat => 'a recspace) => 'a recspace"
"CONSTR ==
%(c::nat) (i::'a::type) r::nat => 'a::type recspace.
mk_rec (ZCONSTR c i (%n::nat. dest_rec (r n)))"
@@ -2862,21 +3046,21 @@
by (import ind_type CONSTR_REC)
consts
- FCONS :: "'a::type => (nat => 'a::type) => nat => 'a::type"
-
-specification (FCONS) FCONS: "(ALL (a::'a::type) f::nat => 'a::type. FCONS a f (0::nat) = a) &
+ FCONS :: "'a => (nat => 'a) => nat => 'a"
+
+specification (FCONS) FCONS: "(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 ind_type FCONS)
constdefs
- FNIL :: "nat => 'a::type"
+ FNIL :: "nat => 'a"
"FNIL == %n::nat. SOME x::'a::type. True"
lemma FNIL: "ALL n::nat. FNIL n = (SOME x::'a::type. True)"
by (import ind_type FNIL)
constdefs
- ISO :: "('a::type => 'b::type) => ('b::type => 'a::type) => bool"
+ ISO :: "('a => 'b) => ('b => 'a) => bool"
"ISO ==
%(f::'a::type => 'b::type) g::'b::type => 'a::type.
(ALL x::'b::type. f (g x) = x) & (ALL y::'a::type. g (f y) = y)"
@@ -2905,16 +3089,16 @@
;setup_theory divides
-lemma ONE_DIVIDES_ALL: "All (op dvd (1::nat))"
+lemma ONE_DIVIDES_ALL: "(All::(nat => bool) => bool) ((op dvd::nat => nat => bool) (1::nat))"
by (import divides ONE_DIVIDES_ALL)
lemma DIVIDES_ADD_2: "ALL (a::nat) (b::nat) c::nat. a dvd b & a dvd b + c --> a dvd c"
by (import divides DIVIDES_ADD_2)
-lemma DIVIDES_FACT: "ALL b>0::nat. b dvd FACT b"
+lemma DIVIDES_FACT: "ALL b>0. b dvd FACT b"
by (import divides DIVIDES_FACT)
-lemma DIVIDES_MULT_LEFT: "ALL (x::nat) xa::nat. (x * xa dvd xa) = (xa = (0::nat) | x = (1::nat))"
+lemma DIVIDES_MULT_LEFT: "ALL (x::nat) xa::nat. (x * xa dvd xa) = (xa = 0 | x = 1)"
by (import divides DIVIDES_MULT_LEFT)
;end_setup
@@ -2925,18 +3109,16 @@
prime :: "nat => bool"
defs
- prime_primdef: "prime.prime ==
-%a::nat. a ~= (1::nat) & (ALL b::nat. b dvd a --> b = a | b = (1::nat))"
+ prime_primdef: "prime.prime == %a::nat. a ~= 1 & (ALL b::nat. b dvd a --> b = a | b = 1)"
lemma prime_def: "ALL a::nat.
- prime.prime a =
- (a ~= (1::nat) & (ALL b::nat. b dvd a --> b = a | b = (1::nat)))"
+ prime.prime a = (a ~= 1 & (ALL b::nat. b dvd a --> b = a | b = 1))"
by (import prime prime_def)
-lemma NOT_PRIME_0: "~ prime.prime (0::nat)"
+lemma NOT_PRIME_0: "~ prime.prime 0"
by (import prime NOT_PRIME_0)
-lemma NOT_PRIME_1: "~ prime.prime (1::nat)"
+lemma NOT_PRIME_1: "~ prime.prime 1"
by (import prime NOT_PRIME_1)
;end_setup
@@ -2944,9 +3126,9 @@
;setup_theory list
consts
- EL :: "nat => 'a::type list => 'a::type"
-
-specification (EL) EL: "(ALL l::'a::type list. EL (0::nat) l = hd l) &
+ EL :: "nat => 'a list => 'a"
+
+specification (EL) EL: "(ALL l::'a::type list. EL 0 l = hd l) &
(ALL (l::'a::type list) n::nat. EL (Suc n) l = EL n (tl l))"
by (import list EL)
@@ -3047,7 +3229,7 @@
by (import list LENGTH_EQ_CONS)
lemma LENGTH_EQ_NIL: "ALL P::'a::type list => bool.
- (ALL l::'a::type list. length l = (0::nat) --> P l) = P []"
+ (ALL l::'a::type list. length l = 0 --> P l) = P []"
by (import list LENGTH_EQ_NIL)
lemma CONS_ACYCLIC: "ALL (l::'a::type list) x::'a::type. l ~= x # l & x # l ~= l"
@@ -3066,8 +3248,7 @@
by (import list APPEND_11)
lemma EL_compute: "ALL n::nat.
- EL n (l::'a::type list) =
- (if n = (0::nat) then hd l else EL (PRE n) (tl l))"
+ EL n (l::'a::type list) = (if n = 0 then hd l else EL (PRE n) (tl l))"
by (import list EL_compute)
lemma WF_LIST_PRED: "WF (%(L1::'a::type list) L2::'a::type list. EX h::'a::type. L2 = h # L1)"
@@ -3245,7 +3426,7 @@
by (import pred_set NUM_SET_WOP)
consts
- GSPEC :: "('b::type => 'a::type * bool) => 'a::type => bool"
+ GSPEC :: "('b => 'a * bool) => 'a => bool"
specification (GSPEC) GSPECIFICATION: "ALL (f::'b::type => 'a::type * bool) v::'a::type.
IN v (GSPEC f) = (EX x::'b::type. (v, True) = f x)"
@@ -3257,7 +3438,7 @@
by (import pred_set SET_MINIMUM)
constdefs
- EMPTY :: "'a::type => bool"
+ EMPTY :: "'a => bool"
"EMPTY == %x::'a::type. False"
lemma EMPTY_DEF: "EMPTY = (%x::'a::type. False)"
@@ -3270,7 +3451,7 @@
by (import pred_set MEMBER_NOT_EMPTY)
consts
- UNIV :: "'a::type => bool"
+ UNIV :: "'a => bool"
defs
UNIV_def: "pred_set.UNIV == %x::'a::type. True"
@@ -3291,7 +3472,7 @@
by (import pred_set EQ_UNIV)
constdefs
- SUBSET :: "('a::type => bool) => ('a::type => bool) => bool"
+ SUBSET :: "('a => bool) => ('a => bool) => bool"
"SUBSET ==
%(s::'a::type => bool) t::'a::type => bool.
ALL x::'a::type. IN x s --> IN x t"
@@ -3324,7 +3505,7 @@
by (import pred_set UNIV_SUBSET)
constdefs
- PSUBSET :: "('a::type => bool) => ('a::type => bool) => bool"
+ PSUBSET :: "('a => bool) => ('a => bool) => bool"
"PSUBSET == %(s::'a::type => bool) t::'a::type => bool. SUBSET s t & s ~= t"
lemma PSUBSET_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
@@ -3349,7 +3530,7 @@
by (import pred_set PSUBSET_UNIV)
consts
- UNION :: "('a::type => bool) => ('a::type => bool) => 'a::type => bool"
+ UNION :: "('a => bool) => ('a => bool) => 'a => bool"
defs
UNION_def: "pred_set.UNION ==
@@ -3403,7 +3584,7 @@
by (import pred_set EMPTY_UNION)
consts
- INTER :: "('a::type => bool) => ('a::type => bool) => 'a::type => bool"
+ INTER :: "('a => bool) => ('a => bool) => 'a => bool"
defs
INTER_def: "pred_set.INTER ==
@@ -3463,7 +3644,7 @@
by (import pred_set INTER_OVER_UNION)
constdefs
- DISJOINT :: "('a::type => bool) => ('a::type => bool) => bool"
+ DISJOINT :: "('a => bool) => ('a => bool) => bool"
"DISJOINT ==
%(s::'a::type => bool) t::'a::type => bool. pred_set.INTER s t = EMPTY"
@@ -3495,7 +3676,7 @@
by (import pred_set DISJOINT_UNION_BOTH)
constdefs
- DIFF :: "('a::type => bool) => ('a::type => bool) => 'a::type => bool"
+ DIFF :: "('a => bool) => ('a => bool) => 'a => bool"
"DIFF ==
%(s::'a::type => bool) t::'a::type => bool.
GSPEC (%x::'a::type. (x, IN x s & ~ IN x t))"
@@ -3525,7 +3706,7 @@
by (import pred_set DIFF_EQ_EMPTY)
constdefs
- INSERT :: "'a::type => ('a::type => bool) => 'a::type => bool"
+ INSERT :: "'a => ('a => bool) => 'a => bool"
"INSERT ==
%(x::'a::type) s::'a::type => bool.
GSPEC (%y::'a::type. (y, y = x | IN y s))"
@@ -3601,7 +3782,7 @@
by (import pred_set INSERT_DIFF)
constdefs
- DELETE :: "('a::type => bool) => 'a::type => 'a::type => bool"
+ DELETE :: "('a => bool) => 'a => 'a => bool"
"DELETE == %(s::'a::type => bool) x::'a::type. DIFF s (INSERT x EMPTY)"
lemma DELETE_DEF: "ALL (s::'a::type => bool) x::'a::type. DELETE s x = DIFF s (INSERT x EMPTY)"
@@ -3669,13 +3850,13 @@
by (import pred_set DISJOINT_DELETE_SYM)
consts
- CHOICE :: "('a::type => bool) => 'a::type"
+ CHOICE :: "('a => bool) => 'a"
specification (CHOICE) CHOICE_DEF: "ALL x::'a::type => bool. x ~= EMPTY --> IN (CHOICE x) x"
by (import pred_set CHOICE_DEF)
constdefs
- REST :: "('a::type => bool) => 'a::type => bool"
+ REST :: "('a => bool) => 'a => bool"
"REST == %s::'a::type => bool. DELETE s (CHOICE s)"
lemma REST_DEF: "ALL s::'a::type => bool. REST s = DELETE s (CHOICE s)"
@@ -3694,7 +3875,7 @@
by (import pred_set REST_PSUBSET)
constdefs
- SING :: "('a::type => bool) => bool"
+ SING :: "('a => bool) => bool"
"SING == %s::'a::type => bool. EX x::'a::type. s = INSERT x EMPTY"
lemma SING_DEF: "ALL s::'a::type => bool. SING s = (EX x::'a::type. s = INSERT x EMPTY)"
@@ -3740,7 +3921,7 @@
by (import pred_set SING_IFF_EMPTY_REST)
constdefs
- IMAGE :: "('a::type => 'b::type) => ('a::type => bool) => 'b::type => bool"
+ IMAGE :: "('a => 'b) => ('a => bool) => 'b => bool"
"IMAGE ==
%(f::'a::type => 'b::type) s::'a::type => bool.
GSPEC (%x::'a::type. (f x, IN x s))"
@@ -3794,7 +3975,7 @@
by (import pred_set IMAGE_INTER)
constdefs
- INJ :: "('a::type => 'b::type) => ('a::type => bool) => ('b::type => bool) => bool"
+ INJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool"
"INJ ==
%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
(ALL x::'a::type. IN x s --> IN (f x) t) &
@@ -3821,7 +4002,7 @@
by (import pred_set INJ_EMPTY)
constdefs
- SURJ :: "('a::type => 'b::type) => ('a::type => bool) => ('b::type => bool) => bool"
+ SURJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool"
"SURJ ==
%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
(ALL x::'a::type. IN x s --> IN (f x) t) &
@@ -3851,7 +4032,7 @@
by (import pred_set IMAGE_SURJ)
constdefs
- BIJ :: "('a::type => 'b::type) => ('a::type => bool) => ('b::type => bool) => bool"
+ BIJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool"
"BIJ ==
%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
INJ f s t & SURJ f s t"
@@ -3874,21 +4055,21 @@
by (import pred_set BIJ_COMPOSE)
consts
- LINV :: "('a::type => 'b::type) => ('a::type => bool) => 'b::type => 'a::type"
+ LINV :: "('a => 'b) => ('a => bool) => 'b => 'a"
specification (LINV) LINV_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
INJ f s t --> (ALL x::'a::type. IN x s --> LINV f s (f x) = x)"
by (import pred_set LINV_DEF)
consts
- RINV :: "('a::type => 'b::type) => ('a::type => bool) => 'b::type => 'a::type"
+ RINV :: "('a => 'b) => ('a => bool) => 'b => 'a"
specification (RINV) RINV_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
SURJ f s t --> (ALL x::'b::type. IN x t --> f (RINV f s x) = x)"
by (import pred_set RINV_DEF)
constdefs
- FINITE :: "('a::type => bool) => bool"
+ FINITE :: "('a => bool) => bool"
"FINITE ==
%s::'a::type => bool.
ALL P::('a::type => bool) => bool.
@@ -3954,7 +4135,7 @@
by (import pred_set IMAGE_FINITE)
consts
- CARD :: "('a::type => bool) => nat"
+ CARD :: "('a => bool) => nat"
specification (CARD) CARD_DEF: "(op &::bool => bool => bool)
((op =::nat => nat => bool)
@@ -3977,7 +4158,7 @@
((CARD::('a::type => bool) => nat) s)))))))"
by (import pred_set CARD_DEF)
-lemma CARD_EMPTY: "CARD EMPTY = (0::nat)"
+lemma CARD_EMPTY: "CARD EMPTY = 0"
by (import pred_set CARD_EMPTY)
lemma CARD_INSERT: "ALL s::'a::type => bool.
@@ -3986,13 +4167,13 @@
CARD (INSERT x s) = (if IN x s then CARD s else Suc (CARD s)))"
by (import pred_set CARD_INSERT)
-lemma CARD_EQ_0: "ALL s::'a::type => bool. FINITE s --> (CARD s = (0::nat)) = (s = EMPTY)"
+lemma CARD_EQ_0: "ALL s::'a::type => bool. FINITE s --> (CARD s = 0) = (s = EMPTY)"
by (import pred_set CARD_EQ_0)
lemma CARD_DELETE: "ALL s::'a::type => bool.
FINITE s -->
(ALL x::'a::type.
- CARD (DELETE s x) = (if IN x s then CARD s - (1::nat) else CARD s))"
+ CARD (DELETE s x) = (if IN x s then CARD s - 1 else CARD s))"
by (import pred_set CARD_DELETE)
lemma CARD_INTER_LESS_EQ: "ALL s::'a::type => bool.
@@ -4016,10 +4197,10 @@
FINITE s --> (ALL t::'a::type => bool. PSUBSET t s --> CARD t < CARD s)"
by (import pred_set CARD_PSUBSET)
-lemma CARD_SING: "ALL x::'a::type. CARD (INSERT x EMPTY) = (1::nat)"
+lemma CARD_SING: "ALL x::'a::type. CARD (INSERT x EMPTY) = 1"
by (import pred_set CARD_SING)
-lemma SING_IFF_CARD1: "ALL x::'a::type => bool. SING x = (CARD x = (1::nat) & FINITE x)"
+lemma SING_IFF_CARD1: "ALL x::'a::type => bool. SING x = (CARD x = 1 & FINITE x)"
by (import pred_set SING_IFF_CARD1)
lemma CARD_DIFF: "ALL t::'a::type => bool.
@@ -4031,7 +4212,7 @@
lemma LESS_CARD_DIFF: "ALL t::'a::type => bool.
FINITE t -->
(ALL s::'a::type => bool.
- FINITE s --> CARD t < CARD s --> (0::nat) < CARD (DIFF s t))"
+ FINITE s --> CARD t < CARD s --> 0 < CARD (DIFF s t))"
by (import pred_set LESS_CARD_DIFF)
lemma FINITE_COMPLETE_INDUCTION: "ALL P::('a::type => bool) => bool.
@@ -4042,7 +4223,7 @@
by (import pred_set FINITE_COMPLETE_INDUCTION)
constdefs
- INFINITE :: "('a::type => bool) => bool"
+ INFINITE :: "('a => bool) => bool"
"INFINITE == %s::'a::type => bool. ~ FINITE s"
lemma INFINITE_DEF: "ALL s::'a::type => bool. INFINITE s = (~ FINITE s)"
@@ -4143,7 +4324,7 @@
by (import pred_set FINITE_WEAK_ENUMERATE)
constdefs
- BIGUNION :: "(('a::type => bool) => bool) => 'a::type => bool"
+ BIGUNION :: "(('a => bool) => bool) => 'a => bool"
"BIGUNION ==
%P::('a::type => bool) => bool.
GSPEC (%x::'a::type. (x, EX p::'a::type => bool. IN p P & IN x p))"
@@ -4190,7 +4371,7 @@
by (import pred_set FINITE_BIGUNION)
constdefs
- BIGINTER :: "(('a::type => bool) => bool) => 'a::type => bool"
+ BIGINTER :: "(('a => bool) => bool) => 'a => bool"
"BIGINTER ==
%B::('a::type => bool) => bool.
GSPEC (%x::'a::type. (x, ALL P::'a::type => bool. IN P B --> IN x P))"
@@ -4229,7 +4410,7 @@
by (import pred_set DISJOINT_BIGINTER)
constdefs
- CROSS :: "('a::type => bool) => ('b::type => bool) => 'a::type * 'b::type => bool"
+ CROSS :: "('a => bool) => ('b => bool) => 'a * 'b => bool"
"CROSS ==
%(P::'a::type => bool) Q::'b::type => bool.
GSPEC (%p::'a::type * 'b::type. (p, IN (fst p) P & IN (snd p) Q))"
@@ -4283,7 +4464,7 @@
by (import pred_set FINITE_CROSS_EQ)
constdefs
- COMPL :: "('a::type => bool) => 'a::type => bool"
+ COMPL :: "('a => bool) => 'a => bool"
"COMPL == DIFF pred_set.UNIV"
lemma COMPL_DEF: "ALL P::'a::type => bool. COMPL P = DIFF pred_set.UNIV P"
@@ -4323,7 +4504,7 @@
lemma IN_COUNT: "ALL (m::nat) n::nat. IN m (count n) = (m < n)"
by (import pred_set IN_COUNT)
-lemma COUNT_ZERO: "count (0::nat) = EMPTY"
+lemma COUNT_ZERO: "count 0 = EMPTY"
by (import pred_set COUNT_ZERO)
lemma COUNT_SUC: "ALL n::nat. count (Suc n) = INSERT n (count n)"
@@ -4336,8 +4517,7 @@
by (import pred_set CARD_COUNT)
constdefs
- ITSET_tupled :: "('a::type => 'b::type => 'b::type)
-=> ('a::type => bool) * 'b::type => 'b::type"
+ ITSET_tupled :: "('a => 'b => 'b) => ('a => bool) * 'b => 'b"
"ITSET_tupled ==
%f::'a::type => 'b::type => 'b::type.
WFREC
@@ -4370,8 +4550,7 @@
by (import pred_set ITSET_tupled_primitive_def)
constdefs
- ITSET :: "('a::type => 'b::type => 'b::type)
-=> ('a::type => bool) => 'b::type => 'b::type"
+ ITSET :: "('a => 'b => 'b) => ('a => bool) => 'b => 'b"
"ITSET ==
%(f::'a::type => 'b::type => 'b::type) (x::'a::type => bool) x1::'b::type.
ITSET_tupled f (x, x1)"
@@ -4403,7 +4582,7 @@
;setup_theory operator
constdefs
- ASSOC :: "('a::type => 'a::type => 'a::type) => bool"
+ ASSOC :: "('a => 'a => 'a) => bool"
"ASSOC ==
%f::'a::type => 'a::type => 'a::type.
ALL (x::'a::type) (y::'a::type) z::'a::type. f x (f y z) = f (f x y) z"
@@ -4414,7 +4593,7 @@
by (import operator ASSOC_DEF)
constdefs
- COMM :: "('a::type => 'a::type => 'b::type) => bool"
+ COMM :: "('a => 'a => 'b) => bool"
"COMM ==
%f::'a::type => 'a::type => 'b::type.
ALL (x::'a::type) y::'a::type. f x y = f y x"
@@ -4424,8 +4603,7 @@
by (import operator COMM_DEF)
constdefs
- FCOMM :: "('a::type => 'b::type => 'a::type)
-=> ('c::type => 'a::type => 'a::type) => bool"
+ FCOMM :: "('a => 'b => 'a) => ('c => 'a => 'a) => bool"
"FCOMM ==
%(f::'a::type => 'b::type => 'a::type) g::'c::type => 'a::type => 'a::type.
ALL (x::'c::type) (y::'a::type) z::'b::type. g x (f y z) = f (g x y) z"
@@ -4437,7 +4615,7 @@
by (import operator FCOMM_DEF)
constdefs
- RIGHT_ID :: "('a::type => 'b::type => 'a::type) => 'b::type => bool"
+ RIGHT_ID :: "('a => 'b => 'a) => 'b => bool"
"RIGHT_ID ==
%(f::'a::type => 'b::type => 'a::type) e::'b::type.
ALL x::'a::type. f x e = x"
@@ -4447,7 +4625,7 @@
by (import operator RIGHT_ID_DEF)
constdefs
- LEFT_ID :: "('a::type => 'b::type => 'b::type) => 'a::type => bool"
+ LEFT_ID :: "('a => 'b => 'b) => 'a => bool"
"LEFT_ID ==
%(f::'a::type => 'b::type => 'b::type) e::'a::type.
ALL x::'b::type. f e x = x"
@@ -4457,7 +4635,7 @@
by (import operator LEFT_ID_DEF)
constdefs
- MONOID :: "('a::type => 'a::type => 'a::type) => 'a::type => bool"
+ MONOID :: "('a => 'a => 'a) => 'a => bool"
"MONOID ==
%(f::'a::type => 'a::type => 'a::type) e::'a::type.
ASSOC f & RIGHT_ID f e & LEFT_ID f e"
@@ -4486,7 +4664,7 @@
;setup_theory rich_list
consts
- SNOC :: "'a::type => 'a::type list => 'a::type list"
+ SNOC :: "'a => 'a list => 'a list"
specification (SNOC) SNOC: "(ALL x::'a::type. SNOC x [] = [x]) &
(ALL (x::'a::type) (x'::'a::type) l::'a::type list.
@@ -4494,8 +4672,7 @@
by (import rich_list SNOC)
consts
- SCANL :: "('b::type => 'a::type => 'b::type)
-=> 'b::type => 'a::type list => 'b::type list"
+ SCANL :: "('b => 'a => 'b) => 'b => 'a list => 'b list"
specification (SCANL) SCANL: "(ALL (f::'b::type => 'a::type => 'b::type) e::'b::type.
SCANL f e [] = [e]) &
@@ -4504,8 +4681,7 @@
by (import rich_list SCANL)
consts
- SCANR :: "('a::type => 'b::type => 'b::type)
-=> 'b::type => 'a::type list => 'b::type list"
+ SCANR :: "('a => 'b => 'b) => 'b => 'a list => 'b list"
specification (SCANR) SCANR: "(ALL (f::'a::type => 'b::type => 'b::type) e::'b::type.
SCANR f e [] = [e]) &
@@ -4532,27 +4708,27 @@
by (import rich_list OR_EL_DEF)
consts
- FIRSTN :: "nat => 'a::type list => 'a::type list"
-
-specification (FIRSTN) FIRSTN: "(ALL l::'a::type list. FIRSTN (0::nat) l = []) &
+ FIRSTN :: "nat => 'a list => 'a list"
+
+specification (FIRSTN) FIRSTN: "(ALL l::'a::type list. FIRSTN 0 l = []) &
(ALL (n::nat) (x::'a::type) l::'a::type list.
FIRSTN (Suc n) (x # l) = x # FIRSTN n l)"
by (import rich_list FIRSTN)
consts
- BUTFIRSTN :: "nat => 'a::type list => 'a::type list"
-
-specification (BUTFIRSTN) BUTFIRSTN: "(ALL l::'a::type list. BUTFIRSTN (0::nat) l = l) &
+ BUTFIRSTN :: "nat => 'a list => 'a list"
+
+specification (BUTFIRSTN) BUTFIRSTN: "(ALL l::'a::type list. BUTFIRSTN 0 l = l) &
(ALL (n::nat) (x::'a::type) l::'a::type list.
BUTFIRSTN (Suc n) (x # l) = BUTFIRSTN n l)"
by (import rich_list BUTFIRSTN)
consts
- SEG :: "nat => nat => 'a::type list => 'a::type list"
-
-specification (SEG) SEG: "(ALL (k::nat) l::'a::type list. SEG (0::nat) k l = []) &
+ SEG :: "nat => nat => 'a list => 'a list"
+
+specification (SEG) SEG: "(ALL (k::nat) l::'a::type list. SEG 0 k l = []) &
(ALL (m::nat) (x::'a::type) l::'a::type list.
- SEG (Suc m) (0::nat) (x # l) = x # SEG m (0::nat) l) &
+ SEG (Suc m) 0 (x # l) = x # SEG m 0 l) &
(ALL (m::nat) (k::nat) (x::'a::type) l::'a::type list.
SEG (Suc m) (Suc k) (x # l) = SEG (Suc m) k l)"
by (import rich_list SEG)
@@ -4564,34 +4740,34 @@
by (import rich_list BUTLAST)
consts
- LASTN :: "nat => 'a::type list => 'a::type list"
-
-specification (LASTN) LASTN: "(ALL l::'a::type list. LASTN (0::nat) l = []) &
+ LASTN :: "nat => 'a list => 'a list"
+
+specification (LASTN) LASTN: "(ALL l::'a::type list. LASTN 0 l = []) &
(ALL (n::nat) (x::'a::type) l::'a::type list.
LASTN (Suc n) (SNOC x l) = SNOC x (LASTN n l))"
by (import rich_list LASTN)
consts
- BUTLASTN :: "nat => 'a::type list => 'a::type list"
-
-specification (BUTLASTN) BUTLASTN: "(ALL l::'a::type list. BUTLASTN (0::nat) l = l) &
+ BUTLASTN :: "nat => 'a list => 'a list"
+
+specification (BUTLASTN) BUTLASTN: "(ALL l::'a::type list. BUTLASTN 0 l = l) &
(ALL (n::nat) (x::'a::type) l::'a::type list.
BUTLASTN (Suc n) (SNOC x l) = BUTLASTN n l)"
by (import rich_list BUTLASTN)
-lemma EL: "(ALL x::'a::type list. EL (0::nat) x = hd x) &
+lemma EL: "(ALL x::'a::type list. EL 0 x = hd x) &
(ALL (x::nat) xa::'a::type list. EL (Suc x) xa = EL x (tl xa))"
by (import rich_list EL)
consts
- ELL :: "nat => 'a::type list => 'a::type"
-
-specification (ELL) ELL: "(ALL l::'a::type list. ELL (0::nat) l = last l) &
+ ELL :: "nat => 'a list => 'a"
+
+specification (ELL) ELL: "(ALL l::'a::type list. ELL 0 l = last l) &
(ALL (n::nat) l::'a::type list. ELL (Suc n) l = ELL n (butlast l))"
by (import rich_list ELL)
consts
- IS_PREFIX :: "'a::type list => 'a::type list => bool"
+ IS_PREFIX :: "'a list => 'a list => bool"
specification (IS_PREFIX) IS_PREFIX: "(ALL l::'a::type list. IS_PREFIX l [] = True) &
(ALL (x::'a::type) l::'a::type list. IS_PREFIX [] (x # l) = False) &
@@ -4616,7 +4792,7 @@
by (import rich_list SNOC_Axiom)
consts
- IS_SUFFIX :: "'a::type list => 'a::type list => bool"
+ IS_SUFFIX :: "'a list => 'a list => bool"
specification (IS_SUFFIX) IS_SUFFIX: "(ALL l::'a::type list. IS_SUFFIX l [] = True) &
(ALL (x::'a::type) l::'a::type list. IS_SUFFIX [] (SNOC x l) = False) &
@@ -4625,7 +4801,7 @@
by (import rich_list IS_SUFFIX)
consts
- IS_SUBLIST :: "'a::type list => 'a::type list => bool"
+ IS_SUBLIST :: "'a list => 'a list => bool"
specification (IS_SUBLIST) IS_SUBLIST: "(ALL l::'a::type list. IS_SUBLIST l [] = True) &
(ALL (x::'a::type) l::'a::type list. IS_SUBLIST [] (x # l) = False) &
@@ -4635,7 +4811,7 @@
by (import rich_list IS_SUBLIST)
consts
- SPLITP :: "('a::type => bool) => 'a::type list => 'a::type list * 'a::type list"
+ SPLITP :: "('a => bool) => 'a list => 'a list * 'a list"
specification (SPLITP) SPLITP: "(ALL P::'a::type => bool. SPLITP P [] = ([], [])) &
(ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
@@ -4644,7 +4820,7 @@
by (import rich_list SPLITP)
constdefs
- PREFIX :: "('a::type => bool) => 'a::type list => 'a::type list"
+ PREFIX :: "('a => bool) => 'a list => 'a list"
"PREFIX == %(P::'a::type => bool) l::'a::type list. fst (SPLITP (Not o P) l)"
lemma PREFIX_DEF: "ALL (P::'a::type => bool) l::'a::type list.
@@ -4652,7 +4828,7 @@
by (import rich_list PREFIX_DEF)
constdefs
- SUFFIX :: "('a::type => bool) => 'a::type list => 'a::type list"
+ SUFFIX :: "('a => bool) => 'a list => 'a list"
"SUFFIX ==
%P::'a::type => bool.
foldl (%(l'::'a::type list) x::'a::type. if P x then SNOC x l' else [])
@@ -4665,31 +4841,31 @@
by (import rich_list SUFFIX_DEF)
constdefs
- UNZIP_FST :: "('a::type * 'b::type) list => 'a::type list"
+ UNZIP_FST :: "('a * 'b) list => 'a list"
"UNZIP_FST == %l::('a::type * 'b::type) list. fst (unzip l)"
lemma UNZIP_FST_DEF: "ALL l::('a::type * 'b::type) list. UNZIP_FST l = fst (unzip l)"
by (import rich_list UNZIP_FST_DEF)
constdefs
- UNZIP_SND :: "('a::type * 'b::type) list => 'b::type list"
+ UNZIP_SND :: "('a * 'b) list => 'b list"
"UNZIP_SND == %l::('a::type * 'b::type) list. snd (unzip l)"
lemma UNZIP_SND_DEF: "ALL l::('a::type * 'b::type) list. UNZIP_SND l = snd (unzip l)"
by (import rich_list UNZIP_SND_DEF)
consts
- GENLIST :: "(nat => 'a::type) => nat => 'a::type list"
-
-specification (GENLIST) GENLIST: "(ALL f::nat => 'a::type. GENLIST f (0::nat) = []) &
+ GENLIST :: "(nat => 'a) => nat => 'a list"
+
+specification (GENLIST) GENLIST: "(ALL f::nat => 'a::type. GENLIST f 0 = []) &
(ALL (f::nat => 'a::type) n::nat.
GENLIST f (Suc n) = SNOC (f n) (GENLIST f n))"
by (import rich_list GENLIST)
consts
- REPLICATE :: "nat => 'a::type => 'a::type list"
-
-specification (REPLICATE) REPLICATE: "(ALL x::'a::type. REPLICATE (0::nat) x = []) &
+ REPLICATE :: "nat => 'a => 'a list"
+
+specification (REPLICATE) REPLICATE: "(ALL x::'a::type. REPLICATE 0 x = []) &
(ALL (n::nat) x::'a::type. REPLICATE (Suc n) x = x # REPLICATE n x)"
by (import rich_list REPLICATE)
@@ -4706,7 +4882,7 @@
lemma LENGTH_EQ: "ALL (x::'a::type list) y::'a::type list. x = y --> length x = length y"
by (import rich_list LENGTH_EQ)
-lemma LENGTH_NOT_NULL: "ALL l::'a::type list. ((0::nat) < length l) = (~ null l)"
+lemma LENGTH_NOT_NULL: "ALL l::'a::type list. (0 < length l) = (~ null l)"
by (import rich_list LENGTH_NOT_NULL)
lemma SNOC_INDUCT: "ALL P::'a::type list => bool.
@@ -4755,11 +4931,10 @@
MONOID f e --> (ALL l::'a::type list. foldr f l e = foldl f e l)"
by (import rich_list FOLDR_FOLDL)
-lemma LENGTH_FOLDR: "ALL l::'a::type list. length l = foldr (%x::'a::type. Suc) l (0::nat)"
+lemma LENGTH_FOLDR: "ALL l::'a::type list. length l = foldr (%x::'a::type. Suc) l 0"
by (import rich_list LENGTH_FOLDR)
-lemma LENGTH_FOLDL: "ALL l::'a::type list.
- length l = foldl (%(l'::nat) x::'a::type. Suc l') (0::nat) l"
+lemma LENGTH_FOLDL: "ALL l::'a::type list. length l = foldl (%(l'::nat) x::'a::type. Suc l') 0 l"
by (import rich_list LENGTH_FOLDL)
lemma MAP_FOLDR: "ALL (f::'a::type => 'b::type) l::'a::type list.
@@ -4870,7 +5045,7 @@
lemma SUM_SNOC: "ALL (x::nat) l::nat list. sum (SNOC x l) = sum l + x"
by (import rich_list SUM_SNOC)
-lemma SUM_FOLDL: "ALL l::nat list. sum l = foldl op + (0::nat) l"
+lemma SUM_FOLDL: "ALL l::nat list. sum l = foldl op + 0 l"
by (import rich_list SUM_FOLDL)
lemma IS_PREFIX_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
@@ -5077,7 +5252,7 @@
lemma NULL_FOLDL: "ALL l::'a::type list. null l = foldl (%(x::bool) l'::'a::type. False) True l"
by (import rich_list NULL_FOLDL)
-lemma SEG_LENGTH_ID: "ALL l::'a::type list. SEG (length l) (0::nat) l = l"
+lemma SEG_LENGTH_ID: "ALL l::'a::type list. SEG (length l) 0 l = l"
by (import rich_list SEG_LENGTH_ID)
lemma SEG_SUC_CONS: "ALL (m::nat) (n::nat) (l::'a::type list) x::'a::type.
@@ -5085,11 +5260,11 @@
by (import rich_list SEG_SUC_CONS)
lemma SEG_0_SNOC: "ALL (m::nat) (l::'a::type list) x::'a::type.
- m <= length l --> SEG m (0::nat) (SNOC x l) = SEG m (0::nat) l"
+ m <= length l --> SEG m 0 (SNOC x l) = SEG m 0 l"
by (import rich_list SEG_0_SNOC)
lemma BUTLASTN_SEG: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTLASTN n l = SEG (length l - n) (0::nat) l"
+ n <= length l --> BUTLASTN n l = SEG (length l - n) 0 l"
by (import rich_list BUTLASTN_SEG)
lemma LASTN_CONS: "ALL (n::nat) l::'a::type list.
@@ -5160,7 +5335,7 @@
by (import rich_list BUTLASTN_LENGTH_CONS)
lemma LAST_LASTN_LAST: "ALL (n::nat) l::'a::type list.
- n <= length l --> (0::nat) < n --> last (LASTN n l) = last l"
+ n <= length l --> 0 < n --> last (LASTN n l) = last l"
by (import rich_list LAST_LASTN_LAST)
lemma BUTLASTN_LASTN_NIL: "ALL (n::nat) l::'a::type list. n <= length l --> BUTLASTN n (LASTN n l) = []"
@@ -5176,10 +5351,10 @@
BUTLASTN m (LASTN n l) = LASTN (n - m) (BUTLASTN m l)"
by (import rich_list BUTLASTN_LASTN)
-lemma LASTN_1: "ALL l::'a::type list. l ~= [] --> LASTN (1::nat) l = [last l]"
+lemma LASTN_1: "ALL l::'a::type list. l ~= [] --> LASTN 1 l = [last l]"
by (import rich_list LASTN_1)
-lemma BUTLASTN_1: "ALL l::'a::type list. l ~= [] --> BUTLASTN (1::nat) l = butlast l"
+lemma BUTLASTN_1: "ALL l::'a::type list. l ~= [] --> BUTLASTN 1 l = butlast l"
by (import rich_list BUTLASTN_1)
lemma BUTLASTN_APPEND1: "ALL (l2::'a::type list) n::nat.
@@ -5292,8 +5467,7 @@
n <= length l --> LASTN n l = SEG n (length l - n) l"
by (import rich_list LASTN_SEG)
-lemma FIRSTN_SEG: "ALL (n::nat) l::'a::type list.
- n <= length l --> FIRSTN n l = SEG n (0::nat) l"
+lemma FIRSTN_SEG: "ALL (n::nat) l::'a::type list. n <= length l --> FIRSTN n l = SEG n 0 l"
by (import rich_list FIRSTN_SEG)
lemma BUTFIRSTN_SEG: "ALL (n::nat) l::'a::type list.
@@ -5331,11 +5505,10 @@
lemma SEG_APPEND: "ALL (m::nat) (l1::'a::type list) (n::nat) l2::'a::type list.
m < length l1 & length l1 <= n + m & n + m <= length l1 + length l2 -->
SEG n m (l1 @ l2) =
- SEG (length l1 - m) m l1 @ SEG (n + m - length l1) (0::nat) l2"
+ SEG (length l1 - m) m l1 @ SEG (n + m - length l1) 0 l2"
by (import rich_list SEG_APPEND)
-lemma SEG_LENGTH_SNOC: "ALL (x::'a::type list) xa::'a::type.
- SEG (1::nat) (length x) (SNOC xa x) = [xa]"
+lemma SEG_LENGTH_SNOC: "ALL (x::'a::type list) xa::'a::type. SEG 1 (length x) (SNOC xa x) = [xa]"
by (import rich_list SEG_LENGTH_SNOC)
lemma SEG_SNOC: "ALL (n::nat) (m::nat) l::'a::type list.
@@ -5343,7 +5516,7 @@
by (import rich_list SEG_SNOC)
lemma ELL_SEG: "ALL (n::nat) l::'a::type list.
- n < length l --> ELL n l = hd (SEG (1::nat) (PRE (length l - n)) l)"
+ n < length l --> ELL n l = hd (SEG 1 (PRE (length l - n)) l)"
by (import rich_list ELL_SEG)
lemma SNOC_FOLDR: "ALL (x::'a::type) l::'a::type list. SNOC x l = foldr op # l [x]"
@@ -5405,13 +5578,13 @@
(ALL (e::'a::type) l::'a::type list. foldl f e (rev l) = foldl f e l)"
by (import rich_list COMM_ASSOC_FOLDL_REVERSE)
-lemma ELL_LAST: "ALL l::'a::type list. ~ null l --> ELL (0::nat) l = last l"
+lemma ELL_LAST: "ALL l::'a::type list. ~ null l --> ELL 0 l = last l"
by (import rich_list ELL_LAST)
-lemma ELL_0_SNOC: "ALL (l::'a::type list) x::'a::type. ELL (0::nat) (SNOC x l) = x"
+lemma ELL_0_SNOC: "ALL (l::'a::type list) x::'a::type. ELL 0 (SNOC x l) = x"
by (import rich_list ELL_0_SNOC)
-lemma ELL_SNOC: "ALL n>0::nat.
+lemma ELL_SNOC: "ALL n>0.
ALL (x::'a::type) l::'a::type list. ELL n (SNOC x l) = ELL (PRE n) l"
by (import rich_list ELL_SNOC)
@@ -5642,12 +5815,10 @@
(ALL f::'a::type => 'b::type. EL n (map f l) = f (EL n l))"
by (import rich_list EL_MAP)
-lemma EL_CONS: "ALL n>0::nat.
- ALL (x::'a::type) l::'a::type list. EL n (x # l) = EL (PRE n) l"
+lemma EL_CONS: "ALL n>0. ALL (x::'a::type) l::'a::type list. EL n (x # l) = EL (PRE n) l"
by (import rich_list EL_CONS)
-lemma EL_SEG: "ALL (n::nat) l::'a::type list.
- n < length l --> EL n l = hd (SEG (1::nat) n l)"
+lemma EL_SEG: "ALL (n::nat) l::'a::type list. n < length l --> EL n l = hd (SEG 1 n l)"
by (import rich_list EL_SEG)
lemma EL_IS_EL: "ALL (n::nat) l::'a::type list. n < length l --> EL n l mem l"
@@ -5726,7 +5897,7 @@
lemma LENGTH_REPLICATE: "ALL (n::nat) x::'a::type. length (REPLICATE n x) = n"
by (import rich_list LENGTH_REPLICATE)
-lemma IS_EL_REPLICATE: "ALL n>0::nat. ALL x::'a::type. x mem REPLICATE n x"
+lemma IS_EL_REPLICATE: "ALL n>0. ALL x::'a::type. x mem REPLICATE n x"
by (import rich_list IS_EL_REPLICATE)
lemma ALL_EL_REPLICATE: "ALL (x::'a::type) n::nat. list_all (op = x) (REPLICATE n x)"
@@ -5749,7 +5920,7 @@
;setup_theory state_transformer
constdefs
- UNIT :: "'b::type => 'a::type => 'b::type * 'a::type"
+ UNIT :: "'b => 'a => 'b * 'a"
"(op ==::('b::type => 'a::type => 'b::type * 'a::type)
=> ('b::type => 'a::type => 'b::type * 'a::type) => prop)
(UNIT::'b::type => 'a::type => 'b::type * 'a::type)
@@ -5759,9 +5930,7 @@
by (import state_transformer UNIT_DEF)
constdefs
- BIND :: "('a::type => 'b::type * 'a::type)
-=> ('b::type => 'a::type => 'c::type * 'a::type)
- => 'a::type => 'c::type * 'a::type"
+ BIND :: "('a => 'b * 'a) => ('b => 'a => 'c * 'a) => 'a => 'c * 'a"
"(op ==::(('a::type => 'b::type * 'a::type)
=> ('b::type => 'a::type => 'c::type * 'a::type)
=> 'a::type => 'c::type * 'a::type)
@@ -5802,8 +5971,7 @@
by (import state_transformer BIND_DEF)
constdefs
- MMAP :: "('c::type => 'b::type)
-=> ('a::type => 'c::type * 'a::type) => 'a::type => 'b::type * 'a::type"
+ MMAP :: "('c => 'b) => ('a => 'c * 'a) => 'a => 'b * 'a"
"MMAP ==
%(f::'c::type => 'b::type) m::'a::type => 'c::type * 'a::type.
BIND m (UNIT o f)"
@@ -5813,8 +5981,7 @@
by (import state_transformer MMAP_DEF)
constdefs
- JOIN :: "('a::type => ('a::type => 'b::type * 'a::type) * 'a::type)
-=> 'a::type => 'b::type * 'a::type"
+ JOIN :: "('a => ('a => 'b * 'a) * 'a) => 'a => 'b * 'a"
"JOIN ==
%z::'a::type => ('a::type => 'b::type * 'a::type) * 'a::type. BIND z I"
--- a/src/HOL/Import/HOL/HOL4Prob.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/HOL4Prob.thy Mon Sep 26 16:10:19 2005 +0200
@@ -9,14 +9,11 @@
f = (%b::bool. True) | f = (%b::bool. b) | f = Not"
by (import prob_extra BOOL_BOOL_CASES_THM)
-lemma EVEN_ODD_BASIC: "EVEN (0::nat) &
-~ EVEN (1::nat) &
-EVEN (2::nat) & ~ ODD (0::nat) & ODD (1::nat) & ~ ODD (2::nat)"
+lemma EVEN_ODD_BASIC: "EVEN 0 & ~ EVEN 1 & EVEN 2 & ~ ODD 0 & ODD 1 & ~ ODD 2"
by (import prob_extra EVEN_ODD_BASIC)
lemma EVEN_ODD_EXISTS_EQ: "ALL n::nat.
- EVEN n = (EX m::nat. n = (2::nat) * m) &
- ODD n = (EX m::nat. n = Suc ((2::nat) * m))"
+ EVEN n = (EX m::nat. n = 2 * m) & ODD n = (EX m::nat. n = Suc (2 * m))"
by (import prob_extra EVEN_ODD_EXISTS_EQ)
lemma DIV_THEN_MULT: "ALL (p::nat) q::nat. Suc q * (p div Suc q) <= p"
@@ -60,20 +57,42 @@
(bit.B0::bit)))))))))"
by (import prob_extra DIV_TWO_UNIQUE)
-lemma DIVISION_TWO: "ALL n::nat.
- n = (2::nat) * (n div (2::nat)) + n mod (2::nat) &
- (n mod (2::nat) = (0::nat) | n mod (2::nat) = (1::nat))"
+lemma DIVISION_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2 & (n mod 2 = 0 | n mod 2 = 1)"
by (import prob_extra DIVISION_TWO)
-lemma DIV_TWO: "ALL n::nat. n = (2::nat) * (n div (2::nat)) + n mod (2::nat)"
+lemma DIV_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2"
by (import prob_extra DIV_TWO)
-lemma MOD_TWO: "(ALL (n::nat).
- ((n mod (2::nat)) = (if (EVEN n) then (0::nat) else (1::nat))))"
+lemma MOD_TWO: "ALL n::nat. n mod 2 = (if EVEN n then 0 else 1)"
by (import prob_extra MOD_TWO)
-lemma DIV_TWO_BASIC: "(0::nat) div (2::nat) = (0::nat) &
-(1::nat) div (2::nat) = (0::nat) & (2::nat) div (2::nat) = (1::nat)"
+lemma DIV_TWO_BASIC: "(op &::bool => bool => bool)
+ ((op =::nat => nat => bool)
+ ((op div::nat => nat => nat) (0::nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ (0::nat))
+ ((op &::bool => bool => bool)
+ ((op =::nat => nat => bool)
+ ((op div::nat => nat => nat) (1::nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ (0::nat))
+ ((op =::nat => nat => bool)
+ ((op div::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ (1::nat)))"
by (import prob_extra DIV_TWO_BASIC)
lemma DIV_TWO_MONO: "(All::(nat => bool) => bool)
@@ -119,18 +138,35 @@
((op <::nat => nat => bool) m n))))"
by (import prob_extra DIV_TWO_MONO_EVEN)
-lemma DIV_TWO_CANCEL: "ALL n::nat.
- (2::nat) * n div (2::nat) = n & Suc ((2::nat) * n) div (2::nat) = n"
+lemma DIV_TWO_CANCEL: "ALL n::nat. 2 * n div 2 = n & Suc (2 * n) div 2 = n"
by (import prob_extra DIV_TWO_CANCEL)
-lemma EXP_DIV_TWO: "ALL n::nat. (2::nat) ^ Suc n div (2::nat) = (2::nat) ^ n"
+lemma EXP_DIV_TWO: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op =::nat => nat => bool)
+ ((op div::nat => nat => nat)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ ((Suc::nat => nat) n))
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ n))"
by (import prob_extra EXP_DIV_TWO)
-lemma EVEN_EXP_TWO: "ALL n::nat. EVEN ((2::nat) ^ n) = (n ~= (0::nat))"
+lemma EVEN_EXP_TWO: "ALL n::nat. EVEN (2 ^ n) = (n ~= 0)"
by (import prob_extra EVEN_EXP_TWO)
-lemma DIV_TWO_EXP: "ALL (n::nat) k::nat.
- (k div (2::nat) < (2::nat) ^ n) = (k < (2::nat) ^ Suc n)"
+lemma DIV_TWO_EXP: "ALL (n::nat) k::nat. (k div 2 < 2 ^ n) = (k < 2 ^ Suc n)"
by (import prob_extra DIV_TWO_EXP)
consts
@@ -195,13 +231,38 @@
z)))"
by (import prob_extra REAL_INF_MIN)
-lemma HALF_POS: "(0::real) < (1::real) / (2::real)"
+lemma HALF_POS: "(op <::real => real => bool) (0::real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))"
by (import prob_extra HALF_POS)
-lemma HALF_CANCEL: "(2::real) * ((1::real) / (2::real)) = (1::real)"
+lemma HALF_CANCEL: "(op =::real => real => bool)
+ ((op *::real => real => real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))))
+ (1::real)"
by (import prob_extra HALF_CANCEL)
-lemma POW_HALF_POS: "ALL n::nat. (0::real) < ((1::real) / (2::real)) ^ n"
+lemma POW_HALF_POS: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op <::real => real => bool) (0::real)
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ n))"
by (import prob_extra POW_HALF_POS)
lemma POW_HALF_MONO: "(All::(nat => bool) => bool)
@@ -228,11 +289,32 @@
m))))"
by (import prob_extra POW_HALF_MONO)
-lemma POW_HALF_TWICE: "ALL n::nat.
- ((1::real) / (2::real)) ^ n = (2::real) * ((1::real) / (2::real)) ^ Suc n"
+lemma POW_HALF_TWICE: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op =::real => real => bool)
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ n)
+ ((op *::real => real => real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit))))
+ ((Suc::nat => nat) n))))"
by (import prob_extra POW_HALF_TWICE)
-lemma X_HALF_HALF: "ALL x::real. (1::real) / (2::real) * x + (1::real) / (2::real) * x = x"
+lemma X_HALF_HALF: "ALL x::real. 1 / 2 * x + 1 / 2 * x = x"
by (import prob_extra X_HALF_HALF)
lemma REAL_SUP_LE_X: "(All::((real => bool) => bool) => bool)
@@ -271,27 +353,60 @@
by (import prob_extra REAL_X_LE_SUP)
lemma ABS_BETWEEN_LE: "ALL (x::real) (y::real) d::real.
- ((0::real) <= d & x - d <= y & y <= x + d) = (abs (y - x) <= d)"
+ (0 <= d & x - d <= y & y <= x + d) = (abs (y - x) <= d)"
by (import prob_extra ABS_BETWEEN_LE)
-lemma ONE_MINUS_HALF: "(1::real) - (1::real) / (2::real) = (1::real) / (2::real)"
+lemma ONE_MINUS_HALF: "(op =::real => real => bool)
+ ((op -::real => real => real) (1::real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))))
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))"
by (import prob_extra ONE_MINUS_HALF)
-lemma HALF_LT_1: "(1::real) / (2::real) < (1::real)"
+lemma HALF_LT_1: "(op <::real => real => bool)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ (1::real)"
by (import prob_extra HALF_LT_1)
-lemma POW_HALF_EXP: "ALL n::nat. ((1::real) / (2::real)) ^ n = inverse (real ((2::nat) ^ n))"
+lemma POW_HALF_EXP: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op =::real => real => bool)
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ n)
+ ((inverse::real => real)
+ ((real::nat => real)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ n))))"
by (import prob_extra POW_HALF_EXP)
-lemma INV_SUC_POS: "ALL n::nat. (0::real) < (1::real) / real (Suc n)"
+lemma INV_SUC_POS: "ALL n::nat. 0 < 1 / real (Suc n)"
by (import prob_extra INV_SUC_POS)
-lemma INV_SUC_MAX: "ALL x::nat. (1::real) / real (Suc x) <= (1::real)"
+lemma INV_SUC_MAX: "ALL x::nat. 1 / real (Suc x) <= 1"
by (import prob_extra INV_SUC_MAX)
-lemma INV_SUC: "ALL n::nat.
- (0::real) < (1::real) / real (Suc n) &
- (1::real) / real (Suc n) <= (1::real)"
+lemma INV_SUC: "ALL n::nat. 0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
by (import prob_extra INV_SUC)
lemma ABS_UNIT_INTERVAL: "(All::(real => bool) => bool)
@@ -894,12 +1009,10 @@
defs
alg_longest_primdef: "alg_longest ==
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t)
- (0::nat)"
+FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
lemma alg_longest_def: "alg_longest =
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t)
- (0::nat)"
+FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
by (import prob_canon alg_longest_def)
consts
@@ -1948,9 +2061,9 @@
SHD :: "(nat => bool) => bool"
defs
- SHD_primdef: "SHD == %f::nat => bool. f (0::nat)"
-
-lemma SHD_def: "ALL f::nat => bool. SHD f = f (0::nat)"
+ SHD_primdef: "SHD == %f::nat => bool. f 0"
+
+lemma SHD_def: "ALL f::nat => bool. SHD f = f 0"
by (import boolean_sequence SHD_def)
consts
@@ -1965,7 +2078,7 @@
consts
SCONS :: "bool => (nat => bool) => nat => bool"
-specification (SCONS_primdef: SCONS) SCONS_def: "(ALL (h::bool) t::nat => bool. SCONS h t (0::nat) = h) &
+specification (SCONS_primdef: SCONS) SCONS_def: "(ALL (h::bool) t::nat => bool. SCONS h t 0 = h) &
(ALL (h::bool) (t::nat => bool) n::nat. SCONS h t (Suc n) = t n)"
by (import boolean_sequence SCONS_def)
@@ -1990,14 +2103,14 @@
consts
STAKE :: "nat => (nat => bool) => bool list"
-specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s::nat => bool. STAKE (0::nat) s = []) &
+specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s::nat => bool. STAKE 0 s = []) &
(ALL (n::nat) s::nat => bool. STAKE (Suc n) s = SHD s # STAKE n (STL s))"
by (import boolean_sequence STAKE_def)
consts
SDROP :: "nat => (nat => bool) => nat => bool"
-specification (SDROP_primdef: SDROP) SDROP_def: "SDROP (0::nat) = I & (ALL n::nat. SDROP (Suc n) = SDROP n o STL)"
+specification (SDROP_primdef: SDROP) SDROP_def: "SDROP 0 = I & (ALL n::nat. SDROP (Suc n) = SDROP n o STL)"
by (import boolean_sequence SDROP_def)
lemma SCONS_SURJ: "ALL x::nat => bool. EX (xa::bool) t::nat => bool. x = SCONS xa t"
@@ -2271,10 +2384,9 @@
consts
alg_measure :: "bool list list => real"
-specification (alg_measure_primdef: alg_measure) alg_measure_def: "alg_measure [] = (0::real) &
+specification (alg_measure_primdef: alg_measure) alg_measure_def: "alg_measure [] = 0 &
(ALL (l::bool list) rest::bool list list.
- alg_measure (l # rest) =
- ((1::real) / (2::real)) ^ length l + alg_measure rest)"
+ alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
by (import prob alg_measure_def)
consts
@@ -2311,18 +2423,42 @@
algebra_measure b = r & SUBSET (algebra_embed b) s)"
by (import prob prob_def)
-lemma ALG_TWINS_MEASURE: "ALL l::bool list.
- ((1::real) / (2::real)) ^ length (SNOC True l) +
- ((1::real) / (2::real)) ^ length (SNOC False l) =
- ((1::real) / (2::real)) ^ length l"
+lemma ALG_TWINS_MEASURE: "(All::(bool list => bool) => bool)
+ (%l::bool list.
+ (op =::real => real => bool)
+ ((op +::real => real => real)
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit))))
+ ((size::bool list => nat)
+ ((SNOC::bool => bool list => bool list) (True::bool) l)))
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit))))
+ ((size::bool list => nat)
+ ((SNOC::bool => bool list => bool list) (False::bool) l))))
+ ((op ^::real => nat => real)
+ ((op /::real => real => real) (1::real)
+ ((number_of::bin => real)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))
+ ((size::bool list => nat) l)))"
by (import prob ALG_TWINS_MEASURE)
-lemma ALG_MEASURE_BASIC: "alg_measure [] = (0::real) &
-alg_measure [[]] = (1::real) &
-(ALL b::bool. alg_measure [[b]] = (1::real) / (2::real))"
+lemma ALG_MEASURE_BASIC: "alg_measure [] = 0 &
+alg_measure [[]] = 1 & (ALL b::bool. alg_measure [[b]] = 1 / 2)"
by (import prob ALG_MEASURE_BASIC)
-lemma ALG_MEASURE_POS: "ALL l::bool list list. (0::real) <= alg_measure l"
+lemma ALG_MEASURE_POS: "ALL l::bool list list. 0 <= alg_measure l"
by (import prob ALG_MEASURE_POS)
lemma ALG_MEASURE_APPEND: "ALL (l1::bool list list) l2::bool list list.
@@ -2330,7 +2466,7 @@
by (import prob ALG_MEASURE_APPEND)
lemma ALG_MEASURE_TLS: "ALL (l::bool list list) b::bool.
- (2::real) * alg_measure (map (op # b) l) = alg_measure l"
+ 2 * alg_measure (map (op # b) l) = alg_measure l"
by (import prob ALG_MEASURE_TLS)
lemma ALG_CANON_PREFS_MONO: "ALL (l::bool list) b::bool list list.
@@ -2357,9 +2493,8 @@
lemma ALGEBRA_MEASURE_DEF_ALT: "ALL l::bool list list. algebra_measure l = alg_measure (alg_canon l)"
by (import prob ALGEBRA_MEASURE_DEF_ALT)
-lemma ALGEBRA_MEASURE_BASIC: "algebra_measure [] = (0::real) &
-algebra_measure [[]] = (1::real) &
-(ALL b::bool. algebra_measure [[b]] = (1::real) / (2::real))"
+lemma ALGEBRA_MEASURE_BASIC: "algebra_measure [] = 0 &
+algebra_measure [[]] = 1 & (ALL b::bool. algebra_measure [[b]] = 1 / 2)"
by (import prob ALGEBRA_MEASURE_BASIC)
lemma ALGEBRA_CANON_MEASURE_MAX: "(All::(bool list list => bool) => bool)
@@ -2370,7 +2505,7 @@
((alg_measure::bool list list => real) l) (1::real)))"
by (import prob ALGEBRA_CANON_MEASURE_MAX)
-lemma ALGEBRA_MEASURE_MAX: "ALL l::bool list list. algebra_measure l <= (1::real)"
+lemma ALGEBRA_MEASURE_MAX: "ALL l::bool list list. algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_MAX)
lemma ALGEBRA_MEASURE_MONO_EMBED: "(All::(bool list list => bool) => bool)
@@ -2458,9 +2593,9 @@
lemma PROB_ALGEBRA: "ALL l::bool list list. prob (algebra_embed l) = algebra_measure l"
by (import prob PROB_ALGEBRA)
-lemma PROB_BASIC: "prob EMPTY = (0::real) &
-prob pred_set.UNIV = (1::real) &
-(ALL b::bool. prob (%s::nat => bool. SHD s = b) = (1::real) / (2::real))"
+lemma PROB_BASIC: "prob EMPTY = 0 &
+prob pred_set.UNIV = 1 &
+(ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2)"
by (import prob PROB_BASIC)
lemma PROB_ADDITIVE: "(All::(((nat => bool) => bool) => bool) => bool)
@@ -2578,7 +2713,7 @@
((prob::((nat => bool) => bool) => real) t))))"
by (import prob PROB_SUBSET_MONO)
-lemma PROB_ALG: "ALL x::bool list. prob (alg_embed x) = ((1::real) / (2::real)) ^ length x"
+lemma PROB_ALG: "ALL x::bool list. prob (alg_embed x) = (1 / 2) ^ length x"
by (import prob PROB_ALG)
lemma PROB_STL: "(All::(((nat => bool) => bool) => bool) => bool)
@@ -2662,26 +2797,25 @@
((prob::((nat => bool) => bool) => real) p)))))"
by (import prob PROB_INTER_SHD)
-lemma ALGEBRA_MEASURE_POS: "ALL l::bool list list. (0::real) <= algebra_measure l"
+lemma ALGEBRA_MEASURE_POS: "ALL l::bool list list. 0 <= algebra_measure l"
by (import prob ALGEBRA_MEASURE_POS)
-lemma ALGEBRA_MEASURE_RANGE: "ALL l::bool list list.
- (0::real) <= algebra_measure l & algebra_measure l <= (1::real)"
+lemma ALGEBRA_MEASURE_RANGE: "ALL l::bool list list. 0 <= algebra_measure l & algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_RANGE)
-lemma PROB_POS: "ALL p::(nat => bool) => bool. (0::real) <= prob p"
+lemma PROB_POS: "ALL p::(nat => bool) => bool. 0 <= prob p"
by (import prob PROB_POS)
-lemma PROB_MAX: "ALL p::(nat => bool) => bool. prob p <= (1::real)"
+lemma PROB_MAX: "ALL p::(nat => bool) => bool. prob p <= 1"
by (import prob PROB_MAX)
-lemma PROB_RANGE: "ALL p::(nat => bool) => bool. (0::real) <= prob p & prob p <= (1::real)"
+lemma PROB_RANGE: "ALL p::(nat => bool) => bool. 0 <= prob p & prob p <= 1"
by (import prob PROB_RANGE)
lemma ABS_PROB: "ALL p::(nat => bool) => bool. abs (prob p) = prob p"
by (import prob ABS_PROB)
-lemma PROB_SHD: "ALL b::bool. prob (%s::nat => bool. SHD s = b) = (1::real) / (2::real)"
+lemma PROB_SHD: "ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2"
by (import prob PROB_SHD)
lemma PROB_COMPL_LE1: "(All::(((nat => bool) => bool) => bool) => bool)
@@ -2719,11 +2853,10 @@
defs
pseudo_linear_tl_primdef: "pseudo_linear_tl ==
-%(a::nat) (b::nat) (n::nat) x::nat.
- (a * x + b) mod ((2::nat) * n + (1::nat))"
+%(a::nat) (b::nat) (n::nat) x::nat. (a * x + b) mod (2 * n + 1)"
lemma pseudo_linear_tl_def: "ALL (a::nat) (b::nat) (n::nat) x::nat.
- pseudo_linear_tl a b n x = (a * x + b) mod ((2::nat) * n + (1::nat))"
+ pseudo_linear_tl a b n x = (a * x + b) mod (2 * n + 1)"
by (import prob_pseudo pseudo_linear_tl_def)
lemma PSEUDO_LINEAR1_EXECUTE: "EX x::nat => nat => bool.
@@ -2828,7 +2961,7 @@
by (import prob_indep alg_cover_def)
consts
- indep :: "((nat => bool) => 'a::type * (nat => bool)) => bool"
+ indep :: "((nat => bool) => 'a * (nat => bool)) => bool"
defs
indep_primdef: "indep ==
@@ -3520,21 +3653,19 @@
defs
unif_bound_primdef: "unif_bound ==
WFREC
- (SOME R::nat => nat => bool.
- WF R & (ALL v::nat. R (Suc v div (2::nat)) (Suc v)))
+ (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
(%unif_bound::nat => nat.
- nat_case (0::nat) (%v1::nat. Suc (unif_bound (Suc v1 div (2::nat)))))"
+ nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
lemma unif_bound_primitive_def: "unif_bound =
WFREC
- (SOME R::nat => nat => bool.
- WF R & (ALL v::nat. R (Suc v div (2::nat)) (Suc v)))
+ (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
(%unif_bound::nat => nat.
- nat_case (0::nat) (%v1::nat. Suc (unif_bound (Suc v1 div (2::nat)))))"
+ nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
by (import prob_uniform unif_bound_primitive_def)
-lemma unif_bound_def: "unif_bound (0::nat) = (0::nat) &
-unif_bound (Suc (v::nat)) = Suc (unif_bound (Suc v div (2::nat)))"
+lemma unif_bound_def: "unif_bound 0 = 0 &
+unif_bound (Suc (v::nat)) = Suc (unif_bound (Suc v div 2))"
by (import prob_uniform unif_bound_def)
lemma unif_bound_ind: "(All::((nat => bool) => bool) => bool)
@@ -3559,30 +3690,24 @@
"unif_tupled ==
WFREC
(SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R &
- (ALL (s::nat => bool) v2::nat. R (Suc v2 div (2::nat), s) (Suc v2, s)))
+ WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
(%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
v1::nat => bool).
- case v of 0 => (0::nat, v1)
+ case v of 0 => (0, v1)
| Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) =
- unif_tupled (Suc v3 div (2::nat), v1)
- in (if SHD s' then (2::nat) * m + (1::nat) else (2::nat) * m,
- STL s'))"
+ let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
lemma unif_tupled_primitive_def: "unif_tupled =
WFREC
(SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R &
- (ALL (s::nat => bool) v2::nat. R (Suc v2 div (2::nat), s) (Suc v2, s)))
+ WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
(%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
v1::nat => bool).
- case v of 0 => (0::nat, v1)
+ case v of 0 => (0, v1)
| Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) =
- unif_tupled (Suc v3 div (2::nat), v1)
- in (if SHD s' then (2::nat) * m + (1::nat) else (2::nat) * m,
- STL s'))"
+ let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_tupled_primitive_def)
consts
@@ -3594,10 +3719,10 @@
lemma unif_curried_def: "ALL (x::nat) x1::nat => bool. unif x x1 = unif_tupled (x, x1)"
by (import prob_uniform unif_curried_def)
-lemma unif_def: "unif (0::nat) (s::nat => bool) = (0::nat, s) &
+lemma unif_def: "unif 0 (s::nat => bool) = (0, s) &
unif (Suc (v2::nat)) s =
-(let (m::nat, s'::nat => bool) = unif (Suc v2 div (2::nat)) s
- in (if SHD s' then (2::nat) * m + (1::nat) else (2::nat) * m, STL s'))"
+(let (m::nat, s'::nat => bool) = unif (Suc v2 div 2) s
+ in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_def)
lemma unif_ind: "(All::((nat => (nat => bool) => bool) => bool) => bool)
@@ -3871,19 +3996,19 @@
(%xa::nat. (All::((nat => bool) => bool) => bool) (P x xa)))))"
by (import prob_uniform uniform_ind)
-lemma uniform_def: "uniform (0::nat) (Suc (n::nat)) (s::nat => bool) = (0::nat, s) &
+lemma uniform_def: "uniform 0 (Suc (n::nat)) (s::nat => bool) = (0, s) &
uniform (Suc (t::nat)) (Suc n) s =
(let (xa::nat, x::nat => bool) = unif n s
in if xa < Suc n then (xa, x) else uniform t (Suc n) x)"
by (import prob_uniform uniform_def)
-lemma SUC_DIV_TWO_ZERO: "ALL n::nat. (Suc n div (2::nat) = (0::nat)) = (n = (0::nat))"
+lemma SUC_DIV_TWO_ZERO: "ALL n::nat. (Suc n div 2 = 0) = (n = 0)"
by (import prob_uniform SUC_DIV_TWO_ZERO)
-lemma UNIF_BOUND_LOWER: "ALL n::nat. n < (2::nat) ^ unif_bound n"
+lemma UNIF_BOUND_LOWER: "ALL n::nat. n < 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER)
-lemma UNIF_BOUND_LOWER_SUC: "ALL n::nat. Suc n <= (2::nat) ^ unif_bound n"
+lemma UNIF_BOUND_LOWER_SUC: "ALL n::nat. Suc n <= 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER_SUC)
lemma UNIF_BOUND_UPPER: "(All::(nat => bool) => bool)
@@ -3905,20 +4030,18 @@
n)))"
by (import prob_uniform UNIF_BOUND_UPPER)
-lemma UNIF_BOUND_UPPER_SUC: "ALL n::nat. (2::nat) ^ unif_bound n <= Suc ((2::nat) * n)"
+lemma UNIF_BOUND_UPPER_SUC: "ALL n::nat. 2 ^ unif_bound n <= Suc (2 * n)"
by (import prob_uniform UNIF_BOUND_UPPER_SUC)
-lemma UNIF_DEF_MONAD: "unif (0::nat) = UNIT (0::nat) &
+lemma UNIF_DEF_MONAD: "unif 0 = UNIT 0 &
(ALL n::nat.
unif (Suc n) =
- BIND (unif (Suc n div (2::nat)))
+ BIND (unif (Suc n div 2))
(%m::nat.
- BIND SDEST
- (%b::bool.
- UNIT (if b then (2::nat) * m + (1::nat) else (2::nat) * m))))"
+ BIND SDEST (%b::bool. UNIT (if b then 2 * m + 1 else 2 * m))))"
by (import prob_uniform UNIF_DEF_MONAD)
-lemma UNIFORM_DEF_MONAD: "(ALL x::nat. uniform (0::nat) (Suc x) = UNIT (0::nat)) &
+lemma UNIFORM_DEF_MONAD: "(ALL x::nat. uniform 0 (Suc x) = UNIT 0) &
(ALL (x::nat) xa::nat.
uniform (Suc x) (Suc xa) =
BIND (unif xa)
@@ -3931,45 +4054,18 @@
lemma INDEP_UNIFORM: "ALL (t::nat) n::nat. indep (uniform t (Suc n))"
by (import prob_uniform INDEP_UNIFORM)
-lemma PROB_UNIF: "(All::(nat => bool) => bool)
- (%n::nat.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op =::real => real => bool)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op =::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((unif::nat => (nat => bool) => nat * (nat => bool)) n
- s))
- k))
- ((If::bool => real => real => real)
- ((op <::nat => nat => bool) k
- ((op ^::nat => nat => nat)
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
- (bit.B1::bit))
- (bit.B0::bit)))
- ((unif_bound::nat => nat) n)))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of::bin => real)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
- (bit.B1::bit))
- (bit.B0::bit))))
- ((unif_bound::nat => nat) n))
- (0::real))))"
+lemma PROB_UNIF: "ALL (n::nat) k::nat.
+ prob (%s::nat => bool. fst (unif n s) = k) =
+ (if k < 2 ^ unif_bound n then (1 / 2) ^ unif_bound n else 0)"
by (import prob_uniform PROB_UNIF)
-lemma UNIF_RANGE: "ALL (n::nat) s::nat => bool. fst (unif n s) < (2::nat) ^ unif_bound n"
+lemma UNIF_RANGE: "ALL (n::nat) s::nat => bool. fst (unif n s) < 2 ^ unif_bound n"
by (import prob_uniform UNIF_RANGE)
lemma PROB_UNIF_PAIR: "ALL (n::nat) (k::nat) k'::nat.
(prob (%s::nat => bool. fst (unif n s) = k) =
prob (%s::nat => bool. fst (unif n s) = k')) =
- ((k < (2::nat) ^ unif_bound n) = (k' < (2::nat) ^ unif_bound n))"
+ ((k < 2 ^ unif_bound n) = (k' < 2 ^ unif_bound n))"
by (import prob_uniform PROB_UNIF_PAIR)
lemma PROB_UNIF_BOUND: "(All::(nat => bool) => bool)
@@ -4004,8 +4100,7 @@
((unif_bound::nat => nat) n))))))"
by (import prob_uniform PROB_UNIF_BOUND)
-lemma PROB_UNIF_GOOD: "ALL n::nat.
- (1::real) / (2::real) <= prob (%s::nat => bool. fst (unif n s) < Suc n)"
+lemma PROB_UNIF_GOOD: "ALL n::nat. 1 / 2 <= prob (%s::nat => bool. fst (unif n s) < Suc n)"
by (import prob_uniform PROB_UNIF_GOOD)
lemma UNIFORM_RANGE: "ALL (t::nat) (n::nat) s::nat => bool. fst (uniform t (Suc n) s) < Suc n"
--- a/src/HOL/Import/HOL/HOL4Real.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/HOL4Real.thy Mon Sep 26 16:10:19 2005 +0200
@@ -267,19 +267,19 @@
;setup_theory real
-lemma REAL_0: "(0::real) = (0::real)"
+lemma REAL_0: "(op =::real => real => bool) (0::real) (0::real)"
by (import real REAL_0)
-lemma REAL_1: "(1::real) = (1::real)"
+lemma REAL_1: "(op =::real => real => bool) (1::real) (1::real)"
by (import real REAL_1)
-lemma REAL_ADD_LID_UNIQ: "ALL (x::real) y::real. (x + y = y) = (x = (0::real))"
+lemma REAL_ADD_LID_UNIQ: "ALL (x::real) y::real. (x + y = y) = (x = 0)"
by (import real REAL_ADD_LID_UNIQ)
-lemma REAL_ADD_RID_UNIQ: "ALL (x::real) y::real. (x + y = x) = (y = (0::real))"
+lemma REAL_ADD_RID_UNIQ: "ALL (x::real) y::real. (x + y = x) = (y = 0)"
by (import real REAL_ADD_RID_UNIQ)
-lemma REAL_LNEG_UNIQ: "ALL (x::real) y::real. (x + y = (0::real)) = (x = - y)"
+lemma REAL_LNEG_UNIQ: "ALL (x::real) y::real. (x + y = 0) = (x = - y)"
by (import real REAL_LNEG_UNIQ)
lemma REAL_LT_ANTISYM: "ALL (x::real) y::real. ~ (x < y & y < x)"
@@ -294,10 +294,10 @@
lemma REAL_LTE_ANTSYM: "ALL (x::real) y::real. ~ (x <= y & y < x)"
by (import real REAL_LTE_ANTSYM)
-lemma REAL_LT_NEGTOTAL: "ALL x::real. x = (0::real) | (0::real) < x | (0::real) < - x"
+lemma REAL_LT_NEGTOTAL: "ALL x::real. x = 0 | 0 < x | 0 < - x"
by (import real REAL_LT_NEGTOTAL)
-lemma REAL_LE_NEGTOTAL: "ALL x::real. (0::real) <= x | (0::real) <= - x"
+lemma REAL_LE_NEGTOTAL: "ALL x::real. 0 <= x | 0 <= - x"
by (import real REAL_LE_NEGTOTAL)
lemma REAL_LT_ADDNEG: "ALL (x::real) (y::real) z::real. (y < x + - z) = (y + z < x)"
@@ -306,16 +306,16 @@
lemma REAL_LT_ADDNEG2: "ALL (x::real) (y::real) z::real. (x + - y < z) = (x < z + y)"
by (import real REAL_LT_ADDNEG2)
-lemma REAL_LT_ADD1: "ALL (x::real) y::real. x <= y --> x < y + (1::real)"
+lemma REAL_LT_ADD1: "ALL (x::real) y::real. x <= y --> x < y + 1"
by (import real REAL_LT_ADD1)
lemma REAL_SUB_ADD2: "ALL (x::real) y::real. y + (x - y) = x"
by (import real REAL_SUB_ADD2)
-lemma REAL_SUB_LT: "ALL (x::real) y::real. ((0::real) < x - y) = (y < x)"
+lemma REAL_SUB_LT: "ALL (x::real) y::real. (0 < x - y) = (y < x)"
by (import real REAL_SUB_LT)
-lemma REAL_SUB_LE: "ALL (x::real) y::real. ((0::real) <= x - y) = (y <= x)"
+lemma REAL_SUB_LE: "ALL (x::real) y::real. (0 <= x - y) = (y <= x)"
by (import real REAL_SUB_LE)
lemma REAL_ADD_SUB: "ALL (x::real) y::real. x + y - x = y"
@@ -324,72 +324,79 @@
lemma REAL_NEG_EQ: "ALL (x::real) y::real. (- x = y) = (x = - y)"
by (import real REAL_NEG_EQ)
-lemma REAL_NEG_MINUS1: "ALL x::real. - x = - (1::real) * x"
+lemma REAL_NEG_MINUS1: "ALL x::real. - x = - 1 * x"
by (import real REAL_NEG_MINUS1)
-lemma REAL_LT_LMUL_0: "ALL (x::real) y::real.
- (0::real) < x --> ((0::real) < x * y) = ((0::real) < y)"
+lemma REAL_LT_LMUL_0: "ALL (x::real) y::real. 0 < x --> (0 < x * y) = (0 < y)"
by (import real REAL_LT_LMUL_0)
-lemma REAL_LT_RMUL_0: "ALL (x::real) y::real.
- (0::real) < y --> ((0::real) < x * y) = ((0::real) < x)"
+lemma REAL_LT_RMUL_0: "ALL (x::real) y::real. 0 < y --> (0 < x * y) = (0 < x)"
by (import real REAL_LT_RMUL_0)
-lemma REAL_LT_LMUL: "ALL (x::real) (y::real) z::real. (0::real) < x --> (x * y < x * z) = (y < z)"
+lemma REAL_LT_LMUL: "ALL (x::real) (y::real) z::real. 0 < x --> (x * y < x * z) = (y < z)"
by (import real REAL_LT_LMUL)
-lemma REAL_LINV_UNIQ: "ALL (x::real) y::real. x * y = (1::real) --> x = inverse y"
+lemma REAL_LINV_UNIQ: "ALL (x::real) y::real. x * y = 1 --> x = inverse y"
by (import real REAL_LINV_UNIQ)
-lemma REAL_LE_INV: "ALL x>=0::real. (0::real) <= inverse x"
+lemma REAL_LE_INV: "(All::(real => bool) => bool)
+ (%x::real.
+ (op -->::bool => bool => bool)
+ ((op <=::real => real => bool) (0::real) x)
+ ((op <=::real => real => bool) (0::real) ((inverse::real => real) x)))"
by (import real REAL_LE_INV)
-lemma REAL_LE_ADDR: "ALL (x::real) y::real. (x <= x + y) = ((0::real) <= y)"
+lemma REAL_LE_ADDR: "ALL (x::real) y::real. (x <= x + y) = (0 <= y)"
by (import real REAL_LE_ADDR)
-lemma REAL_LE_ADDL: "ALL (x::real) y::real. (y <= x + y) = ((0::real) <= x)"
+lemma REAL_LE_ADDL: "ALL (x::real) y::real. (y <= x + y) = (0 <= x)"
by (import real REAL_LE_ADDL)
-lemma REAL_LT_ADDR: "ALL (x::real) y::real. (x < x + y) = ((0::real) < y)"
+lemma REAL_LT_ADDR: "ALL (x::real) y::real. (x < x + y) = (0 < y)"
by (import real REAL_LT_ADDR)
-lemma REAL_LT_ADDL: "ALL (x::real) y::real. (y < x + y) = ((0::real) < x)"
+lemma REAL_LT_ADDL: "ALL (x::real) y::real. (y < x + y) = (0 < x)"
by (import real REAL_LT_ADDL)
-lemma REAL_LT_NZ: "ALL n::nat. (real n ~= (0::real)) = ((0::real) < real n)"
+lemma REAL_LT_NZ: "ALL n::nat. (real n ~= 0) = (0 < real n)"
by (import real REAL_LT_NZ)
-lemma REAL_NZ_IMP_LT: "ALL n::nat. n ~= (0::nat) --> (0::real) < real n"
+lemma REAL_NZ_IMP_LT: "ALL n::nat. n ~= 0 --> 0 < real n"
by (import real REAL_NZ_IMP_LT)
-lemma REAL_LT_RDIV_0: "ALL (y::real) z::real.
- (0::real) < z --> ((0::real) < y / z) = ((0::real) < y)"
+lemma REAL_LT_RDIV_0: "ALL (y::real) z::real. 0 < z --> (0 < y / z) = (0 < y)"
by (import real REAL_LT_RDIV_0)
-lemma REAL_LT_RDIV: "ALL (x::real) (y::real) z::real. (0::real) < z --> (x / z < y / z) = (x < y)"
+lemma REAL_LT_RDIV: "ALL (x::real) (y::real) z::real. 0 < z --> (x / z < y / z) = (x < y)"
by (import real REAL_LT_RDIV)
-lemma REAL_LT_FRACTION_0: "ALL (n::nat) d::real.
- n ~= (0::nat) --> ((0::real) < d / real n) = ((0::real) < d)"
+lemma REAL_LT_FRACTION_0: "ALL (n::nat) d::real. n ~= 0 --> (0 < d / real n) = (0 < d)"
by (import real REAL_LT_FRACTION_0)
-lemma REAL_LT_MULTIPLE: "ALL (x::nat) xa::real.
- (1::nat) < x --> (xa < real x * xa) = ((0::real) < xa)"
+lemma REAL_LT_MULTIPLE: "ALL (x::nat) xa::real. 1 < x --> (xa < real x * xa) = (0 < xa)"
by (import real REAL_LT_MULTIPLE)
-lemma REAL_LT_FRACTION: "ALL (n::nat) d::real. (1::nat) < n --> (d / real n < d) = ((0::real) < d)"
+lemma REAL_LT_FRACTION: "ALL (n::nat) d::real. 1 < n --> (d / real n < d) = (0 < d)"
by (import real REAL_LT_FRACTION)
-lemma REAL_LT_HALF2: "ALL d::real. (d / (2::real) < d) = ((0::real) < d)"
+lemma REAL_LT_HALF2: "ALL d::real. (d / 2 < d) = (0 < d)"
by (import real REAL_LT_HALF2)
-lemma REAL_DIV_LMUL: "ALL (x::real) y::real. y ~= (0::real) --> y * (x / y) = x"
+lemma REAL_DIV_LMUL: "ALL (x::real) y::real. y ~= 0 --> y * (x / y) = x"
by (import real REAL_DIV_LMUL)
-lemma REAL_DIV_RMUL: "ALL (x::real) y::real. y ~= (0::real) --> x / y * y = x"
+lemma REAL_DIV_RMUL: "ALL (x::real) y::real. y ~= 0 --> x / y * y = x"
by (import real REAL_DIV_RMUL)
-lemma REAL_DOWN: "ALL x>0::real. EX xa>0::real. xa < x"
+lemma REAL_DOWN: "(All::(real => bool) => bool)
+ (%x::real.
+ (op -->::bool => bool => bool)
+ ((op <::real => real => bool) (0::real) x)
+ ((Ex::(real => bool) => bool)
+ (%xa::real.
+ (op &::bool => bool => bool)
+ ((op <::real => real => bool) (0::real) xa)
+ ((op <::real => real => bool) xa x))))"
by (import real REAL_DOWN)
lemma REAL_SUB_SUB: "ALL (x::real) y::real. x - y - x = - y"
@@ -408,13 +415,11 @@
by (import real REAL_SUB_TRIANGLE)
lemma REAL_INV_MUL: "ALL (x::real) y::real.
- x ~= (0::real) & y ~= (0::real) -->
- inverse (x * y) = inverse x * inverse y"
+ x ~= 0 & y ~= 0 --> inverse (x * y) = inverse x * inverse y"
by (import real REAL_INV_MUL)
lemma REAL_SUB_INV2: "ALL (x::real) y::real.
- x ~= (0::real) & y ~= (0::real) -->
- inverse x - inverse y = (y - x) / (x * y)"
+ x ~= 0 & y ~= 0 --> inverse x - inverse y = (y - x) / (x * y)"
by (import real REAL_SUB_INV2)
lemma REAL_SUB_SUB2: "ALL (x::real) y::real. x - (x - y) = y"
@@ -424,75 +429,74 @@
by (import real REAL_ADD_SUB2)
lemma REAL_LE_MUL2: "ALL (x1::real) (x2::real) (y1::real) y2::real.
- (0::real) <= x1 & (0::real) <= y1 & x1 <= x2 & y1 <= y2 -->
- x1 * y1 <= x2 * y2"
+ 0 <= x1 & 0 <= y1 & x1 <= x2 & y1 <= y2 --> x1 * y1 <= x2 * y2"
by (import real REAL_LE_MUL2)
-lemma REAL_LE_DIV: "ALL (x::real) xa::real.
- (0::real) <= x & (0::real) <= xa --> (0::real) <= x / xa"
+lemma REAL_LE_DIV: "ALL (x::real) xa::real. 0 <= x & 0 <= xa --> 0 <= x / xa"
by (import real REAL_LE_DIV)
-lemma REAL_LT_1: "ALL (x::real) y::real. (0::real) <= x & x < y --> x / y < (1::real)"
+lemma REAL_LT_1: "ALL (x::real) y::real. 0 <= x & x < y --> x / y < 1"
by (import real REAL_LT_1)
-lemma REAL_POS_NZ: "ALL x>0::real. x ~= (0::real)"
+lemma REAL_POS_NZ: "(All::(real => bool) => bool)
+ (%x::real.
+ (op -->::bool => bool => bool)
+ ((op <::real => real => bool) (0::real) x)
+ ((Not::bool => bool) ((op =::real => real => bool) x (0::real))))"
by (import real REAL_POS_NZ)
-lemma REAL_EQ_LMUL_IMP: "ALL (x::real) (xa::real) xb::real.
- x ~= (0::real) & x * xa = x * xb --> xa = xb"
+lemma REAL_EQ_LMUL_IMP: "ALL (x::real) (xa::real) xb::real. x ~= 0 & x * xa = x * xb --> xa = xb"
by (import real REAL_EQ_LMUL_IMP)
-lemma REAL_FACT_NZ: "ALL n::nat. real (FACT n) ~= (0::real)"
+lemma REAL_FACT_NZ: "ALL n::nat. real (FACT n) ~= 0"
by (import real REAL_FACT_NZ)
lemma REAL_DIFFSQ: "ALL (x::real) y::real. (x + y) * (x - y) = x * x - y * y"
by (import real REAL_DIFFSQ)
-lemma REAL_POASQ: "ALL x::real. ((0::real) < x * x) = (x ~= (0::real))"
+lemma REAL_POASQ: "ALL x::real. (0 < x * x) = (x ~= 0)"
by (import real REAL_POASQ)
-lemma REAL_SUMSQ: "ALL (x::real) y::real.
- (x * x + y * y = (0::real)) = (x = (0::real) & y = (0::real))"
+lemma REAL_SUMSQ: "ALL (x::real) y::real. (x * x + y * y = 0) = (x = 0 & y = 0)"
by (import real REAL_SUMSQ)
lemma REAL_DIV_MUL2: "ALL (x::real) z::real.
- x ~= (0::real) & z ~= (0::real) -->
- (ALL y::real. y / z = x * y / (x * z))"
+ x ~= 0 & z ~= 0 --> (ALL y::real. y / z = x * y / (x * z))"
by (import real REAL_DIV_MUL2)
-lemma REAL_MIDDLE1: "ALL (a::real) b::real. a <= b --> a <= (a + b) / (2::real)"
+lemma REAL_MIDDLE1: "ALL (a::real) b::real. a <= b --> a <= (a + b) / 2"
by (import real REAL_MIDDLE1)
-lemma REAL_MIDDLE2: "ALL (a::real) b::real. a <= b --> (a + b) / (2::real) <= b"
+lemma REAL_MIDDLE2: "ALL (a::real) b::real. a <= b --> (a + b) / 2 <= b"
by (import real REAL_MIDDLE2)
lemma ABS_LT_MUL2: "ALL (w::real) (x::real) (y::real) z::real.
abs w < y & abs x < z --> abs (w * x) < y * z"
by (import real ABS_LT_MUL2)
-lemma ABS_REFL: "ALL x::real. (abs x = x) = ((0::real) <= x)"
+lemma ABS_REFL: "ALL x::real. (abs x = x) = (0 <= x)"
by (import real ABS_REFL)
lemma ABS_BETWEEN: "ALL (x::real) (y::real) d::real.
- ((0::real) < d & x - d < y & y < x + d) = (abs (y - x) < d)"
+ (0 < d & x - d < y & y < x + d) = (abs (y - x) < d)"
by (import real ABS_BETWEEN)
lemma ABS_BOUND: "ALL (x::real) (y::real) d::real. abs (x - y) < d --> y < x + d"
by (import real ABS_BOUND)
-lemma ABS_STILLNZ: "ALL (x::real) y::real. abs (x - y) < abs y --> x ~= (0::real)"
+lemma ABS_STILLNZ: "ALL (x::real) y::real. abs (x - y) < abs y --> x ~= 0"
by (import real ABS_STILLNZ)
-lemma ABS_CASES: "ALL x::real. x = (0::real) | (0::real) < abs x"
+lemma ABS_CASES: "ALL x::real. x = 0 | 0 < abs x"
by (import real ABS_CASES)
lemma ABS_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & abs (y - x) < z - x --> y < z"
by (import real ABS_BETWEEN1)
-lemma ABS_SIGN: "ALL (x::real) y::real. abs (x - y) < y --> (0::real) < x"
+lemma ABS_SIGN: "ALL (x::real) y::real. abs (x - y) < y --> 0 < x"
by (import real ABS_SIGN)
-lemma ABS_SIGN2: "ALL (x::real) y::real. abs (x - y) < - y --> x < (0::real)"
+lemma ABS_SIGN2: "ALL (x::real) y::real. abs (x - y) < - y --> x < 0"
by (import real ABS_SIGN2)
lemma ABS_CIRCLE: "ALL (x::real) (y::real) h::real.
@@ -500,44 +504,62 @@
by (import real ABS_CIRCLE)
lemma ABS_BETWEEN2: "ALL (x0::real) (x::real) (y0::real) y::real.
- x0 < y0 &
- abs (x - x0) < (y0 - x0) / (2::real) &
- abs (y - y0) < (y0 - x0) / (2::real) -->
+ x0 < y0 & abs (x - x0) < (y0 - x0) / 2 & abs (y - y0) < (y0 - x0) / 2 -->
x < y"
by (import real ABS_BETWEEN2)
-lemma POW_PLUS1: "ALL e>0::real. ALL n::nat. (1::real) + real n * e <= ((1::real) + e) ^ n"
+lemma POW_PLUS1: "ALL e>0. ALL n::nat. 1 + real n * e <= (1 + e) ^ n"
by (import real POW_PLUS1)
-lemma POW_M1: "ALL n::nat. abs ((- (1::real)) ^ n) = (1::real)"
+lemma POW_M1: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op =::real => real => bool)
+ ((abs::real => real)
+ ((op ^::real => nat => real) ((uminus::real => real) (1::real)) n))
+ (1::real))"
by (import real POW_M1)
-lemma REAL_LE1_POW2: "ALL x>=1::real. (1::real) <= x ^ 2"
+lemma REAL_LE1_POW2: "(All::(real => bool) => bool)
+ (%x::real.
+ (op -->::bool => bool => bool)
+ ((op <=::real => real => bool) (1::real) x)
+ ((op <=::real => real => bool) (1::real)
+ ((op ^::real => nat => real) x
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))))"
by (import real REAL_LE1_POW2)
-lemma REAL_LT1_POW2: "ALL x>1::real. (1::real) < x ^ 2"
+lemma REAL_LT1_POW2: "(All::(real => bool) => bool)
+ (%x::real.
+ (op -->::bool => bool => bool)
+ ((op <::real => real => bool) (1::real) x)
+ ((op <::real => real => bool) (1::real)
+ ((op ^::real => nat => real) x
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit))))))"
by (import real REAL_LT1_POW2)
-lemma POW_POS_LT: "ALL (x::real) n::nat. (0::real) < x --> (0::real) < x ^ Suc n"
+lemma POW_POS_LT: "ALL (x::real) n::nat. 0 < x --> 0 < x ^ Suc n"
by (import real POW_POS_LT)
-lemma POW_LT: "ALL (n::nat) (x::real) y::real.
- (0::real) <= x & x < y --> x ^ Suc n < y ^ Suc n"
+lemma POW_LT: "ALL (n::nat) (x::real) y::real. 0 <= x & x < y --> x ^ Suc n < y ^ Suc n"
by (import real POW_LT)
-lemma POW_ZERO_EQ: "ALL (n::nat) x::real. (x ^ Suc n = (0::real)) = (x = (0::real))"
+lemma POW_ZERO_EQ: "ALL (n::nat) x::real. (x ^ Suc n = 0) = (x = 0)"
by (import real POW_ZERO_EQ)
-lemma REAL_POW_LT2: "ALL (n::nat) (x::real) y::real.
- n ~= (0::nat) & (0::real) <= x & x < y --> x ^ n < y ^ n"
+lemma REAL_POW_LT2: "ALL (n::nat) (x::real) y::real. n ~= 0 & 0 <= x & x < y --> x ^ n < y ^ n"
by (import real REAL_POW_LT2)
-lemma REAL_POW_MONO_LT: "ALL (m::nat) (n::nat) x::real. (1::real) < x & m < n --> x ^ m < x ^ n"
+lemma REAL_POW_MONO_LT: "ALL (m::nat) (n::nat) x::real. 1 < x & m < n --> x ^ m < x ^ n"
by (import real REAL_POW_MONO_LT)
lemma REAL_SUP_SOMEPOS: "ALL P::real => bool.
- (EX x::real. P x & (0::real) < x) &
- (EX z::real. ALL x::real. P x --> x < z) -->
+ (EX x::real. P x & 0 < x) & (EX z::real. ALL x::real. P x --> x < z) -->
(EX s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
by (import real REAL_SUP_SOMEPOS)
@@ -546,8 +568,7 @@
(ALL y::real. (EX x::real. P x & y < x) = (y < s + d))"
by (import real SUP_LEMMA1)
-lemma SUP_LEMMA2: "ALL P::real => bool.
- Ex P --> (EX (d::real) x::real. P (x + d) & (0::real) < x)"
+lemma SUP_LEMMA2: "ALL P::real => bool. Ex P --> (EX (d::real) x::real. P (x + d) & 0 < x)"
by (import real SUP_LEMMA2)
lemma SUP_LEMMA3: "ALL d::real.
@@ -595,14 +616,13 @@
(ALL y::real. P y --> y <= sup P)"
by (import real REAL_SUP_UBOUND_LE)
-lemma REAL_ARCH_LEAST: "ALL y>0::real.
- ALL x>=0::real. EX n::nat. real n * y <= x & x < real (Suc n) * y"
+lemma REAL_ARCH_LEAST: "ALL y>0. ALL x>=0. EX n::nat. real n * y <= x & x < real (Suc n) * y"
by (import real REAL_ARCH_LEAST)
consts
sumc :: "nat => nat => (nat => real) => real"
-specification (sumc) sumc: "(ALL (n::nat) f::nat => real. sumc n (0::nat) f = (0::real)) &
+specification (sumc) sumc: "(ALL (n::nat) f::nat => real. sumc n 0 f = 0) &
(ALL (n::nat) (m::nat) f::nat => real.
sumc n (Suc m) f = sumc n m f + f (n + m))"
by (import real sumc)
@@ -622,16 +642,16 @@
by (import real SUM_DEF)
lemma sum: "ALL (x::nat => real) (xa::nat) xb::nat.
- real.sum (xa, 0::nat) x = (0::real) &
+ real.sum (xa, 0) x = 0 &
real.sum (xa, Suc xb) x = real.sum (xa, xb) x + x (xa + xb)"
by (import real sum)
lemma SUM_TWO: "ALL (f::nat => real) (n::nat) p::nat.
- real.sum (0::nat, n) f + real.sum (n, p) f = real.sum (0::nat, n + p) f"
+ real.sum (0, n) f + real.sum (n, p) f = real.sum (0, n + p) f"
by (import real SUM_TWO)
lemma SUM_DIFF: "ALL (f::nat => real) (m::nat) n::nat.
- real.sum (m, n) f = real.sum (0::nat, m + n) f - real.sum (0::nat, m) f"
+ real.sum (m, n) f = real.sum (0, m + n) f - real.sum (0, m) f"
by (import real SUM_DIFF)
lemma ABS_SUM: "ALL (f::nat => real) (m::nat) n::nat.
@@ -649,13 +669,12 @@
by (import real SUM_EQ)
lemma SUM_POS: "ALL f::nat => real.
- (ALL n::nat. (0::real) <= f n) -->
- (ALL (m::nat) n::nat. (0::real) <= real.sum (m, n) f)"
+ (ALL n::nat. 0 <= f n) --> (ALL (m::nat) n::nat. 0 <= real.sum (m, n) f)"
by (import real SUM_POS)
lemma SUM_POS_GEN: "ALL (f::nat => real) m::nat.
- (ALL n::nat. m <= n --> (0::real) <= f n) -->
- (ALL n::nat. (0::real) <= real.sum (m, n) f)"
+ (ALL n::nat. m <= n --> 0 <= f n) -->
+ (ALL n::nat. 0 <= real.sum (m, n) f)"
by (import real SUM_POS_GEN)
lemma SUM_ABS: "ALL (f::nat => real) (m::nat) x::nat.
@@ -668,8 +687,8 @@
by (import real SUM_ABS_LE)
lemma SUM_ZERO: "ALL (f::nat => real) N::nat.
- (ALL n::nat. N <= n --> f n = (0::real)) -->
- (ALL (m::nat) n::nat. N <= m --> real.sum (m, n) f = (0::real))"
+ (ALL n::nat. N <= n --> f n = 0) -->
+ (ALL (m::nat) n::nat. N <= m --> real.sum (m, n) f = 0)"
by (import real SUM_ZERO)
lemma SUM_ADD: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
@@ -696,8 +715,7 @@
by (import real SUM_SUBST)
lemma SUM_NSUB: "ALL (n::nat) (f::nat => real) c::real.
- real.sum (0::nat, n) f - real n * c =
- real.sum (0::nat, n) (%p::nat. f p - c)"
+ real.sum (0, n) f - real n * c = real.sum (0, n) (%p::nat. f p - c)"
by (import real SUM_NSUB)
lemma SUM_BOUND: "ALL (f::nat => real) (k::real) (m::nat) n::nat.
@@ -706,26 +724,25 @@
by (import real SUM_BOUND)
lemma SUM_GROUP: "ALL (n::nat) (k::nat) f::nat => real.
- real.sum (0::nat, n) (%m::nat. real.sum (m * k, k) f) =
- real.sum (0::nat, n * k) f"
+ real.sum (0, n) (%m::nat. real.sum (m * k, k) f) = real.sum (0, n * k) f"
by (import real SUM_GROUP)
-lemma SUM_1: "ALL (f::nat => real) n::nat. real.sum (n, 1::nat) f = f n"
+lemma SUM_1: "ALL (f::nat => real) n::nat. real.sum (n, 1) f = f n"
by (import real SUM_1)
-lemma SUM_2: "ALL (f::nat => real) n::nat. real.sum (n, 2::nat) f = f n + f (n + (1::nat))"
+lemma SUM_2: "ALL (f::nat => real) n::nat. real.sum (n, 2) f = f n + f (n + 1)"
by (import real SUM_2)
lemma SUM_OFFSET: "ALL (f::nat => real) (n::nat) k::nat.
- real.sum (0::nat, n) (%m::nat. f (m + k)) =
- real.sum (0::nat, n + k) f - real.sum (0::nat, k) f"
+ real.sum (0, n) (%m::nat. f (m + k)) =
+ real.sum (0, n + k) f - real.sum (0, k) f"
by (import real SUM_OFFSET)
lemma SUM_REINDEX: "ALL (f::nat => real) (m::nat) (k::nat) n::nat.
real.sum (m + k, n) f = real.sum (m, n) (%r::nat. f (r + k))"
by (import real SUM_REINDEX)
-lemma SUM_0: "ALL (m::nat) n::nat. real.sum (m, n) (%r::nat. 0::real) = (0::real)"
+lemma SUM_0: "ALL (m::nat) n::nat. real.sum (m, n) (%r::nat. 0) = 0"
by (import real SUM_0)
lemma SUM_PERMUTE_0: "(All::(nat => bool) => bool)
@@ -756,12 +773,10 @@
real.sum (n, d) (%n::nat. f (Suc n) - f n) = f (n + d) - f n"
by (import real SUM_CANCEL)
-lemma REAL_EQ_RDIV_EQ: "ALL (x::real) (xa::real) xb::real.
- (0::real) < xb --> (x = xa / xb) = (x * xb = xa)"
+lemma REAL_EQ_RDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x = xa / xb) = (x * xb = xa)"
by (import real REAL_EQ_RDIV_EQ)
-lemma REAL_EQ_LDIV_EQ: "ALL (x::real) (xa::real) xb::real.
- (0::real) < xb --> (x / xb = xa) = (x = xa * xb)"
+lemma REAL_EQ_LDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x / xb = xa) = (x = xa * xb)"
by (import real REAL_EQ_LDIV_EQ)
;end_setup
@@ -769,7 +784,7 @@
;setup_theory topology
constdefs
- re_Union :: "(('a::type => bool) => bool) => 'a::type => bool"
+ re_Union :: "(('a => bool) => bool) => 'a => bool"
"re_Union ==
%(P::('a::type => bool) => bool) x::'a::type.
EX s::'a::type => bool. P s & s x"
@@ -779,7 +794,7 @@
by (import topology re_Union)
constdefs
- re_union :: "('a::type => bool) => ('a::type => bool) => 'a::type => bool"
+ re_union :: "('a => bool) => ('a => bool) => 'a => bool"
"re_union ==
%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x | Q x"
@@ -788,7 +803,7 @@
by (import topology re_union)
constdefs
- re_intersect :: "('a::type => bool) => ('a::type => bool) => 'a::type => bool"
+ re_intersect :: "('a => bool) => ('a => bool) => 'a => bool"
"re_intersect ==
%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x & Q x"
@@ -797,21 +812,21 @@
by (import topology re_intersect)
constdefs
- re_null :: "'a::type => bool"
+ re_null :: "'a => bool"
"re_null == %x::'a::type. False"
lemma re_null: "re_null = (%x::'a::type. False)"
by (import topology re_null)
constdefs
- re_universe :: "'a::type => bool"
+ re_universe :: "'a => bool"
"re_universe == %x::'a::type. True"
lemma re_universe: "re_universe = (%x::'a::type. True)"
by (import topology re_universe)
constdefs
- re_subset :: "('a::type => bool) => ('a::type => bool) => bool"
+ re_subset :: "('a => bool) => ('a => bool) => bool"
"re_subset ==
%(P::'a::type => bool) Q::'a::type => bool. ALL x::'a::type. P x --> Q x"
@@ -820,7 +835,7 @@
by (import topology re_subset)
constdefs
- re_compl :: "('a::type => bool) => 'a::type => bool"
+ re_compl :: "('a => bool) => 'a => bool"
"re_compl == %(P::'a::type => bool) x::'a::type. ~ P x"
lemma re_compl: "ALL P::'a::type => bool. re_compl P = (%x::'a::type. ~ P x)"
@@ -841,7 +856,7 @@
by (import topology SUBSET_TRANS)
constdefs
- istopology :: "(('a::type => bool) => bool) => bool"
+ istopology :: "(('a => bool) => bool) => bool"
"istopology ==
%L::('a::type => bool) => bool.
L re_null &
@@ -867,8 +882,8 @@
lemmas topology_TY_DEF = typedef_hol2hol4 [OF type_definition_topology]
consts
- topology :: "(('a::type => bool) => bool) => 'a::type topology"
- "open" :: "'a::type topology => ('a::type => bool) => bool"
+ topology :: "(('a => bool) => bool) => 'a topology"
+ "open" :: "'a topology => ('a => bool) => bool"
specification ("open" topology) topology_tybij: "(ALL a::'a::type topology. topology (open a) = a) &
(ALL r::('a::type => bool) => bool. istopology r = (open (topology r) = r))"
@@ -888,7 +903,7 @@
by (import topology TOPOLOGY_UNION)
constdefs
- neigh :: "'a::type topology => ('a::type => bool) * 'a::type => bool"
+ neigh :: "'a topology => ('a => bool) * 'a => bool"
"neigh ==
%(top::'a::type topology) (N::'a::type => bool, x::'a::type).
EX P::'a::type => bool. open top P & re_subset P N & P x"
@@ -920,7 +935,7 @@
by (import topology OPEN_NEIGH)
constdefs
- closed :: "'a::type topology => ('a::type => bool) => bool"
+ closed :: "'a topology => ('a => bool) => bool"
"closed == %(L::'a::type topology) S'::'a::type => bool. open L (re_compl S')"
lemma closed: "ALL (L::'a::type topology) S'::'a::type => bool.
@@ -928,7 +943,7 @@
by (import topology closed)
constdefs
- limpt :: "'a::type topology => 'a::type => ('a::type => bool) => bool"
+ limpt :: "'a topology => 'a => ('a => bool) => bool"
"limpt ==
%(top::'a::type topology) (x::'a::type) S'::'a::type => bool.
ALL N::'a::type => bool.
@@ -945,16 +960,16 @@
by (import topology CLOSED_LIMPT)
constdefs
- ismet :: "('a::type * 'a::type => real) => bool"
+ ismet :: "('a * 'a => real) => bool"
"ismet ==
%m::'a::type * 'a::type => real.
- (ALL (x::'a::type) y::'a::type. (m (x, y) = (0::real)) = (x = y)) &
+ (ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
(ALL (x::'a::type) (y::'a::type) z::'a::type.
m (y, z) <= m (x, y) + m (x, z))"
lemma ismet: "ALL m::'a::type * 'a::type => real.
ismet m =
- ((ALL (x::'a::type) y::'a::type. (m (x, y) = (0::real)) = (x = y)) &
+ ((ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
(ALL (x::'a::type) (y::'a::type) z::'a::type.
m (y, z) <= m (x, y) + m (x, z)))"
by (import topology ismet)
@@ -967,8 +982,8 @@
lemmas metric_TY_DEF = typedef_hol2hol4 [OF type_definition_metric]
consts
- metric :: "('a::type * 'a::type => real) => 'a::type metric"
- dist :: "'a::type metric => 'a::type * 'a::type => real"
+ metric :: "('a * 'a => real) => 'a metric"
+ dist :: "'a metric => 'a * 'a => real"
specification (dist metric) metric_tybij: "(ALL a::'a::type metric. metric (dist a) = a) &
(ALL r::'a::type * 'a::type => real. ismet r = (dist (metric r) = r))"
@@ -978,14 +993,13 @@
by (import topology METRIC_ISMET)
lemma METRIC_ZERO: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- (dist m (x, y) = (0::real)) = (x = y)"
+ (dist m (x, y) = 0) = (x = y)"
by (import topology METRIC_ZERO)
-lemma METRIC_SAME: "ALL (m::'a::type metric) x::'a::type. dist m (x, x) = (0::real)"
+lemma METRIC_SAME: "ALL (m::'a::type metric) x::'a::type. dist m (x, x) = 0"
by (import topology METRIC_SAME)
-lemma METRIC_POS: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- (0::real) <= dist m (x, y)"
+lemma METRIC_POS: "ALL (m::'a::type metric) (x::'a::type) y::'a::type. 0 <= dist m (x, y)"
by (import topology METRIC_POS)
lemma METRIC_SYM: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
@@ -997,45 +1011,41 @@
by (import topology METRIC_TRIANGLE)
lemma METRIC_NZ: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- x ~= y --> (0::real) < dist m (x, y)"
+ x ~= y --> 0 < dist m (x, y)"
by (import topology METRIC_NZ)
constdefs
- mtop :: "'a::type metric => 'a::type topology"
+ mtop :: "'a metric => 'a topology"
"mtop ==
%m::'a::type metric.
topology
(%S'::'a::type => bool.
ALL x::'a::type.
- S' x -->
- (EX e>0::real. ALL y::'a::type. dist m (x, y) < e --> S' y))"
+ S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
lemma mtop: "ALL m::'a::type metric.
mtop m =
topology
(%S'::'a::type => bool.
ALL x::'a::type.
- S' x -->
- (EX e>0::real. ALL y::'a::type. dist m (x, y) < e --> S' y))"
+ S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
by (import topology mtop)
lemma mtop_istopology: "ALL m::'a::type metric.
istopology
(%S'::'a::type => bool.
ALL x::'a::type.
- S' x -->
- (EX e>0::real. ALL y::'a::type. dist m (x, y) < e --> S' y))"
+ S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
by (import topology mtop_istopology)
lemma MTOP_OPEN: "ALL (S'::'a::type => bool) x::'a::type metric.
open (mtop x) S' =
(ALL xa::'a::type.
- S' xa -->
- (EX e>0::real. ALL y::'a::type. dist x (xa, y) < e --> S' y))"
+ S' xa --> (EX e>0. ALL y::'a::type. dist x (xa, y) < e --> S' y))"
by (import topology MTOP_OPEN)
constdefs
- B :: "'a::type metric => 'a::type * real => 'a::type => bool"
+ B :: "'a metric => 'a * real => 'a => bool"
"B ==
%(m::'a::type metric) (x::'a::type, e::real) y::'a::type. dist m (x, y) < e"
@@ -1044,16 +1054,16 @@
by (import topology ball)
lemma BALL_OPEN: "ALL (m::'a::type metric) (x::'a::type) e::real.
- (0::real) < e --> open (mtop m) (B m (x, e))"
+ 0 < e --> open (mtop m) (B m (x, e))"
by (import topology BALL_OPEN)
lemma BALL_NEIGH: "ALL (m::'a::type metric) (x::'a::type) e::real.
- (0::real) < e --> neigh (mtop m) (B m (x, e), x)"
+ 0 < e --> neigh (mtop m) (B m (x, e), x)"
by (import topology BALL_NEIGH)
lemma MTOP_LIMPT: "ALL (m::'a::type metric) (x::'a::type) S'::'a::type => bool.
limpt (mtop m) x S' =
- (ALL e>0::real. EX y::'a::type. x ~= y & S' y & dist m (x, y) < e)"
+ (ALL e>0. EX y::'a::type. x ~= y & S' y & dist m (x, y) < e)"
by (import topology MTOP_LIMPT)
lemma ISMET_R1: "ismet (%(x::real, y::real). abs (y - x))"
@@ -1075,16 +1085,16 @@
lemma MR1_SUB: "ALL (x::real) d::real. dist mr1 (x, x - d) = abs d"
by (import topology MR1_SUB)
-lemma MR1_ADD_POS: "ALL (x::real) d::real. (0::real) <= d --> dist mr1 (x, x + d) = d"
+lemma MR1_ADD_POS: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x + d) = d"
by (import topology MR1_ADD_POS)
-lemma MR1_SUB_LE: "ALL (x::real) d::real. (0::real) <= d --> dist mr1 (x, x - d) = d"
+lemma MR1_SUB_LE: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x - d) = d"
by (import topology MR1_SUB_LE)
-lemma MR1_ADD_LT: "ALL (x::real) d::real. (0::real) < d --> dist mr1 (x, x + d) = d"
+lemma MR1_ADD_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x + d) = d"
by (import topology MR1_ADD_LT)
-lemma MR1_SUB_LT: "ALL (x::real) d::real. (0::real) < d --> dist mr1 (x, x - d) = d"
+lemma MR1_SUB_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x - d) = d"
by (import topology MR1_SUB_LT)
lemma MR1_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & dist mr1 (x, y) < z - x --> y < z"
@@ -1098,7 +1108,7 @@
;setup_theory nets
constdefs
- dorder :: "('a::type => 'a::type => bool) => bool"
+ dorder :: "('a => 'a => bool) => bool"
"dorder ==
%g::'a::type => 'a::type => bool.
ALL (x::'a::type) y::'a::type.
@@ -1113,8 +1123,7 @@
by (import nets dorder)
constdefs
- tends :: "('b::type => 'a::type)
-=> 'a::type => 'a::type topology * ('b::type => 'b::type => bool) => bool"
+ tends :: "('b => 'a) => 'a => 'a topology * ('b => 'b => bool) => bool"
"tends ==
%(s::'b::type => 'a::type) (l::'a::type) (top::'a::type topology,
g::'b::type => 'b::type => bool).
@@ -1131,8 +1140,7 @@
by (import nets tends)
constdefs
- bounded :: "'a::type metric * ('b::type => 'b::type => bool)
-=> ('b::type => 'a::type) => bool"
+ bounded :: "'a metric * ('b => 'b => bool) => ('b => 'a) => bool"
"bounded ==
%(m::'a::type metric, g::'b::type => 'b::type => bool)
f::'b::type => 'a::type.
@@ -1147,14 +1155,13 @@
by (import nets bounded)
constdefs
- tendsto :: "'a::type metric * 'a::type => 'a::type => 'a::type => bool"
+ tendsto :: "'a metric * 'a => 'a => 'a => bool"
"tendsto ==
%(m::'a::type metric, x::'a::type) (y::'a::type) z::'a::type.
- (0::real) < dist m (x, y) & dist m (x, y) <= dist m (x, z)"
+ 0 < dist m (x, y) & dist m (x, y) <= dist m (x, z)"
lemma tendsto: "ALL (m::'a::type metric) (x::'a::type) (y::'a::type) z::'a::type.
- tendsto (m, x) y z =
- ((0::real) < dist m (x, y) & dist m (x, y) <= dist m (x, z))"
+ tendsto (m, x) y z = (0 < dist m (x, y) & dist m (x, y) <= dist m (x, z))"
by (import nets tendsto)
lemma DORDER_LEMMA: "ALL g::'a::type => 'a::type => bool.
@@ -1174,7 +1181,7 @@
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>0::real.
+ (ALL e>0.
EX n::'b::type.
g n n & (ALL m::'b::type. g m n --> dist d (x m, x0) < e))"
by (import nets MTOP_TENDS)
@@ -1188,18 +1195,17 @@
lemma SEQ_TENDS: "ALL (d::'a::type metric) (x::nat => 'a::type) x0::'a::type.
tends x x0 (mtop d, nat_ge) =
- (ALL xa>0::real.
- EX xb::nat. ALL xc::nat. xb <= xc --> dist d (x xc, x0) < xa)"
+ (ALL xa>0. EX xb::nat. ALL xc::nat. xb <= xc --> dist d (x xc, x0) < xa)"
by (import nets 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>0::real.
- EX d>0::real.
+ (ALL e>0.
+ EX d>0.
ALL x::'a::type.
- (0::real) < dist m1 (x, x0) & dist m1 (x, x0) <= d -->
+ 0 < dist m1 (x, x0) & dist m1 (x, x0) <= d -->
dist m2 (f x, y0) < e)"
by (import nets LIM_TENDS)
@@ -1207,10 +1213,10 @@
(x0::'a::type) y0::'b::type.
limpt (mtop m1) x0 re_universe -->
tends f y0 (mtop m2, tendsto (m1, x0)) =
- (ALL e>0::real.
- EX d>0::real.
+ (ALL e>0.
+ EX d>0.
ALL x::'a::type.
- (0::real) < dist m1 (x, x0) & dist m1 (x, x0) < d -->
+ 0 < dist m1 (x, x0) & dist m1 (x, x0) < d -->
dist m2 (f x, y0) < e)"
by (import nets LIM_TENDS2)
@@ -1221,8 +1227,7 @@
by (import nets MR1_BOUNDED)
lemma NET_NULL: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) =
- tends (%n::'a::type. x n - x0) (0::real) (mtop mr1, g)"
+ tends x x0 (mtop mr1, g) = tends (%n::'a::type. x n - x0) 0 (mtop mr1, g)"
by (import nets NET_NULL)
lemma NET_CONV_BOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
@@ -1230,32 +1235,31 @@
by (import nets NET_CONV_BOUNDED)
lemma NET_CONV_NZ: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= (0::real) -->
- (EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n ~= (0::real)))"
+ tends x x0 (mtop mr1, g) & x0 ~= 0 -->
+ (EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n ~= 0))"
by (import nets NET_CONV_NZ)
lemma NET_CONV_IBOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= (0::real) -->
+ tends x x0 (mtop mr1, g) & x0 ~= 0 -->
bounded (mr1, g) (%n::'a::type. inverse (x n))"
by (import nets NET_CONV_IBOUNDED)
lemma NET_NULL_ADD: "ALL g::'a::type => 'a::type => bool.
dorder g -->
(ALL (x::'a::type => real) y::'a::type => real.
- tends x (0::real) (mtop mr1, g) & tends y (0::real) (mtop mr1, g) -->
- tends (%n::'a::type. x n + y n) (0::real) (mtop mr1, g))"
+ tends x 0 (mtop mr1, g) & tends y 0 (mtop mr1, g) -->
+ tends (%n::'a::type. x n + y n) 0 (mtop mr1, g))"
by (import nets NET_NULL_ADD)
lemma NET_NULL_MUL: "ALL g::'a::type => 'a::type => bool.
dorder g -->
(ALL (x::'a::type => real) y::'a::type => real.
- bounded (mr1, g) x & tends y (0::real) (mtop mr1, g) -->
- tends (%n::'a::type. x n * y n) (0::real) (mtop mr1, g))"
+ bounded (mr1, g) x & tends y 0 (mtop mr1, g) -->
+ tends (%n::'a::type. x n * y n) 0 (mtop mr1, g))"
by (import nets NET_NULL_MUL)
lemma NET_NULL_CMUL: "ALL (g::'a::type => 'a::type => bool) (k::real) x::'a::type => real.
- tends x (0::real) (mtop mr1, g) -->
- tends (%n::'a::type. k * x n) (0::real) (mtop mr1, g)"
+ tends x 0 (mtop mr1, g) --> tends (%n::'a::type. k * x n) 0 (mtop mr1, g)"
by (import nets NET_NULL_CMUL)
lemma NET_ADD: "ALL g::'a::type => 'a::type => bool.
@@ -1289,15 +1293,14 @@
lemma NET_INV: "ALL g::'a::type => 'a::type => bool.
dorder g -->
(ALL (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= (0::real) -->
+ tends x x0 (mtop mr1, g) & x0 ~= 0 -->
tends (%n::'a::type. inverse (x n)) (inverse x0) (mtop mr1, g))"
by (import nets NET_INV)
lemma NET_DIV: "ALL g::'a::type => 'a::type => bool.
dorder g -->
(ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
- tends x x0 (mtop mr1, g) &
- tends y y0 (mtop mr1, g) & y0 ~= (0::real) -->
+ tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) & y0 ~= 0 -->
tends (%xa::'a::type. x xa / y xa) (x0 / y0) (mtop mr1, g))"
by (import nets NET_DIV)
@@ -1764,7 +1767,7 @@
lemma SEQ_SUC: "ALL (f::nat => real) l::real. --> f l = --> (%n::nat. f (Suc n)) l"
by (import seq SEQ_SUC)
-lemma SEQ_ABS: "ALL f::nat => real. --> (%n::nat. abs (f n)) (0::real) = --> f (0::real)"
+lemma SEQ_ABS: "ALL f::nat => real. --> (%n::nat. abs (f n)) 0 = --> f 0"
by (import seq SEQ_ABS)
lemma SEQ_ABS_IMP: "(All::((nat => real) => bool) => bool)
@@ -1929,10 +1932,9 @@
constdefs
sums :: "(nat => real) => real => bool"
- "sums == %f::nat => real. --> (%n::nat. real.sum (0::nat, n) f)"
-
-lemma sums: "ALL (f::nat => real) s::real.
- sums f s = --> (%n::nat. real.sum (0::nat, n) f) s"
+ "sums == %f::nat => real. --> (%n::nat. real.sum (0, n) f)"
+
+lemma sums: "ALL (f::nat => real) s::real. sums f s = --> (%n::nat. real.sum (0, n) f) s"
by (import seq sums)
constdefs
@@ -2538,7 +2540,7 @@
by (import lim LIM_DIV)
lemma LIM_NULL: "ALL (f::real => real) (l::real) x::real.
- tends_real_real f l x = tends_real_real (%x::real. f x - l) (0::real) x"
+ tends_real_real f l x = tends_real_real (%x::real. f x - l) 0 x"
by (import lim LIM_NULL)
lemma LIM_X: "ALL x0::real. tends_real_real (%x::real. x) x0 x0"
@@ -2614,21 +2616,19 @@
diffl :: "(real => real) => real => real => bool"
"diffl ==
%(f::real => real) (l::real) x::real.
- tends_real_real (%h::real. (f (x + h) - f x) / h) l (0::real)"
+ tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
lemma diffl: "ALL (f::real => real) (l::real) x::real.
- diffl f l x =
- tends_real_real (%h::real. (f (x + h) - f x) / h) l (0::real)"
+ diffl f l x = tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
by (import lim diffl)
constdefs
contl :: "(real => real) => real => bool"
"contl ==
-%(f::real => real) x::real.
- tends_real_real (%h::real. f (x + h)) (f x) (0::real)"
+%(f::real => real) x::real. tends_real_real (%h::real. f (x + h)) (f x) 0"
lemma contl: "ALL (f::real => real) x::real.
- contl f x = tends_real_real (%h::real. f (x + h)) (f x) (0::real)"
+ contl f x = tends_real_real (%h::real. f (x + h)) (f x) 0"
by (import lim contl)
constdefs
@@ -2839,7 +2839,7 @@
((op =::real => real => bool) (f x) y))))))))"
by (import lim IVT2)
-lemma DIFF_CONST: "ALL k::real. All (diffl (%x::real. k) (0::real))"
+lemma DIFF_CONST: "ALL k::real. All (diffl (%x::real. k) 0)"
by (import lim DIFF_CONST)
lemma DIFF_ADD: "(All::((real => real) => bool) => bool)
@@ -2960,11 +2960,10 @@
((op *::real => real => real) l m) x))))))"
by (import lim DIFF_CHAIN)
-lemma DIFF_X: "All (diffl (%x::real. x) (1::real))"
+lemma DIFF_X: "All (diffl (%x::real. x) 1)"
by (import lim DIFF_X)
-lemma DIFF_POW: "ALL (n::nat) x::real.
- diffl (%x::real. x ^ n) (real n * x ^ (n - (1::nat))) x"
+lemma DIFF_POW: "ALL (n::nat) x::real. diffl (%x::real. x ^ n) (real n * x ^ (n - 1)) x"
by (import lim DIFF_POW)
lemma DIFF_XM1: "(All::(real => bool) => bool)
@@ -3877,18 +3876,18 @@
;setup_theory powser
lemma POWDIFF_LEMMA: "ALL (n::nat) (x::real) y::real.
- real.sum (0::nat, Suc n) (%p::nat. x ^ p * y ^ (Suc n - p)) =
- y * real.sum (0::nat, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
+ real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (Suc n - p)) =
+ y * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
by (import powser POWDIFF_LEMMA)
lemma POWDIFF: "ALL (n::nat) (x::real) y::real.
x ^ Suc n - y ^ Suc n =
- (x - y) * real.sum (0::nat, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
+ (x - y) * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
by (import powser POWDIFF)
lemma POWREV: "ALL (n::nat) (x::real) y::real.
- real.sum (0::nat, Suc n) (%xa::nat. x ^ xa * y ^ (n - xa)) =
- real.sum (0::nat, Suc n) (%xa::nat. x ^ (n - xa) * y ^ xa)"
+ real.sum (0, Suc n) (%xa::nat. x ^ xa * y ^ (n - xa)) =
+ real.sum (0, Suc n) (%xa::nat. x ^ (n - xa) * y ^ xa)"
by (import powser POWREV)
lemma POWSER_INSIDEA: "(All::((nat => real) => bool) => bool)
@@ -3943,15 +3942,15 @@
by (import powser DIFFS_NEG)
lemma DIFFS_LEMMA: "ALL (n::nat) (c::nat => real) x::real.
- real.sum (0::nat, n) (%n::nat. diffs c n * x ^ n) =
- real.sum (0::nat, n) (%n::nat. real n * (c n * x ^ (n - (1::nat)))) +
- real n * (c n * x ^ (n - (1::nat)))"
+ real.sum (0, n) (%n::nat. diffs c n * x ^ n) =
+ real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) +
+ real n * (c n * x ^ (n - 1))"
by (import powser DIFFS_LEMMA)
lemma DIFFS_LEMMA2: "ALL (n::nat) (c::nat => real) x::real.
- real.sum (0::nat, n) (%n::nat. real n * (c n * x ^ (n - (1::nat)))) =
- real.sum (0::nat, n) (%n::nat. diffs c n * x ^ n) -
- real n * (c n * x ^ (n - (1::nat)))"
+ real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) =
+ real.sum (0, n) (%n::nat. diffs c n * x ^ n) -
+ real n * (c n * x ^ (n - 1))"
by (import powser DIFFS_LEMMA2)
lemma DIFFS_EQUIV: "(All::((nat => real) => bool) => bool)
@@ -3978,8 +3977,8 @@
by (import powser DIFFS_EQUIV)
lemma TERMDIFF_LEMMA1: "ALL (m::nat) (z::real) h::real.
- real.sum (0::nat, m) (%p::nat. (z + h) ^ (m - p) * z ^ p - z ^ m) =
- real.sum (0::nat, m) (%p::nat. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
+ real.sum (0, m) (%p::nat. (z + h) ^ (m - p) * z ^ p - z ^ m) =
+ real.sum (0, m) (%p::nat. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
by (import powser TERMDIFF_LEMMA1)
lemma TERMDIFF_LEMMA2: "(All::(real => bool) => bool)
@@ -4185,23 +4184,17 @@
constdefs
cos :: "real => real"
- "(cos ==
- (%(x::real).
- (suminf
- (%(n::nat).
- ((if (EVEN n)
- then (((- (1::real)) ^ (n div (2::nat))) / (real (FACT n)))
- else (0::real)) *
- (x ^ n))))))"
-
-lemma cos: "(ALL (x::real).
- ((cos x) =
- (suminf
- (%(n::nat).
- ((if (EVEN n)
- then (((- (1::real)) ^ (n div (2::nat))) / (real (FACT n)))
- else (0::real)) *
- (x ^ n))))))"
+ "cos ==
+%x::real.
+ suminf
+ (%n::nat.
+ (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
+
+lemma cos: "ALL x::real.
+ cos x =
+ suminf
+ (%n::nat.
+ (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
by (import transc cos)
constdefs
@@ -4210,18 +4203,14 @@
%x::real.
suminf
(%n::nat.
- (if EVEN n then 0::real
- else (- (1::real)) ^ ((n - (1::nat)) div (2::nat)) /
- real (FACT n)) *
+ (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
x ^ n)"
lemma sin: "ALL x::real.
sin x =
suminf
(%n::nat.
- (if EVEN n then 0::real
- else (- (1::real)) ^ ((n - (1::nat)) div (2::nat)) /
- real (FACT n)) *
+ (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
x ^ n)"
by (import transc sin)
@@ -4231,56 +4220,36 @@
lemma SIN_CONVERGES: "ALL x::real.
sums
(%n::nat.
- (if EVEN n then 0::real
- else (- (1::real)) ^ ((n - (1::nat)) div (2::nat)) /
- real (FACT n)) *
+ (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
x ^ n)
(sin x)"
by (import transc SIN_CONVERGES)
-lemma COS_CONVERGES: "(ALL (x::real).
- (sums
- (%(n::nat).
- ((if (EVEN n)
- then (((- (1::real)) ^ (n div (2::nat))) / (real (FACT n)))
- else (0::real)) *
- (x ^ n)))
- (cos x)))"
+lemma COS_CONVERGES: "ALL x::real.
+ sums
+ (%n::nat.
+ (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)
+ (cos x)"
by (import transc COS_CONVERGES)
lemma EXP_FDIFF: "diffs (%n::nat. inverse (real (FACT n))) =
(%n::nat. inverse (real (FACT n)))"
by (import transc EXP_FDIFF)
-lemma SIN_FDIFF: "((diffs
- (%(n::nat).
- (if (EVEN n) then (0::real)
- else (((- (1::real)) ^ ((n - (1::nat)) div (2::nat))) /
- (real (FACT n)))))) =
- (%(n::nat).
- (if (EVEN n)
- then (((- (1::real)) ^ (n div (2::nat))) / (real (FACT n)))
- else (0::real))))"
+lemma SIN_FDIFF: "diffs
+ (%n::nat. if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) =
+(%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0)"
by (import transc SIN_FDIFF)
-lemma COS_FDIFF: "((diffs
- (%(n::nat).
- (if (EVEN n)
- then (((- (1::real)) ^ (n div (2::nat))) / (real (FACT n)))
- else (0::real)))) =
- (%(n::nat).
- (- (if (EVEN n) then (0::real)
- else (((- (1::real)) ^ ((n - (1::nat)) div (2::nat))) /
- (real (FACT n)))))))"
+lemma COS_FDIFF: "diffs (%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) =
+(%n::nat. - (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)))"
by (import transc COS_FDIFF)
lemma SIN_NEGLEMMA: "ALL x::real.
- sin x =
suminf
(%n::nat.
- - ((if EVEN n then 0::real
- else (- (1::real)) ^ ((n - (1::nat)) div (2::nat)) /
- real (FACT n)) *
+ - ((if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
x ^ n))"
by (import transc SIN_NEGLEMMA)
@@ -4402,22 +4371,22 @@
x))))))))))"
by (import transc DIFF_COMPOSITE)
-lemma EXP_0: "exp (0::real) = (1::real)"
+lemma EXP_0: "exp 0 = 1"
by (import transc EXP_0)
-lemma EXP_LE_X: "ALL x>=0::real. (1::real) + x <= exp x"
+lemma EXP_LE_X: "ALL x>=0. 1 + x <= exp x"
by (import transc EXP_LE_X)
-lemma EXP_LT_1: "ALL x>0::real. (1::real) < exp x"
+lemma EXP_LT_1: "ALL x>0. 1 < exp x"
by (import transc EXP_LT_1)
lemma EXP_ADD_MUL: "ALL (x::real) y::real. exp (x + y) * exp (- x) = exp y"
by (import transc EXP_ADD_MUL)
-lemma EXP_NEG_MUL: "ALL x::real. exp x * exp (- x) = (1::real)"
+lemma EXP_NEG_MUL: "ALL x::real. exp x * exp (- x) = 1"
by (import transc EXP_NEG_MUL)
-lemma EXP_NEG_MUL2: "ALL x::real. exp (- x) * exp x = (1::real)"
+lemma EXP_NEG_MUL2: "ALL x::real. exp (- x) * exp x = 1"
by (import transc EXP_NEG_MUL2)
lemma EXP_NEG: "ALL x::real. exp (- x) = inverse (exp x)"
@@ -4426,13 +4395,13 @@
lemma EXP_ADD: "ALL (x::real) y::real. exp (x + y) = exp x * exp y"
by (import transc EXP_ADD)
-lemma EXP_POS_LE: "ALL x::real. (0::real) <= exp x"
+lemma EXP_POS_LE: "ALL x::real. 0 <= exp x"
by (import transc EXP_POS_LE)
-lemma EXP_NZ: "ALL x::real. exp x ~= (0::real)"
+lemma EXP_NZ: "ALL x::real. exp x ~= 0"
by (import transc EXP_NZ)
-lemma EXP_POS_LT: "ALL x::real. (0::real) < exp x"
+lemma EXP_POS_LT: "ALL x::real. 0 < exp x"
by (import transc EXP_POS_LT)
lemma EXP_N: "ALL (n::nat) x::real. exp (real n * x) = exp x ^ n"
@@ -4459,10 +4428,10 @@
lemma EXP_INJ: "ALL (x::real) y::real. (exp x = exp y) = (x = y)"
by (import transc EXP_INJ)
-lemma EXP_TOTAL_LEMMA: "ALL y>=1::real. EX x>=0::real. x <= y - (1::real) & exp x = y"
+lemma EXP_TOTAL_LEMMA: "ALL y>=1. EX x>=0. x <= y - 1 & exp x = y"
by (import transc EXP_TOTAL_LEMMA)
-lemma EXP_TOTAL: "ALL y>0::real. EX x::real. exp x = y"
+lemma EXP_TOTAL: "ALL y>0. EX x::real. exp x = y"
by (import transc EXP_TOTAL)
constdefs
@@ -4475,7 +4444,7 @@
lemma LN_EXP: "ALL x::real. ln (exp x) = x"
by (import transc LN_EXP)
-lemma EXP_LN: "ALL x::real. (exp (ln x) = x) = ((0::real) < x)"
+lemma EXP_LN: "ALL x::real. (exp (ln x) = x) = (0 < x)"
by (import transc EXP_LN)
lemma LN_MUL: "(All::(real => bool) => bool)
@@ -4506,10 +4475,10 @@
((op =::real => real => bool) x y))))"
by (import transc LN_INJ)
-lemma LN_1: "ln (1::real) = (0::real)"
+lemma LN_1: "ln 1 = 0"
by (import transc LN_1)
-lemma LN_INV: "ALL x>0::real. ln (inverse x) = - ln x"
+lemma LN_INV: "ALL x>0. ln (inverse x) = - ln x"
by (import transc LN_INV)
lemma LN_DIV: "(All::(real => bool) => bool)
@@ -4566,13 +4535,13 @@
((ln::real => real) x)))))"
by (import transc LN_POW)
-lemma LN_LE: "ALL x>=0::real. ln ((1::real) + x) <= x"
+lemma LN_LE: "ALL x>=0. ln (1 + x) <= x"
by (import transc LN_LE)
-lemma LN_LT_X: "ALL x>0::real. ln x < x"
+lemma LN_LT_X: "ALL x>0. ln x < x"
by (import transc LN_LT_X)
-lemma LN_POS: "ALL x>=1::real. (0::real) <= ln x"
+lemma LN_POS: "ALL x>=1. 0 <= ln x"
by (import transc LN_POS)
constdefs
@@ -4606,9 +4575,9 @@
constdefs
sqrt :: "real => real"
- "sqrt == root (2::nat)"
-
-lemma sqrt: "ALL x::real. sqrt x = root (2::nat) x"
+ "sqrt == root 2"
+
+lemma sqrt: "ALL x::real. sqrt x = root 2 x"
by (import transc sqrt)
lemma ROOT_LT_LEMMA: "(All::(nat => bool) => bool)
@@ -4639,10 +4608,10 @@
((real::nat => real) ((Suc::nat => nat) n)))))))"
by (import transc ROOT_LN)
-lemma ROOT_0: "ALL n::nat. root (Suc n) (0::real) = (0::real)"
+lemma ROOT_0: "ALL n::nat. root (Suc n) 0 = 0"
by (import transc ROOT_0)
-lemma ROOT_1: "ALL n::nat. root (Suc n) (1::real) = (1::real)"
+lemma ROOT_1: "ALL n::nat. root (Suc n) 1 = 1"
by (import transc ROOT_1)
lemma ROOT_POS_LT: "(All::(nat => bool) => bool)
@@ -4773,22 +4742,22 @@
((root::nat => real => real) ((Suc::nat => nat) n) y))))"
by (import transc ROOT_MONO_LE)
-lemma SQRT_0: "sqrt (0::real) = (0::real)"
+lemma SQRT_0: "sqrt 0 = 0"
by (import transc SQRT_0)
-lemma SQRT_1: "sqrt (1::real) = (1::real)"
+lemma SQRT_1: "sqrt 1 = 1"
by (import transc SQRT_1)
-lemma SQRT_POS_LT: "ALL x>0::real. (0::real) < sqrt x"
+lemma SQRT_POS_LT: "ALL x>0. 0 < sqrt x"
by (import transc SQRT_POS_LT)
-lemma SQRT_POS_LE: "ALL x>=0::real. (0::real) <= sqrt x"
+lemma SQRT_POS_LE: "ALL x>=0. 0 <= sqrt x"
by (import transc SQRT_POS_LE)
-lemma SQRT_POW2: "ALL x::real. (sqrt x ^ 2 = x) = ((0::real) <= x)"
+lemma SQRT_POW2: "ALL x::real. (sqrt x ^ 2 = x) = (0 <= x)"
by (import transc SQRT_POW2)
-lemma SQRT_POW_2: "ALL x>=0::real. sqrt x ^ 2 = x"
+lemma SQRT_POW_2: "ALL x>=0. sqrt x ^ 2 = x"
by (import transc SQRT_POW_2)
lemma POW_2_SQRT: "(op -->::bool => bool => bool)
@@ -4837,7 +4806,7 @@
((sqrt::real => real) xa)))))"
by (import transc SQRT_MUL)
-lemma SQRT_INV: "ALL x>=0::real. sqrt (inverse x) = inverse (sqrt x)"
+lemma SQRT_INV: "ALL x>=0. sqrt (inverse x) = inverse (sqrt x)"
by (import transc SQRT_INV)
lemma SQRT_DIV: "(All::(real => bool) => bool)
@@ -4891,7 +4860,7 @@
(bit.B0::bit)))))))"
by (import transc SQRT_EVEN_POW2)
-lemma REAL_DIV_SQRT: "ALL x>=0::real. x / sqrt x = sqrt x"
+lemma REAL_DIV_SQRT: "ALL x>=0. x / sqrt x = sqrt x"
by (import transc REAL_DIV_SQRT)
lemma SQRT_EQ: "(All::(real => bool) => bool)
@@ -4912,34 +4881,34 @@
((op =::real => real => bool) x ((sqrt::real => real) y))))"
by (import transc SQRT_EQ)
-lemma SIN_0: "sin (0::real) = (0::real)"
+lemma SIN_0: "sin 0 = 0"
by (import transc SIN_0)
-lemma COS_0: "cos (0::real) = (1::real)"
+lemma COS_0: "cos 0 = 1"
by (import transc COS_0)
-lemma SIN_CIRCLE: "ALL x::real. sin x ^ 2 + cos x ^ 2 = (1::real)"
+lemma SIN_CIRCLE: "ALL x::real. sin x ^ 2 + cos x ^ 2 = 1"
by (import transc SIN_CIRCLE)
-lemma SIN_BOUND: "ALL x::real. abs (sin x) <= (1::real)"
+lemma SIN_BOUND: "ALL x::real. abs (sin x) <= 1"
by (import transc SIN_BOUND)
-lemma SIN_BOUNDS: "ALL x::real. - (1::real) <= sin x & sin x <= (1::real)"
+lemma SIN_BOUNDS: "ALL x::real. - 1 <= sin x & sin x <= 1"
by (import transc SIN_BOUNDS)
-lemma COS_BOUND: "ALL x::real. abs (cos x) <= (1::real)"
+lemma COS_BOUND: "ALL x::real. abs (cos x) <= 1"
by (import transc COS_BOUND)
-lemma COS_BOUNDS: "ALL x::real. - (1::real) <= cos x & cos x <= (1::real)"
+lemma COS_BOUNDS: "ALL x::real. - 1 <= cos x & cos x <= 1"
by (import transc COS_BOUNDS)
lemma SIN_COS_ADD: "ALL (x::real) y::real.
(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 =
- (0::real)"
+ 0"
by (import transc SIN_COS_ADD)
-lemma SIN_COS_NEG: "ALL x::real. (sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2 = (0::real)"
+lemma SIN_COS_NEG: "ALL x::real. (sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2 = 0"
by (import transc SIN_COS_NEG)
lemma SIN_ADD: "ALL (x::real) y::real. sin (x + y) = sin x * cos y + cos x * sin y"
@@ -4954,17 +4923,14 @@
lemma COS_NEG: "ALL x::real. cos (- x) = cos x"
by (import transc COS_NEG)
-lemma SIN_DOUBLE: "ALL x::real. sin ((2::real) * x) = (2::real) * (sin x * cos x)"
+lemma SIN_DOUBLE: "ALL x::real. sin (2 * x) = 2 * (sin x * cos x)"
by (import transc SIN_DOUBLE)
-lemma COS_DOUBLE: "ALL x::real. cos ((2::real) * x) = cos x ^ 2 - sin x ^ 2"
+lemma COS_DOUBLE: "ALL x::real. cos (2 * x) = cos x ^ 2 - sin x ^ 2"
by (import transc COS_DOUBLE)
lemma SIN_PAIRED: "ALL x::real.
- sums
- (%n::nat.
- (- (1::real)) ^ n / real (FACT ((2::nat) * n + (1::nat))) *
- x ^ ((2::nat) * n + (1::nat)))
+ sums (%n::nat. (- 1) ^ n / real (FACT (2 * n + 1)) * x ^ (2 * n + 1))
(sin x)"
by (import transc SIN_PAIRED)
@@ -4982,55 +4948,47 @@
by (import transc SIN_POS)
lemma COS_PAIRED: "ALL x::real.
- sums
- (%n::nat.
- (- (1::real)) ^ n / real (FACT ((2::nat) * n)) * x ^ ((2::nat) * n))
- (cos x)"
+ sums (%n::nat. (- 1) ^ n / real (FACT (2 * n)) * x ^ (2 * n)) (cos x)"
by (import transc COS_PAIRED)
-lemma COS_2: "cos (2::real) < (0::real)"
+lemma COS_2: "cos 2 < 0"
by (import transc COS_2)
-lemma COS_ISZERO: "EX! x::real. (0::real) <= x & x <= (2::real) & cos x = (0::real)"
+lemma COS_ISZERO: "EX! x::real. 0 <= x & x <= 2 & cos x = 0"
by (import transc COS_ISZERO)
constdefs
pi :: "real"
- "pi ==
-(2::real) *
-(SOME x::real. (0::real) <= x & x <= (2::real) & cos x = (0::real))"
-
-lemma pi: "pi =
-(2::real) *
-(SOME x::real. (0::real) <= x & x <= (2::real) & cos x = (0::real))"
+ "pi == 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
+
+lemma pi: "pi = 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
by (import transc pi)
-lemma PI2: "pi / (2::real) =
-(SOME x::real. (0::real) <= x & x <= (2::real) & cos x = (0::real))"
+lemma PI2: "pi / 2 = (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
by (import transc PI2)
-lemma COS_PI2: "cos (pi / (2::real)) = (0::real)"
+lemma COS_PI2: "cos (pi / 2) = 0"
by (import transc COS_PI2)
-lemma PI2_BOUNDS: "(0::real) < pi / (2::real) & pi / (2::real) < (2::real)"
+lemma PI2_BOUNDS: "0 < pi / 2 & pi / 2 < 2"
by (import transc PI2_BOUNDS)
-lemma PI_POS: "(0::real) < pi"
+lemma PI_POS: "0 < pi"
by (import transc PI_POS)
-lemma SIN_PI2: "sin (pi / (2::real)) = (1::real)"
+lemma SIN_PI2: "sin (pi / 2) = 1"
by (import transc SIN_PI2)
-lemma COS_PI: "cos pi = - (1::real)"
+lemma COS_PI: "cos pi = - 1"
by (import transc COS_PI)
-lemma SIN_PI: "sin pi = (0::real)"
+lemma SIN_PI: "sin pi = 0"
by (import transc SIN_PI)
-lemma SIN_COS: "ALL x::real. sin x = cos (pi / (2::real) - x)"
+lemma SIN_COS: "ALL x::real. sin x = cos (pi / 2 - x)"
by (import transc SIN_COS)
-lemma COS_SIN: "ALL x::real. cos x = sin (pi / (2::real) - x)"
+lemma COS_SIN: "ALL x::real. cos x = sin (pi / 2 - x)"
by (import transc COS_SIN)
lemma SIN_PERIODIC_PI: "ALL x::real. sin (x + pi) = - sin x"
@@ -5039,16 +4997,16 @@
lemma COS_PERIODIC_PI: "ALL x::real. cos (x + pi) = - cos x"
by (import transc COS_PERIODIC_PI)
-lemma SIN_PERIODIC: "ALL x::real. sin (x + (2::real) * pi) = sin x"
+lemma SIN_PERIODIC: "ALL x::real. sin (x + 2 * pi) = sin x"
by (import transc SIN_PERIODIC)
-lemma COS_PERIODIC: "ALL x::real. cos (x + (2::real) * pi) = cos x"
+lemma COS_PERIODIC: "ALL x::real. cos (x + 2 * pi) = cos x"
by (import transc COS_PERIODIC)
-lemma COS_NPI: "ALL n::nat. cos (real n * pi) = (- (1::real)) ^ n"
+lemma COS_NPI: "ALL n::nat. cos (real n * pi) = (- 1) ^ n"
by (import transc COS_NPI)
-lemma SIN_NPI: "ALL n::nat. sin (real n * pi) = (0::real)"
+lemma SIN_NPI: "ALL n::nat. sin (real n * pi) = 0"
by (import transc SIN_NPI)
lemma SIN_POS_PI2: "(All::(real => bool) => bool)
@@ -5259,15 +5217,15 @@
by (import transc SIN_ZERO_LEMMA)
lemma COS_ZERO: "ALL x::real.
- (cos x = (0::real)) =
- ((EX n::nat. ~ EVEN n & x = real n * (pi / (2::real))) |
- (EX n::nat. ~ EVEN n & x = - (real n * (pi / (2::real)))))"
+ (cos x = 0) =
+ ((EX n::nat. ~ EVEN n & x = real n * (pi / 2)) |
+ (EX n::nat. ~ EVEN n & x = - (real n * (pi / 2))))"
by (import transc COS_ZERO)
lemma SIN_ZERO: "ALL x::real.
- (sin x = (0::real)) =
- ((EX n::nat. EVEN n & x = real n * (pi / (2::real))) |
- (EX n::nat. EVEN n & x = - (real n * (pi / (2::real)))))"
+ (sin x = 0) =
+ ((EX n::nat. EVEN n & x = real n * (pi / 2)) |
+ (EX n::nat. EVEN n & x = - (real n * (pi / 2))))"
by (import transc SIN_ZERO)
constdefs
@@ -5277,19 +5235,19 @@
lemma tan: "ALL x::real. tan x = sin x / cos x"
by (import transc tan)
-lemma TAN_0: "tan (0::real) = (0::real)"
+lemma TAN_0: "tan 0 = 0"
by (import transc TAN_0)
-lemma TAN_PI: "tan pi = (0::real)"
+lemma TAN_PI: "tan pi = 0"
by (import transc TAN_PI)
-lemma TAN_NPI: "ALL n::nat. tan (real n * pi) = (0::real)"
+lemma TAN_NPI: "ALL n::nat. tan (real n * pi) = 0"
by (import transc TAN_NPI)
lemma TAN_NEG: "ALL x::real. tan (- x) = - tan x"
by (import transc TAN_NEG)
-lemma TAN_PERIODIC: "ALL x::real. tan (x + (2::real) * pi) = tan x"
+lemma TAN_PERIODIC: "ALL x::real. tan (x + 2 * pi) = tan x"
by (import transc TAN_PERIODIC)
lemma TAN_ADD: "(All::(real => bool) => bool)
@@ -5393,43 +5351,35 @@
x))"
by (import transc DIFF_TAN)
-lemma TAN_TOTAL_LEMMA: "ALL y>0::real. EX x>0::real. x < pi / (2::real) & y < tan x"
+lemma TAN_TOTAL_LEMMA: "ALL y>0. EX x>0. x < pi / 2 & y < tan x"
by (import transc TAN_TOTAL_LEMMA)
-lemma TAN_TOTAL_POS: "ALL y>=0::real. EX x>=0::real. x < pi / (2::real) & tan x = y"
+lemma TAN_TOTAL_POS: "ALL y>=0. EX x>=0. x < pi / 2 & tan x = y"
by (import transc TAN_TOTAL_POS)
-lemma TAN_TOTAL: "ALL y::real.
- EX! x::real. - (pi / (2::real)) < x & x < pi / (2::real) & tan x = y"
+lemma TAN_TOTAL: "ALL y::real. EX! x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
by (import transc TAN_TOTAL)
constdefs
asn :: "real => real"
- "asn ==
-%y::real.
- SOME x::real. - (pi / (2::real)) <= x & x <= pi / (2::real) & sin x = y"
+ "asn == %y::real. SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y"
lemma asn: "ALL y::real.
- asn y =
- (SOME x::real. - (pi / (2::real)) <= x & x <= pi / (2::real) & sin x = y)"
+ asn y = (SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y)"
by (import transc asn)
constdefs
acs :: "real => real"
- "acs == %y::real. SOME x::real. (0::real) <= x & x <= pi & cos x = y"
-
-lemma acs: "ALL y::real. acs y = (SOME x::real. (0::real) <= x & x <= pi & cos x = y)"
+ "acs == %y::real. SOME x::real. 0 <= x & x <= pi & cos x = y"
+
+lemma acs: "ALL y::real. acs y = (SOME x::real. 0 <= x & x <= pi & cos x = y)"
by (import transc acs)
constdefs
atn :: "real => real"
- "atn ==
-%y::real.
- SOME x::real. - (pi / (2::real)) < x & x < pi / (2::real) & tan x = y"
-
-lemma atn: "ALL y::real.
- atn y =
- (SOME x::real. - (pi / (2::real)) < x & x < pi / (2::real) & tan x = y)"
+ "atn == %y::real. SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
+
+lemma atn: "ALL y::real. atn y = (SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y)"
by (import transc atn)
lemma ASN: "(All::(real => bool) => bool)
@@ -5600,14 +5550,13 @@
((acs::real => real) ((cos::real => real) x)) x))"
by (import transc COS_ACS)
-lemma ATN: "ALL y::real.
- - (pi / (2::real)) < atn y & atn y < pi / (2::real) & tan (atn y) = y"
+lemma ATN: "ALL y::real. - (pi / 2) < atn y & atn y < pi / 2 & tan (atn y) = y"
by (import transc ATN)
lemma ATN_TAN: "ALL x::real. tan (atn x) = x"
by (import transc ATN_TAN)
-lemma ATN_BOUNDS: "ALL x::real. - (pi / (2::real)) < atn x & atn x < pi / (2::real)"
+lemma ATN_BOUNDS: "ALL x::real. - (pi / 2) < atn x & atn x < pi / 2"
by (import transc ATN_BOUNDS)
lemma TAN_ATN: "(All::(real => bool) => bool)
@@ -5703,7 +5652,7 @@
(bit.B0::bit))))))))"
by (import transc COS_SIN_SQ)
-lemma COS_ATN_NZ: "ALL x::real. cos (atn x) ~= (0::real)"
+lemma COS_ATN_NZ: "ALL x::real. cos (atn x) ~= 0"
by (import transc COS_ATN_NZ)
lemma COS_ASN_NZ: "(All::(real => bool) => bool)
@@ -5758,7 +5707,7 @@
(bit.B0::bit))))))))"
by (import transc SIN_COS_SQRT)
-lemma DIFF_LN: "ALL x>0::real. diffl ln (inverse x) x"
+lemma DIFF_LN: "ALL x>0. diffl ln (inverse x) x"
by (import transc DIFF_LN)
lemma DIFF_ASN_LEMMA: "(All::(real => bool) => bool)
@@ -5825,7 +5774,7 @@
x))"
by (import transc DIFF_ACS)
-lemma DIFF_ATN: "ALL x::real. diffl atn (inverse ((1::real) + x ^ 2)) x"
+lemma DIFF_ATN: "ALL x::real. diffl atn (inverse (1 + x ^ 2)) x"
by (import transc DIFF_ATN)
constdefs
@@ -6002,11 +5951,11 @@
rsum :: "(nat => real) * (nat => real) => (real => real) => real"
"rsum ==
%(D::nat => real, p::nat => real) f::real => real.
- real.sum (0::nat, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
+ real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
lemma rsum: "ALL (D::nat => real) (p::nat => real) f::real => real.
rsum (D, p) f =
- real.sum (0::nat, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
+ real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
by (import transc rsum)
constdefs
@@ -6519,7 +6468,7 @@
((op =::real => real => bool) k1 k2))))))"
by (import transc DINT_UNIQ)
-lemma INTEGRAL_NULL: "ALL (f::real => real) a::real. Dint (a, a) f (0::real)"
+lemma INTEGRAL_NULL: "ALL (f::real => real) a::real. Dint (a, a) f 0"
by (import transc INTEGRAL_NULL)
lemma FTC1: "(All::((real => real) => bool) => bool)
@@ -6795,7 +6744,7 @@
EX xa::real.
abs xa <= abs x &
exp x =
- real.sum (0::nat, n) (%m::nat. x ^ m / real (FACT m)) +
+ real.sum (0, n) (%m::nat. x ^ m / real (FACT m)) +
exp xa / real (FACT n) * x ^ n"
by (import transc MCLAURIN_EXP_LE)
@@ -6823,7 +6772,7 @@
consts
poly :: "real list => real => real"
-specification (poly_primdef: poly) poly_def: "(ALL x::real. poly [] x = (0::real)) &
+specification (poly_primdef: poly) poly_def: "(ALL x::real. poly [] x = 0) &
(ALL (h::real) (t::real list) x::real. poly (h # t) x = h + x * poly t x)"
by (import poly poly_def)
@@ -6847,9 +6796,9 @@
poly_neg :: "real list => real list"
defs
- poly_neg_primdef: "poly_neg == ## (- (1::real))"
-
-lemma poly_neg_def: "poly_neg = ## (- (1::real))"
+ poly_neg_primdef: "poly_neg == ## (- 1)"
+
+lemma poly_neg_def: "poly_neg = ## (- 1)"
by (import poly poly_neg_def)
consts
@@ -6858,14 +6807,13 @@
specification (poly_mul_primdef: poly_mul) poly_mul_def: "(ALL l2::real list. poly_mul [] l2 = []) &
(ALL (h::real) (t::real list) l2::real list.
poly_mul (h # t) l2 =
- (if t = [] then ## h l2
- else poly_add (## h l2) ((0::real) # poly_mul t l2)))"
+ (if t = [] then ## h l2 else poly_add (## h l2) (0 # poly_mul t l2)))"
by (import poly poly_mul_def)
consts
poly_exp :: "real list => nat => real list"
-specification (poly_exp_primdef: poly_exp) poly_exp_def: "(ALL p::real list. poly_exp p (0::nat) = [1::real]) &
+specification (poly_exp_primdef: poly_exp) poly_exp_def: "(ALL p::real list. poly_exp p 0 = [1]) &
(ALL (p::real list) n::nat. poly_exp p (Suc n) = poly_mul p (poly_exp p n))"
by (import poly poly_exp_def)
@@ -6879,10 +6827,9 @@
constdefs
diff :: "real list => real list"
- "diff == %l::real list. if l = [] then [] else poly_diff_aux (1::nat) (tl l)"
-
-lemma poly_diff_def: "ALL l::real list.
- diff l = (if l = [] then [] else poly_diff_aux (1::nat) (tl l))"
+ "diff == %l::real list. if l = [] then [] else poly_diff_aux 1 (tl l)"
+
+lemma poly_diff_def: "ALL l::real list. diff l = (if l = [] then [] else poly_diff_aux 1 (tl l))"
by (import poly poly_diff_def)
lemma POLY_ADD_CLAUSES: "poly_add [] (p2::real list) = p2 &
@@ -6900,12 +6847,11 @@
lemma POLY_MUL_CLAUSES: "poly_mul [] (p2::real list) = [] &
poly_mul [h1::real] p2 = ## h1 p2 &
poly_mul (h1 # (k1::real) # (t1::real list)) p2 =
-poly_add (## h1 p2) ((0::real) # poly_mul (k1 # t1) p2)"
+poly_add (## h1 p2) (0 # poly_mul (k1 # t1) p2)"
by (import poly POLY_MUL_CLAUSES)
lemma POLY_DIFF_CLAUSES: "diff [] = [] &
-diff [c::real] = [] &
-diff ((h::real) # (t::real list)) = poly_diff_aux (1::nat) t"
+diff [c::real] = [] & diff ((h::real) # (t::real list)) = poly_diff_aux 1 t"
by (import poly POLY_DIFF_CLAUSES)
lemma POLY_ADD: "ALL (t::real list) (p2::real list) x::real.
@@ -7054,7 +7000,7 @@
by (import poly POLY_DIFF_NEG)
lemma POLY_DIFF_MUL_LEMMA: "ALL (x::real list) xa::real.
- poly (diff (xa # x)) = poly (poly_add ((0::real) # diff x) x)"
+ poly (diff (xa # x)) = poly (poly_add (0 # diff x) x)"
by (import poly POLY_DIFF_MUL_LEMMA)
lemma POLY_DIFF_MUL: "ALL (t::real list) p2::real list.
@@ -7068,22 +7014,20 @@
by (import poly POLY_DIFF_EXP)
lemma POLY_DIFF_EXP_PRIME: "ALL (n::nat) a::real.
- poly (diff (poly_exp [- a, 1::real] (Suc n))) =
- poly (## (real (Suc n)) (poly_exp [- a, 1::real] n))"
+ poly (diff (poly_exp [- a, 1] (Suc n))) =
+ poly (## (real (Suc n)) (poly_exp [- a, 1] n))"
by (import poly POLY_DIFF_EXP_PRIME)
lemma POLY_LINEAR_REM: "ALL (t::real list) h::real.
EX (q::real list) r::real.
- h # t = poly_add [r] (poly_mul [- (a::real), 1::real] q)"
+ h # t = poly_add [r] (poly_mul [- (a::real), 1] q)"
by (import poly POLY_LINEAR_REM)
lemma POLY_LINEAR_DIVIDES: "ALL (a::real) t::real list.
- (poly t a = (0::real)) =
- (t = [] | (EX q::real list. t = poly_mul [- a, 1::real] q))"
+ (poly t a = 0) = (t = [] | (EX q::real list. t = poly_mul [- a, 1] q))"
by (import poly POLY_LINEAR_DIVIDES)
-lemma POLY_LENGTH_MUL: "ALL x::real list.
- length (poly_mul [- (a::real), 1::real] x) = Suc (length x)"
+lemma POLY_LENGTH_MUL: "ALL x::real list. length (poly_mul [- (a::real), 1] x) = Suc (length x)"
by (import poly POLY_LENGTH_MUL)
lemma POLY_ROOTS_INDEX_LEMMA: "(All::(nat => bool) => bool)
@@ -7233,13 +7177,13 @@
by (import poly POLY_MUL_LCANCEL)
lemma POLY_EXP_EQ_0: "ALL (p::real list) n::nat.
- (poly (poly_exp p n) = poly []) = (poly p = poly [] & n ~= (0::nat))"
+ (poly (poly_exp p n) = poly []) = (poly p = poly [] & n ~= 0)"
by (import poly POLY_EXP_EQ_0)
-lemma POLY_PRIME_EQ_0: "ALL a::real. poly [a, 1::real] ~= poly []"
+lemma POLY_PRIME_EQ_0: "ALL a::real. poly [a, 1] ~= poly []"
by (import poly POLY_PRIME_EQ_0)
-lemma POLY_EXP_PRIME_EQ_0: "ALL (a::real) n::nat. poly (poly_exp [a, 1::real] n) ~= poly []"
+lemma POLY_EXP_PRIME_EQ_0: "ALL (a::real) n::nat. poly (poly_exp [a, 1] n) ~= poly []"
by (import poly POLY_EXP_PRIME_EQ_0)
lemma POLY_ZERO_LEMMA: "(All::(real => bool) => bool)
@@ -7258,12 +7202,12 @@
((poly::real list => real => real) ([]::real list))))))"
by (import poly POLY_ZERO_LEMMA)
-lemma POLY_ZERO: "ALL t::real list. (poly t = poly []) = list_all (%c::real. c = (0::real)) t"
+lemma POLY_ZERO: "ALL t::real list. (poly t = poly []) = list_all (%c::real. c = 0) t"
by (import poly POLY_ZERO)
lemma POLY_DIFF_AUX_ISZERO: "ALL (t::real list) n::nat.
- list_all (%c::real. c = (0::real)) (poly_diff_aux (Suc n) t) =
- list_all (%c::real. c = (0::real)) t"
+ list_all (%c::real. c = 0) (poly_diff_aux (Suc n) t) =
+ list_all (%c::real. c = 0) t"
by (import poly POLY_DIFF_AUX_ISZERO)
lemma POLY_DIFF_ISZERO: "(All::(real list => bool) => bool)
@@ -7320,8 +7264,8 @@
by (import poly poly_divides)
lemma POLY_PRIMES: "ALL (a::real) (p::real list) q::real list.
- poly_divides [a, 1::real] (poly_mul p q) =
- (poly_divides [a, 1::real] p | poly_divides [a, 1::real] q)"
+ poly_divides [a, 1] (poly_mul p q) =
+ (poly_divides [a, 1] p | poly_divides [a, 1] q)"
by (import poly POLY_PRIMES)
lemma POLY_DIVIDES_REFL: "ALL p::real list. poly_divides p p"
@@ -7496,19 +7440,19 @@
"poly_order ==
%(a::real) p::real list.
SOME n::nat.
- poly_divides (poly_exp [- a, 1::real] n) p &
- ~ poly_divides (poly_exp [- a, 1::real] (Suc n)) p"
+ poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p"
lemma poly_order: "ALL (a::real) p::real list.
poly_order a p =
(SOME n::nat.
- poly_divides (poly_exp [- a, 1::real] n) p &
- ~ poly_divides (poly_exp [- a, 1::real] (Suc n)) p)"
+ poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
by (import poly poly_order)
lemma ORDER: "ALL (p::real list) (a::real) n::nat.
- (poly_divides (poly_exp [- a, 1::real] n) p &
- ~ poly_divides (poly_exp [- a, 1::real] (Suc n)) p) =
+ (poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p) =
(n = poly_order a p & poly p ~= poly [])"
by (import poly ORDER)
@@ -7592,11 +7536,11 @@
by (import poly ORDER_POLY)
lemma ORDER_ROOT: "ALL (p::real list) a::real.
- (poly p a = (0::real)) = (poly p = poly [] | poly_order a p ~= (0::nat))"
+ (poly p a = 0) = (poly p = poly [] | poly_order a p ~= 0)"
by (import poly ORDER_ROOT)
lemma ORDER_DIVIDES: "ALL (p::real list) (a::real) n::nat.
- poly_divides (poly_exp [- a, 1::real] n) p =
+ poly_divides (poly_exp [- a, 1] n) p =
(poly p = poly [] | n <= poly_order a p)"
by (import poly ORDER_DIVIDES)
@@ -7729,17 +7673,16 @@
"rsquarefree ==
%p::real list.
poly p ~= poly [] &
- (ALL a::real. poly_order a p = (0::nat) | poly_order a p = (1::nat))"
+ (ALL a::real. poly_order a p = 0 | poly_order a p = 1)"
lemma rsquarefree: "ALL p::real list.
rsquarefree p =
(poly p ~= poly [] &
- (ALL a::real. poly_order a p = (0::nat) | poly_order a p = (1::nat)))"
+ (ALL a::real. poly_order a p = 0 | poly_order a p = 1))"
by (import poly rsquarefree)
lemma RSQUAREFREE_ROOTS: "ALL p::real list.
- rsquarefree p =
- (ALL a::real. ~ (poly p a = (0::real) & poly (diff p) a = (0::real)))"
+ rsquarefree p = (ALL a::real. ~ (poly p a = 0 & poly (diff p) a = 0))"
by (import poly RSQUAREFREE_ROOTS)
lemma RSQUAREFREE_DECOMP: "(All::(real list => bool) => bool)
@@ -7827,7 +7770,7 @@
specification (normalize) normalize: "normalize [] = [] &
(ALL (h::real) t::real list.
normalize (h # t) =
- (if normalize t = [] then if h = (0::real) then [] else [h]
+ (if normalize t = [] then if h = 0 then [] else [h]
else h # normalize t))"
by (import poly normalize)
--- a/src/HOL/Import/HOL/HOL4Vec.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/HOL4Vec.thy Mon Sep 26 16:10:19 2005 +0200
@@ -142,50 +142,47 @@
lemmas word_TY_DEF = typedef_hol2hol4 [OF type_definition_word]
consts
- mk_word :: "'a::type list recspace => 'a::type word"
- dest_word :: "'a::type word => 'a::type list recspace"
+ mk_word :: "'a list recspace => 'a word"
+ dest_word :: "'a word => 'a list recspace"
specification (dest_word mk_word) word_repfns: "(ALL a::'a::type word. mk_word (dest_word a) = a) &
(ALL r::'a::type list recspace.
(ALL word::'a::type list recspace => bool.
(ALL a0::'a::type list recspace.
- (EX a::'a::type list.
- a0 = CONSTR (0::nat) a (%n::nat. BOTTOM)) -->
+ (EX a::'a::type list. a0 = CONSTR 0 a (%n::nat. BOTTOM)) -->
word a0) -->
word r) =
(dest_word (mk_word r) = r))"
by (import word_base word_repfns)
consts
- word_base0 :: "'a::type list => 'a::type word"
+ word_base0 :: "'a list => 'a word"
defs
- word_base0_primdef: "word_base0 ==
-%a::'a::type list. mk_word (CONSTR (0::nat) a (%n::nat. BOTTOM))"
+ word_base0_primdef: "word_base0 == %a::'a::type list. mk_word (CONSTR 0 a (%n::nat. BOTTOM))"
-lemma word_base0_def: "word_base0 =
-(%a::'a::type list. mk_word (CONSTR (0::nat) a (%n::nat. BOTTOM)))"
+lemma word_base0_def: "word_base0 = (%a::'a::type list. mk_word (CONSTR 0 a (%n::nat. BOTTOM)))"
by (import word_base word_base0_def)
constdefs
- WORD :: "'a::type list => 'a::type word"
+ WORD :: "'a list => 'a word"
"WORD == word_base0"
lemma WORD: "WORD = word_base0"
by (import word_base WORD)
consts
- word_case :: "('a::type list => 'b::type) => 'a::type word => 'b::type"
+ word_case :: "('a list => 'b) => 'a word => 'b"
specification (word_case_primdef: word_case) word_case_def: "ALL (f::'a::type list => 'b::type) a::'a::type list.
word_case f (WORD a) = f a"
by (import word_base word_case_def)
consts
- word_size :: "('a::type => nat) => 'a::type word => nat"
+ word_size :: "('a => nat) => 'a word => nat"
specification (word_size_primdef: word_size) word_size_def: "ALL (f::'a::type => nat) a::'a::type list.
- word_size f (WORD a) = (1::nat) + list_size f a"
+ word_size f (WORD a) = 1 + list_size f a"
by (import word_base word_size_def)
lemma word_11: "ALL (a::'a::type list) a'::'a::type list. (WORD a = WORD a') = (a = a')"
@@ -222,13 +219,13 @@
by (import word_base word_cases)
consts
- WORDLEN :: "'a::type word => nat"
+ WORDLEN :: "'a word => nat"
specification (WORDLEN) WORDLEN_DEF: "ALL l::'a::type list. WORDLEN (WORD l) = length l"
by (import word_base WORDLEN_DEF)
consts
- PWORDLEN :: "nat => 'a::type word => bool"
+ PWORDLEN :: "nat => 'a word => bool"
defs
PWORDLEN_primdef: "PWORDLEN == %n::nat. GSPEC (%w::'a::type word. (w, WORDLEN w = n))"
@@ -242,20 +239,20 @@
lemma PWORDLEN: "ALL (n::nat) w::'a::type word. IN w (PWORDLEN n) = (WORDLEN w = n)"
by (import word_base PWORDLEN)
-lemma PWORDLEN0: "ALL w::'a::type word. IN w (PWORDLEN (0::nat)) --> w = WORD []"
+lemma PWORDLEN0: "ALL w::'a::type word. IN w (PWORDLEN 0) --> w = WORD []"
by (import word_base PWORDLEN0)
-lemma PWORDLEN1: "ALL x::'a::type. IN (WORD [x]) (PWORDLEN (1::nat))"
+lemma PWORDLEN1: "ALL x::'a::type. IN (WORD [x]) (PWORDLEN 1)"
by (import word_base PWORDLEN1)
consts
- WSEG :: "nat => nat => 'a::type word => 'a::type word"
+ WSEG :: "nat => nat => 'a word => 'a word"
specification (WSEG) WSEG_DEF: "ALL (m::nat) (k::nat) l::'a::type list.
WSEG m k (WORD l) = WORD (LASTN m (BUTLASTN k l))"
by (import word_base WSEG_DEF)
-lemma WSEG0: "ALL (k::nat) w::'a::type word. WSEG (0::nat) k w = WORD []"
+lemma WSEG0: "ALL (k::nat) w::'a::type word. WSEG 0 k w = WORD []"
by (import word_base WSEG0)
lemma WSEG_PWORDLEN: "ALL n::nat.
@@ -271,17 +268,16 @@
xb + xc <= x --> WORDLEN (WSEG xb xc xa) = xb)"
by (import word_base WSEG_WORDLEN)
-lemma WSEG_WORD_LENGTH: "ALL n::nat.
- RES_FORALL (PWORDLEN n) (%w::'a::type word. WSEG n (0::nat) w = w)"
+lemma WSEG_WORD_LENGTH: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::'a::type word. WSEG n 0 w = w)"
by (import word_base WSEG_WORD_LENGTH)
consts
- bit :: "nat => 'a::type word => 'a::type"
+ bit :: "nat => 'a word => 'a"
specification (bit) BIT_DEF: "ALL (k::nat) l::'a::type list. bit k (WORD l) = ELL k l"
by (import word_base BIT_DEF)
-lemma BIT0: "ALL x::'a::type. bit (0::nat) (WORD [x]) = x"
+lemma BIT0: "ALL x::'a::type. bit 0 (WORD [x]) = x"
by (import word_base BIT0)
lemma WSEG_BIT: "(All::(nat => bool) => bool)
@@ -311,36 +307,35 @@
by (import word_base BIT_WSEG)
consts
- MSB :: "'a::type word => 'a::type"
+ MSB :: "'a word => 'a"
specification (MSB) MSB_DEF: "ALL l::'a::type list. MSB (WORD l) = hd l"
by (import word_base MSB_DEF)
lemma MSB: "ALL n::nat.
RES_FORALL (PWORDLEN n)
- (%w::'a::type word. (0::nat) < n --> MSB w = bit (PRE n) w)"
+ (%w::'a::type word. 0 < n --> MSB w = bit (PRE n) w)"
by (import word_base MSB)
consts
- LSB :: "'a::type word => 'a::type"
+ LSB :: "'a word => 'a"
specification (LSB) LSB_DEF: "ALL l::'a::type list. LSB (WORD l) = last l"
by (import word_base LSB_DEF)
lemma LSB: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word. (0::nat) < n --> LSB w = bit (0::nat) w)"
+ RES_FORALL (PWORDLEN n) (%w::'a::type word. 0 < n --> LSB w = bit 0 w)"
by (import word_base LSB)
consts
- WSPLIT :: "nat => 'a::type word => 'a::type word * 'a::type word"
+ WSPLIT :: "nat => 'a word => 'a word * 'a word"
specification (WSPLIT) WSPLIT_DEF: "ALL (m::nat) l::'a::type list.
WSPLIT m (WORD l) = (WORD (BUTLASTN m l), WORD (LASTN m l))"
by (import word_base WSPLIT_DEF)
consts
- WCAT :: "'a::type word * 'a::type word => 'a::type word"
+ WCAT :: "'a word * 'a word => 'a word"
specification (WCAT) WCAT_DEF: "ALL (l1::'a::type list) l2::'a::type list.
WCAT (WORD l1, WORD l2) = WORD (l1 @ l2)"
@@ -451,7 +446,7 @@
lemma WORDLEN_SUC_WCAT: "ALL (n::nat) w::'a::type word.
IN w (PWORDLEN (Suc n)) -->
- RES_EXISTS (PWORDLEN (1::nat))
+ RES_EXISTS (PWORDLEN 1)
(%b::'a::type word.
RES_EXISTS (PWORDLEN n) (%w'::'a::type word. w = WCAT (b, w')))"
by (import word_base WORDLEN_SUC_WCAT)
@@ -538,25 +533,25 @@
lemma WORD_SPLIT: "ALL (x::nat) xa::nat.
RES_FORALL (PWORDLEN (x + xa))
- (%w::'a::type word. w = WCAT (WSEG x xa w, WSEG xa (0::nat) w))"
+ (%w::'a::type word. w = WCAT (WSEG x xa w, WSEG xa 0 w))"
by (import word_base WORD_SPLIT)
lemma WORDLEN_SUC_WCAT_WSEG_WSEG: "RES_FORALL (PWORDLEN (Suc (n::nat)))
- (%w::'a::type word. w = WCAT (WSEG (1::nat) n w, WSEG n (0::nat) w))"
+ (%w::'a::type word. w = WCAT (WSEG 1 n w, WSEG n 0 w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG)
lemma WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT: "RES_FORALL (PWORDLEN (Suc (n::nat)))
- (%w::'a::type word. w = WCAT (WSEG n (1::nat) w, WSEG (1::nat) (0::nat) w))"
+ (%w::'a::type word. w = WCAT (WSEG n 1 w, WSEG 1 0 w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT)
lemma WORDLEN_SUC_WCAT_BIT_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word. w = WCAT (WORD [bit n w], WSEG n (0::nat) w))"
+ (%w::'a::type word. w = WCAT (WORD [bit n w], WSEG n 0 w))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG)
lemma WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word. w = WCAT (WSEG n (1::nat) w, WORD [bit (0::nat) w]))"
+ (%w::'a::type word. w = WCAT (WSEG n 1 w, WORD [bit 0 w]))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT)
lemma WSEG_WCAT1: "ALL (n1::nat) n2::nat.
@@ -570,7 +565,7 @@
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word. WSEG n2 (0::nat) (WCAT (w1, w2)) = w2))"
+ (%w2::'a::type word. WSEG n2 0 (WCAT (w1, w2)) = w2))"
by (import word_base WSEG_WCAT2)
lemma WSEG_SUC: "ALL n::nat.
@@ -578,7 +573,7 @@
(%w::'a::type word.
ALL (k::nat) m1::nat.
k + Suc m1 < n -->
- WSEG (Suc m1) k w = WCAT (WSEG (1::nat) (k + m1) w, WSEG m1 k w))"
+ WSEG (Suc m1) k w = WCAT (WSEG 1 (k + m1) w, WSEG m1 k w))"
by (import word_base WSEG_SUC)
lemma WORD_CONS_WCAT: "ALL (x::'a::type) l::'a::type list. WORD (x # l) = WCAT (WORD [x], WORD l)"
@@ -657,7 +652,7 @@
ALL (m::nat) k::nat.
m + k <= n1 + n2 & k < n2 & n2 <= m + k -->
WSEG m k (WCAT (w1, w2)) =
- WCAT (WSEG (m + k - n2) (0::nat) w1, WSEG (n2 - k) k w2)))"
+ WCAT (WSEG (m + k - n2) 0 w1, WSEG (n2 - k) k w2)))"
by (import word_base WSEG_WCAT_WSEG)
lemma BIT_EQ_IMP_WORD_EQ: "(All::(nat => bool) => bool)
@@ -686,23 +681,22 @@
;setup_theory word_num
constdefs
- LVAL :: "('a::type => nat) => nat => 'a::type list => nat"
+ LVAL :: "('a => nat) => nat => 'a list => nat"
"LVAL ==
-%(f::'a::type => nat) b::nat.
- foldl (%(e::nat) x::'a::type. b * e + f x) (0::nat)"
+%(f::'a::type => nat) b::nat. foldl (%(e::nat) x::'a::type. b * e + f x) 0"
lemma LVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
- LVAL f b l = foldl (%(e::nat) x::'a::type. b * e + f x) (0::nat) l"
+ LVAL f b l = foldl (%(e::nat) x::'a::type. b * e + f x) 0 l"
by (import word_num LVAL_DEF)
consts
- NVAL :: "('a::type => nat) => nat => 'a::type word => nat"
+ NVAL :: "('a => nat) => nat => 'a word => nat"
specification (NVAL) NVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
NVAL f b (WORD l) = LVAL f b l"
by (import word_num NVAL_DEF)
-lemma LVAL: "(ALL (x::'a::type => nat) xa::nat. LVAL x xa [] = (0::nat)) &
+lemma LVAL: "(ALL (x::'a::type => nat) xa::nat. LVAL x xa [] = 0) &
(ALL (x::'a::type list) (xa::'a::type => nat) (xb::nat) xc::'a::type.
LVAL xa xb (xc # x) = xa xc * xb ^ length x + LVAL xa xb x)"
by (import word_num LVAL)
@@ -721,16 +715,15 @@
RES_FORALL (PWORDLEN n) (%w::'a::type word. NVAL f b w < b ^ n))"
by (import word_num NVAL_MAX)
-lemma NVAL0: "ALL (x::'a::type => nat) xa::nat. NVAL x xa (WORD []) = (0::nat)"
+lemma NVAL0: "ALL (x::'a::type => nat) xa::nat. NVAL x xa (WORD []) = 0"
by (import word_num NVAL0)
lemma NVAL1: "ALL (x::'a::type => nat) (xa::nat) xb::'a::type.
NVAL x xa (WORD [xb]) = x xb"
by (import word_num NVAL1)
-lemma NVAL_WORDLEN_0: "RES_FORALL (PWORDLEN (0::nat))
- (%w::'a::type word.
- ALL (fv::'a::type => nat) r::nat. NVAL fv r w = (0::nat))"
+lemma NVAL_WORDLEN_0: "RES_FORALL (PWORDLEN 0)
+ (%w::'a::type word. ALL (fv::'a::type => nat) r::nat. NVAL fv r w = 0)"
by (import word_num NVAL_WORDLEN_0)
lemma NVAL_WCAT1: "ALL (w::'a::type word) (f::'a::type => nat) (b::nat) x::'a::type.
@@ -755,17 +748,16 @@
by (import word_num NVAL_WCAT)
consts
- NLIST :: "nat => (nat => 'a::type) => nat => nat => 'a::type list"
+ NLIST :: "nat => (nat => 'a) => nat => nat => 'a list"
-specification (NLIST) NLIST_DEF: "(ALL (frep::nat => 'a::type) (b::nat) m::nat.
- NLIST (0::nat) frep b m = []) &
+specification (NLIST) NLIST_DEF: "(ALL (frep::nat => 'a::type) (b::nat) m::nat. NLIST 0 frep b m = []) &
(ALL (n::nat) (frep::nat => 'a::type) (b::nat) m::nat.
NLIST (Suc n) frep b m =
SNOC (frep (m mod b)) (NLIST n frep b (m div b)))"
by (import word_num NLIST_DEF)
constdefs
- NWORD :: "nat => (nat => 'a::type) => nat => nat => 'a::type word"
+ NWORD :: "nat => (nat => 'a) => nat => nat => 'a word"
"NWORD ==
%(n::nat) (frep::nat => 'a::type) (b::nat) m::nat. WORD (NLIST n frep b m)"
@@ -786,7 +778,7 @@
;setup_theory word_bitop
consts
- PBITOP :: "('a::type word => 'b::type word) => bool"
+ PBITOP :: "('a word => 'b word) => bool"
defs
PBITOP_primdef: "PBITOP ==
@@ -867,7 +859,7 @@
by (import word_bitop PBITOP_BIT)
consts
- PBITBOP :: "('a::type word => 'b::type word => 'c::type word) => bool"
+ PBITBOP :: "('a word => 'b word => 'c word) => bool"
defs
PBITBOP_primdef: "PBITBOP ==
@@ -944,7 +936,7 @@
by (import word_bitop PBITBOP_EXISTS)
consts
- WMAP :: "('a::type => 'b::type) => 'a::type word => 'b::type word"
+ WMAP :: "('a => 'b) => 'a word => 'b word"
specification (WMAP) WMAP_DEF: "ALL (f::'a::type => 'b::type) l::'a::type list.
WMAP f (WORD l) = WORD (map f l)"
@@ -1000,7 +992,7 @@
by (import word_bitop WMAP_o)
consts
- FORALLBITS :: "('a::type => bool) => 'a::type word => bool"
+ FORALLBITS :: "('a => bool) => 'a word => bool"
specification (FORALLBITS) FORALLBITS_DEF: "ALL (P::'a::type => bool) l::'a::type list.
FORALLBITS P (WORD l) = list_all P l"
@@ -1038,7 +1030,7 @@
by (import word_bitop FORALLBITS_WCAT)
consts
- EXISTSABIT :: "('a::type => bool) => 'a::type word => bool"
+ EXISTSABIT :: "('a => bool) => 'a word => bool"
specification (EXISTSABIT) EXISTSABIT_DEF: "ALL (P::'a::type => bool) l::'a::type list.
EXISTSABIT P (WORD l) = list_exists P l"
@@ -1085,37 +1077,33 @@
by (import word_bitop EXISTSABIT_WCAT)
constdefs
- SHR :: "bool => 'a::type => 'a::type word => 'a::type word * 'a::type"
+ SHR :: "bool => 'a => 'a word => 'a word * 'a"
"SHR ==
%(f::bool) (b::'a::type) w::'a::type word.
(WCAT
- (if f then WSEG (1::nat) (PRE (WORDLEN w)) w else WORD [b],
- WSEG (PRE (WORDLEN w)) (1::nat) w),
- bit (0::nat) w)"
+ (if f then WSEG 1 (PRE (WORDLEN w)) w else WORD [b],
+ WSEG (PRE (WORDLEN w)) 1 w),
+ bit 0 w)"
lemma SHR_DEF: "ALL (f::bool) (b::'a::type) w::'a::type word.
SHR f b w =
(WCAT
- (if f then WSEG (1::nat) (PRE (WORDLEN w)) w else WORD [b],
- WSEG (PRE (WORDLEN w)) (1::nat) w),
- bit (0::nat) w)"
+ (if f then WSEG 1 (PRE (WORDLEN w)) w else WORD [b],
+ WSEG (PRE (WORDLEN w)) 1 w),
+ bit 0 w)"
by (import word_bitop SHR_DEF)
constdefs
- SHL :: "bool => 'a::type word => 'a::type => 'a::type * 'a::type word"
+ SHL :: "bool => 'a word => 'a => 'a * 'a word"
"SHL ==
%(f::bool) (w::'a::type word) b::'a::type.
(bit (PRE (WORDLEN w)) w,
- WCAT
- (WSEG (PRE (WORDLEN w)) (0::nat) w,
- if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
+ WCAT (WSEG (PRE (WORDLEN w)) 0 w, if f then WSEG 1 0 w else WORD [b]))"
lemma SHL_DEF: "ALL (f::bool) (w::'a::type word) b::'a::type.
SHL f w b =
(bit (PRE (WORDLEN w)) w,
- WCAT
- (WSEG (PRE (WORDLEN w)) (0::nat) w,
- if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
+ WCAT (WSEG (PRE (WORDLEN w)) 0 w, if f then WSEG 1 0 w else WORD [b]))"
by (import word_bitop SHL_DEF)
lemma SHR_WSEG: "ALL n::nat.
@@ -1123,14 +1111,12 @@
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
- (0::nat) < m -->
+ 0 < m -->
(ALL (f::bool) b::'a::type.
SHR f b (WSEG m k w) =
(if f
- then WCAT
- (WSEG (1::nat) (k + (m - (1::nat))) w,
- WSEG (m - (1::nat)) (k + (1::nat)) w)
- else WCAT (WORD [b], WSEG (m - (1::nat)) (k + (1::nat)) w),
+ then WCAT (WSEG 1 (k + (m - 1)) w, WSEG (m - 1) (k + 1) w)
+ else WCAT (WORD [b], WSEG (m - 1) (k + 1) w),
bit k w)))"
by (import word_bitop SHR_WSEG)
@@ -1139,9 +1125,9 @@
(%w::'a::type word.
ALL (b::'a::type) (m::nat) k::nat.
m + k <= n -->
- (0::nat) < m -->
+ 0 < m -->
SHR False b (WSEG m k w) =
- (WCAT (WORD [b], WSEG (m - (1::nat)) (k + (1::nat)) w), bit k w))"
+ (WCAT (WORD [b], WSEG (m - 1) (k + 1) w), bit k w))"
by (import word_bitop SHR_WSEG_1F)
lemma SHR_WSEG_NF: "ALL n::nat.
@@ -1149,9 +1135,9 @@
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k < n -->
- (0::nat) < m -->
+ 0 < m -->
SHR False (bit (m + k) w) (WSEG m k w) =
- (WSEG m (k + (1::nat)) w, bit k w))"
+ (WSEG m (k + 1) w, bit k w))"
by (import word_bitop SHR_WSEG_NF)
lemma SHL_WSEG: "ALL n::nat.
@@ -1159,12 +1145,12 @@
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
- (0::nat) < m -->
+ 0 < m -->
(ALL (f::bool) b::'a::type.
SHL f (WSEG m k w) b =
- (bit (k + (m - (1::nat))) w,
- if f then WCAT (WSEG (m - (1::nat)) k w, WSEG (1::nat) k w)
- else WCAT (WSEG (m - (1::nat)) k w, WORD [b]))))"
+ (bit (k + (m - 1)) w,
+ if f then WCAT (WSEG (m - 1) k w, WSEG 1 k w)
+ else WCAT (WSEG (m - 1) k w, WORD [b]))))"
by (import word_bitop SHL_WSEG)
lemma SHL_WSEG_1F: "ALL n::nat.
@@ -1172,10 +1158,9 @@
(%w::'a::type word.
ALL (b::'a::type) (m::nat) k::nat.
m + k <= n -->
- (0::nat) < m -->
+ 0 < m -->
SHL False (WSEG m k w) b =
- (bit (k + (m - (1::nat))) w,
- WCAT (WSEG (m - (1::nat)) k w, WORD [b])))"
+ (bit (k + (m - 1)) w, WCAT (WSEG (m - 1) k w, WORD [b])))"
by (import word_bitop SHL_WSEG_1F)
lemma SHL_WSEG_NF: "ALL n::nat.
@@ -1183,31 +1168,28 @@
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
- (0::nat) < m -->
- (0::nat) < k -->
- SHL False (WSEG m k w) (bit (k - (1::nat)) w) =
- (bit (k + (m - (1::nat))) w, WSEG m (k - (1::nat)) w))"
+ 0 < m -->
+ 0 < k -->
+ SHL False (WSEG m k w) (bit (k - 1) w) =
+ (bit (k + (m - 1)) w, WSEG m (k - 1) w))"
by (import word_bitop SHL_WSEG_NF)
lemma WSEG_SHL: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word.
ALL (m::nat) k::nat.
- (0::nat) < k & m + k <= Suc n -->
+ 0 < k & m + k <= Suc n -->
(ALL b::'a::type.
- WSEG m k (snd (SHL (f::bool) w b)) =
- WSEG m (k - (1::nat)) w))"
+ WSEG m k (snd (SHL (f::bool) w b)) = WSEG m (k - 1) w))"
by (import word_bitop WSEG_SHL)
lemma WSEG_SHL_0: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word.
ALL (m::nat) b::'a::type.
- (0::nat) < m & m <= Suc n -->
- WSEG m (0::nat) (snd (SHL (f::bool) w b)) =
- WCAT
- (WSEG (m - (1::nat)) (0::nat) w,
- if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
+ 0 < m & m <= Suc n -->
+ WSEG m 0 (snd (SHL (f::bool) w b)) =
+ WCAT (WSEG (m - 1) 0 w, if f then WSEG 1 0 w else WORD [b]))"
by (import word_bitop WSEG_SHL_0)
;end_setup
@@ -1216,24 +1198,24 @@
constdefs
BV :: "bool => nat"
- "(BV == (%(b::bool). (if b then (Suc (0::nat)) else (0::nat))))"
+ "BV == %b::bool. if b then Suc 0 else 0"
-lemma BV_DEF: "(ALL (b::bool). ((BV b) = (if b then (Suc (0::nat)) else (0::nat))))"
+lemma BV_DEF: "ALL b::bool. BV b = (if b then Suc 0 else 0)"
by (import bword_num BV_DEF)
consts
BNVAL :: "bool word => nat"
-specification (BNVAL) BNVAL_DEF: "ALL l::bool list. BNVAL (WORD l) = LVAL BV (2::nat) l"
+specification (BNVAL) BNVAL_DEF: "ALL l::bool list. BNVAL (WORD l) = LVAL BV 2 l"
by (import bword_num BNVAL_DEF)
-lemma BV_LESS_2: "ALL x::bool. BV x < (2::nat)"
+lemma BV_LESS_2: "ALL x::bool. BV x < 2"
by (import bword_num BV_LESS_2)
-lemma BNVAL_NVAL: "ALL w::bool word. BNVAL w = NVAL BV (2::nat) w"
+lemma BNVAL_NVAL: "ALL w::bool word. BNVAL w = NVAL BV 2 w"
by (import bword_num BNVAL_NVAL)
-lemma BNVAL0: "BNVAL (WORD []) = (0::nat)"
+lemma BNVAL0: "BNVAL (WORD []) = 0"
by (import bword_num BNVAL0)
lemma BNVAL_11: "ALL (w1::bool word) w2::bool word.
@@ -1243,20 +1225,19 @@
lemma BNVAL_ONTO: "ALL w::bool word. Ex (op = (BNVAL w))"
by (import bword_num BNVAL_ONTO)
-lemma BNVAL_MAX: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w < (2::nat) ^ n)"
+lemma BNVAL_MAX: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w < 2 ^ n)"
by (import bword_num BNVAL_MAX)
lemma BNVAL_WCAT1: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::bool word.
- ALL x::bool. BNVAL (WCAT (w, WORD [x])) = BNVAL w * (2::nat) + BV x)"
+ ALL x::bool. BNVAL (WCAT (w, WORD [x])) = BNVAL w * 2 + BV x)"
by (import bword_num BNVAL_WCAT1)
lemma BNVAL_WCAT2: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::bool word.
- ALL x::bool.
- BNVAL (WCAT (WORD [x], w)) = BV x * (2::nat) ^ n + BNVAL w)"
+ ALL x::bool. BNVAL (WCAT (WORD [x], w)) = BV x * 2 ^ n + BNVAL w)"
by (import bword_num BNVAL_WCAT2)
lemma BNVAL_WCAT: "ALL (n::nat) m::nat.
@@ -1264,24 +1245,24 @@
(%w1::bool word.
RES_FORALL (PWORDLEN m)
(%w2::bool word.
- BNVAL (WCAT (w1, w2)) = BNVAL w1 * (2::nat) ^ m + BNVAL w2))"
+ BNVAL (WCAT (w1, w2)) = BNVAL w1 * 2 ^ m + BNVAL w2))"
by (import bword_num BNVAL_WCAT)
constdefs
VB :: "nat => bool"
- "VB == %n::nat. n mod (2::nat) ~= (0::nat)"
+ "VB == %n::nat. n mod 2 ~= 0"
-lemma VB_DEF: "ALL n::nat. VB n = (n mod (2::nat) ~= (0::nat))"
+lemma VB_DEF: "ALL n::nat. VB n = (n mod 2 ~= 0)"
by (import bword_num VB_DEF)
constdefs
NBWORD :: "nat => nat => bool word"
- "NBWORD == %(n::nat) m::nat. WORD (NLIST n VB (2::nat) m)"
+ "NBWORD == %(n::nat) m::nat. WORD (NLIST n VB 2 m)"
-lemma NBWORD_DEF: "ALL (n::nat) m::nat. NBWORD n m = WORD (NLIST n VB (2::nat) m)"
+lemma NBWORD_DEF: "ALL (n::nat) m::nat. NBWORD n m = WORD (NLIST n VB 2 m)"
by (import bword_num NBWORD_DEF)
-lemma NBWORD0: "ALL x::nat. NBWORD (0::nat) x = WORD []"
+lemma NBWORD0: "ALL x::nat. NBWORD 0 x = WORD []"
by (import bword_num NBWORD0)
lemma WORDLEN_NBWORD: "ALL (x::nat) xa::nat. WORDLEN (NBWORD x xa) = x"
@@ -1291,8 +1272,7 @@
by (import bword_num PWORDLEN_NBWORD)
lemma NBWORD_SUC: "ALL (n::nat) m::nat.
- NBWORD (Suc n) m =
- WCAT (NBWORD n (m div (2::nat)), WORD [VB (m mod (2::nat))])"
+ NBWORD (Suc n) m = WCAT (NBWORD n (m div 2), WORD [VB (m mod 2)])"
by (import bword_num NBWORD_SUC)
lemma VB_BV: "ALL x::bool. VB (BV x) = x"
@@ -1313,21 +1293,18 @@
lemma NBWORD_BNVAL: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. NBWORD n (BNVAL w) = w)"
by (import bword_num NBWORD_BNVAL)
-lemma BNVAL_NBWORD: "ALL (n::nat) m::nat. m < (2::nat) ^ n --> BNVAL (NBWORD n m) = m"
+lemma BNVAL_NBWORD: "ALL (n::nat) m::nat. m < 2 ^ n --> BNVAL (NBWORD n m) = m"
by (import bword_num BNVAL_NBWORD)
lemma ZERO_WORD_VAL: "RES_FORALL (PWORDLEN (n::nat))
- (%w::bool word. (w = NBWORD n (0::nat)) = (BNVAL w = (0::nat)))"
+ (%w::bool word. (w = NBWORD n 0) = (BNVAL w = 0))"
by (import bword_num ZERO_WORD_VAL)
-lemma WCAT_NBWORD_0: "ALL (n1::nat) n2::nat.
- WCAT (NBWORD n1 (0::nat), NBWORD n2 (0::nat)) = NBWORD (n1 + n2) (0::nat)"
+lemma WCAT_NBWORD_0: "ALL (n1::nat) n2::nat. WCAT (NBWORD n1 0, NBWORD n2 0) = NBWORD (n1 + n2) 0"
by (import bword_num WCAT_NBWORD_0)
lemma WSPLIT_NBWORD_0: "ALL (n::nat) m::nat.
- m <= n -->
- WSPLIT m (NBWORD n (0::nat)) =
- (NBWORD (n - m) (0::nat), NBWORD m (0::nat))"
+ m <= n --> WSPLIT m (NBWORD n 0) = (NBWORD (n - m) 0, NBWORD m 0)"
by (import bword_num WSPLIT_NBWORD_0)
lemma EQ_NBWORD0_SPLIT: "(All::(nat => bool) => bool)
@@ -1354,43 +1331,41 @@
((NBWORD::nat => nat => bool word) m (0::nat))))))))"
by (import bword_num EQ_NBWORD0_SPLIT)
-lemma NBWORD_MOD: "ALL (n::nat) m::nat. NBWORD n (m mod (2::nat) ^ n) = NBWORD n m"
+lemma NBWORD_MOD: "ALL (n::nat) m::nat. NBWORD n (m mod 2 ^ n) = NBWORD n m"
by (import bword_num NBWORD_MOD)
-lemma WSEG_NBWORD_SUC: "ALL (n::nat) m::nat. WSEG n (0::nat) (NBWORD (Suc n) m) = NBWORD n m"
+lemma WSEG_NBWORD_SUC: "ALL (n::nat) m::nat. WSEG n 0 (NBWORD (Suc n) m) = NBWORD n m"
by (import bword_num WSEG_NBWORD_SUC)
lemma NBWORD_SUC_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
- (%w::bool word. NBWORD n (BNVAL w) = WSEG n (0::nat) w)"
+ (%w::bool word. NBWORD n (BNVAL w) = WSEG n 0 w)"
by (import bword_num NBWORD_SUC_WSEG)
-lemma DOUBL_EQ_SHL: "ALL x>0::nat.
+lemma DOUBL_EQ_SHL: "ALL x>0.
RES_FORALL (PWORDLEN x)
(%xa::bool word.
ALL xb::bool.
NBWORD x (BNVAL xa + BNVAL xa + BV xb) = snd (SHL False xa xb))"
by (import bword_num DOUBL_EQ_SHL)
-lemma MSB_NBWORD: "ALL (n::nat) m::nat.
- bit n (NBWORD (Suc n) m) = VB (m div (2::nat) ^ n mod (2::nat))"
+lemma MSB_NBWORD: "ALL (n::nat) m::nat. bit n (NBWORD (Suc n) m) = VB (m div 2 ^ n mod 2)"
by (import bword_num MSB_NBWORD)
lemma NBWORD_SPLIT: "ALL (n1::nat) (n2::nat) m::nat.
- NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div (2::nat) ^ n2), NBWORD n2 m)"
+ NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div 2 ^ n2), NBWORD n2 m)"
by (import bword_num NBWORD_SPLIT)
lemma WSEG_NBWORD: "ALL (m::nat) (k::nat) n::nat.
m + k <= n -->
- (ALL l::nat. WSEG m k (NBWORD n l) = NBWORD m (l div (2::nat) ^ k))"
+ (ALL l::nat. WSEG m k (NBWORD n l) = NBWORD m (l div 2 ^ k))"
by (import bword_num WSEG_NBWORD)
lemma NBWORD_SUC_FST: "ALL (n::nat) x::nat.
- NBWORD (Suc n) x =
- WCAT (WORD [VB (x div (2::nat) ^ n mod (2::nat))], NBWORD n x)"
+ NBWORD (Suc n) x = WCAT (WORD [VB (x div 2 ^ n mod 2)], NBWORD n x)"
by (import bword_num NBWORD_SUC_FST)
-lemma BIT_NBWORD0: "ALL (k::nat) n::nat. k < n --> bit k (NBWORD n (0::nat)) = False"
+lemma BIT_NBWORD0: "ALL (k::nat) n::nat. k < n --> bit k (NBWORD n 0) = False"
by (import bword_num BIT_NBWORD0)
lemma ADD_BNVAL_LEFT: "ALL n::nat.
@@ -1399,8 +1374,8 @@
RES_FORALL (PWORDLEN (Suc n))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
- (BV (bit n w1) + BV (bit n w2)) * (2::nat) ^ n +
- (BNVAL (WSEG n (0::nat) w1) + BNVAL (WSEG n (0::nat) w2))))"
+ (BV (bit n w1) + BV (bit n w2)) * 2 ^ n +
+ (BNVAL (WSEG n 0 w1) + BNVAL (WSEG n 0 w2))))"
by (import bword_num ADD_BNVAL_LEFT)
lemma ADD_BNVAL_RIGHT: "ALL n::nat.
@@ -1409,9 +1384,8 @@
RES_FORALL (PWORDLEN (Suc n))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
- (BNVAL (WSEG n (1::nat) w1) + BNVAL (WSEG n (1::nat) w2)) *
- (2::nat) +
- (BV (bit (0::nat) w1) + BV (bit (0::nat) w2))))"
+ (BNVAL (WSEG n 1 w1) + BNVAL (WSEG n 1 w2)) * 2 +
+ (BV (bit 0 w1) + BV (bit 0 w2))))"
by (import bword_num ADD_BNVAL_RIGHT)
lemma ADD_BNVAL_SPLIT: "ALL (n1::nat) n2::nat.
@@ -1420,9 +1394,8 @@
RES_FORALL (PWORDLEN (n1 + n2))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
- (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2)) *
- (2::nat) ^ n2 +
- (BNVAL (WSEG n2 (0::nat) w1) + BNVAL (WSEG n2 (0::nat) w2))))"
+ (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2)) * 2 ^ n2 +
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2))))"
by (import bword_num ADD_BNVAL_SPLIT)
;end_setup
@@ -1432,19 +1405,16 @@
consts
ACARRY :: "nat => bool word => bool word => bool => bool"
-specification (ACARRY) ACARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool.
- ACARRY (0::nat) w1 w2 cin = cin) &
+specification (ACARRY) ACARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool. ACARRY 0 w1 w2 cin = cin) &
(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
ACARRY (Suc n) w1 w2 cin =
- VB ((BV (bit n w1) + BV (bit n w2) + BV (ACARRY n w1 w2 cin)) div
- (2::nat)))"
+ VB ((BV (bit n w1) + BV (bit n w2) + BV (ACARRY n w1 w2 cin)) div 2))"
by (import bword_arith ACARRY_DEF)
consts
ICARRY :: "nat => bool word => bool word => bool => bool"
-specification (ICARRY) ICARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool.
- ICARRY (0::nat) w1 w2 cin = cin) &
+specification (ICARRY) ICARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool. ICARRY 0 w1 w2 cin = cin) &
(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
ICARRY (Suc n) w1 w2 cin =
(bit n w1 & bit n w2 | (bit n w1 | bit n w2) & ICARRY n w1 w2 cin))"
@@ -1459,9 +1429,7 @@
k <= n --> ACARRY k w1 w2 cin = ICARRY k w1 w2 cin))"
by (import bword_arith ACARRY_EQ_ICARRY)
-lemma BNVAL_LESS_EQ: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::bool word. BNVAL w <= (2::nat) ^ n - (1::nat))"
+lemma BNVAL_LESS_EQ: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w <= 2 ^ n - 1)"
by (import bword_arith BNVAL_LESS_EQ)
lemma ADD_BNVAL_LESS_EQ1: "ALL (n::nat) cin::bool.
@@ -1469,8 +1437,7 @@
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
- (BNVAL w1 + (BNVAL w2 + BV cin)) div (2::nat) ^ n
- <= Suc (0::nat)))"
+ (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n <= Suc 0))"
by (import bword_arith ADD_BNVAL_LESS_EQ1)
lemma ADD_BV_BNVAL_DIV_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
@@ -1479,9 +1446,9 @@
RES_FORALL (PWORDLEN n)
(%w2::bool word.
(BV x1 + BV x2 +
- (BNVAL w1 + (BNVAL w2 + BV cin)) div (2::nat) ^ n) div
- (2::nat)
- <= (1::nat)))"
+ (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n) div
+ 2
+ <= 1))"
by (import bword_arith ADD_BV_BNVAL_DIV_LESS_EQ1)
lemma ADD_BV_BNVAL_LESS_EQ: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
@@ -1490,7 +1457,7 @@
RES_FORALL (PWORDLEN n)
(%w2::bool word.
BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))
- <= Suc ((2::nat) ^ Suc n)))"
+ <= Suc (2 ^ Suc n)))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ)
lemma ADD_BV_BNVAL_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
@@ -1499,8 +1466,8 @@
RES_FORALL (PWORDLEN n)
(%w2::bool word.
(BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))) div
- (2::nat) ^ Suc n
- <= (1::nat)))"
+ 2 ^ Suc n
+ <= 1))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ1)
lemma ACARRY_EQ_ADD_DIV: "(All::(nat => bool) => bool)
@@ -1552,9 +1519,7 @@
(BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
BV (ACARRY n2 w1 w2 cin)),
NBWORD n2
- (BNVAL (WSEG n2 (0::nat) w1) +
- BNVAL (WSEG n2 (0::nat) w2) +
- BV cin))))"
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) + BV cin))))"
by (import bword_arith ADD_WORD_SPLIT)
lemma WSEG_NBWORD_ADD: "ALL n::nat.
@@ -1577,16 +1542,14 @@
(%w2::bool word.
ALL cin::bool.
(NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
- NBWORD (n1 + n2) (0::nat)) =
+ NBWORD (n1 + n2) 0) =
(NBWORD n1
(BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
BV (ACARRY n2 w1 w2 cin)) =
- NBWORD n1 (0::nat) &
+ NBWORD n1 0 &
NBWORD n2
- (BNVAL (WSEG n2 (0::nat) w1) +
- BNVAL (WSEG n2 (0::nat) w2) +
- BV cin) =
- NBWORD n2 (0::nat))))"
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) + BV cin) =
+ NBWORD n2 0)))"
by (import bword_arith ADD_NBWORD_EQ0_SPLIT)
lemma ACARRY_MSB: "ALL n::nat.
@@ -1606,7 +1569,7 @@
(%w2::bool word.
ALL (cin::bool) (k::nat) m::nat.
k < m & m <= n -->
- ACARRY k (WSEG m (0::nat) w1) (WSEG m (0::nat) w2) cin =
+ ACARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
ACARRY k w1 w2 cin))"
by (import bword_arith ACARRY_WSEG)
@@ -1617,7 +1580,7 @@
(%w2::bool word.
ALL (cin::bool) (k::nat) m::nat.
k < m & m <= n -->
- ICARRY k (WSEG m (0::nat) w1) (WSEG m (0::nat) w2) cin =
+ ICARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
ICARRY k w1 w2 cin))"
by (import bword_arith ICARRY_WSEG)
--- a/src/HOL/Import/HOL/HOL4Word32.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/HOL4Word32.thy Mon Sep 26 16:10:19 2005 +0200
@@ -8,78 +8,54 @@
DIV2 :: "nat => nat"
defs
- DIV2_primdef: "DIV2 == %n::nat. n div (2::nat)"
+ DIV2_primdef: "DIV2 == %n::nat. n div 2"
-lemma DIV2_def: "ALL n::nat. DIV2 n = n div (2::nat)"
+lemma DIV2_def: "ALL n::nat. DIV2 n = n div 2"
by (import bits DIV2_def)
consts
TIMES_2EXP :: "nat => nat => nat"
defs
- TIMES_2EXP_primdef: "TIMES_2EXP == %(x::nat) n::nat. n * (2::nat) ^ x"
+ TIMES_2EXP_primdef: "TIMES_2EXP == %(x::nat) n::nat. n * 2 ^ x"
-lemma TIMES_2EXP_def: "ALL (x::nat) n::nat. TIMES_2EXP x n = n * (2::nat) ^ x"
+lemma TIMES_2EXP_def: "ALL (x::nat) n::nat. TIMES_2EXP x n = n * 2 ^ x"
by (import bits TIMES_2EXP_def)
consts
DIV_2EXP :: "nat => nat => nat"
defs
- DIV_2EXP_primdef: "DIV_2EXP == %(x::nat) n::nat. n div (2::nat) ^ x"
+ DIV_2EXP_primdef: "DIV_2EXP == %(x::nat) n::nat. n div 2 ^ x"
-lemma DIV_2EXP_def: "ALL (x::nat) n::nat. DIV_2EXP x n = n div (2::nat) ^ x"
+lemma DIV_2EXP_def: "ALL (x::nat) n::nat. DIV_2EXP x n = n div 2 ^ x"
by (import bits DIV_2EXP_def)
consts
MOD_2EXP :: "nat => nat => nat"
defs
- MOD_2EXP_primdef: "MOD_2EXP == %(x::nat) n::nat. n mod (2::nat) ^ x"
+ MOD_2EXP_primdef: "MOD_2EXP == %(x::nat) n::nat. n mod 2 ^ x"
-lemma MOD_2EXP_def: "ALL (x::nat) n::nat. MOD_2EXP x n = n mod (2::nat) ^ x"
+lemma MOD_2EXP_def: "ALL (x::nat) n::nat. MOD_2EXP x n = n mod 2 ^ x"
by (import bits MOD_2EXP_def)
consts
DIVMOD_2EXP :: "nat => nat => nat * nat"
defs
- DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %(x::nat) n::nat. (n div (2::nat) ^ x, n mod (2::nat) ^ x)"
+ DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %(x::nat) n::nat. (n div 2 ^ x, n mod 2 ^ x)"
-lemma DIVMOD_2EXP_def: "ALL (x::nat) n::nat.
- DIVMOD_2EXP x n = (n div (2::nat) ^ x, n mod (2::nat) ^ x)"
+lemma DIVMOD_2EXP_def: "ALL (x::nat) n::nat. DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)"
by (import bits DIVMOD_2EXP_def)
consts
SBIT :: "bool => nat => nat"
defs
- SBIT_primdef: "(op ==::(bool => nat => nat) => (bool => nat => nat) => prop)
- (SBIT::bool => nat => nat)
- (%(b::bool) n::nat.
- (If::bool => nat => nat => nat) b
- ((op ^::nat => nat => nat)
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
- (bit.B0::bit)))
- n)
- (0::nat))"
+ SBIT_primdef: "SBIT == %(b::bool) n::nat. if b then 2 ^ n else 0"
-lemma SBIT_def: "(All::(bool => bool) => bool)
- (%b::bool.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool) ((SBIT::bool => nat => nat) b n)
- ((If::bool => nat => nat => nat) b
- ((op ^::nat => nat => nat)
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
- (bit.B1::bit))
- (bit.B0::bit)))
- n)
- (0::nat))))"
+lemma SBIT_def: "ALL (b::bool) n::nat. SBIT b n = (if b then 2 ^ n else 0)"
by (import bits SBIT_def)
consts
@@ -94,9 +70,9 @@
constdefs
bit :: "nat => nat => bool"
- "bit == %(b::nat) n::nat. BITS b b n = (1::nat)"
+ "bit == %(b::nat) n::nat. BITS b b n = 1"
-lemma BIT_def: "ALL (b::nat) n::nat. bit b n = (BITS b b n = (1::nat))"
+lemma BIT_def: "ALL (b::nat) n::nat. bit b n = (BITS b b n = 1)"
by (import bits BIT_def)
consts
@@ -113,65 +89,125 @@
LSBn :: "nat => bool"
defs
- LSBn_primdef: "LSBn == bit (0::nat)"
+ LSBn_primdef: "LSBn == bit 0"
-lemma LSBn_def: "LSBn = bit (0::nat)"
+lemma LSBn_def: "LSBn = bit 0"
by (import bits LSBn_def)
consts
BITWISE :: "nat => (bool => bool => bool) => nat => nat => nat"
-specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL (oper::bool => bool => bool) (x::nat) y::nat.
- BITWISE (0::nat) oper x y = (0::nat)) &
+specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL (oper::bool => bool => bool) (x::nat) y::nat. BITWISE 0 oper x y = 0) &
(ALL (n::nat) (oper::bool => bool => bool) (x::nat) y::nat.
BITWISE (Suc n) oper x y =
BITWISE n oper x y + SBIT (oper (bit n x) (bit n y)) n)"
by (import bits BITWISE_def)
-lemma DIV1: "ALL x::nat. x div (1::nat) = x"
+lemma DIV1: "ALL x::nat. x div 1 = x"
by (import bits DIV1)
-lemma SUC_SUB: "Suc (a::nat) - a = (1::nat)"
+lemma SUC_SUB: "Suc (a::nat) - a = 1"
by (import bits SUC_SUB)
-lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = (1::nat)"
+lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = 1"
by (import bits DIV_MULT_1)
-lemma ZERO_LT_TWOEXP: "ALL n::nat. (0::nat) < (2::nat) ^ n"
+lemma ZERO_LT_TWOEXP: "(All::(nat => bool) => bool)
+ (%n::nat.
+ (op <::nat => nat => bool) (0::nat)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
+ (bit.B0::bit)))
+ n))"
by (import bits ZERO_LT_TWOEXP)
-lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod (2::nat) ^ n < (2::nat) ^ n"
+lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod 2 ^ n < 2 ^ n"
by (import bits MOD_2EXP_LT)
-lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat.
- k = k div (2::nat) ^ n * (2::nat) ^ n + k mod (2::nat) ^ n"
+lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat. k = k div 2 ^ n * 2 ^ n + k mod 2 ^ n"
by (import bits TWOEXP_DIVISION)
-lemma TWOEXP_MONO: "ALL (a::nat) b::nat. a < b --> (2::nat) ^ a < (2::nat) ^ b"
+lemma TWOEXP_MONO: "(All::(nat => bool) => bool)
+ (%a::nat.
+ (All::(nat => bool) => bool)
+ (%b::nat.
+ (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b)
+ ((op <::nat => nat => bool)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ a)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ b))))"
by (import bits TWOEXP_MONO)
-lemma TWOEXP_MONO2: "ALL (a::nat) b::nat. a <= b --> (2::nat) ^ a <= (2::nat) ^ b"
+lemma TWOEXP_MONO2: "(All::(nat => bool) => bool)
+ (%a::nat.
+ (All::(nat => bool) => bool)
+ (%b::nat.
+ (op -->::bool => bool => bool) ((op <=::nat => nat => bool) a b)
+ ((op <=::nat => nat => bool)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ a)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ b))))"
by (import bits TWOEXP_MONO2)
-lemma EXP_SUB_LESS_EQ: "ALL (a::nat) b::nat. (2::nat) ^ (a - b) <= (2::nat) ^ a"
+lemma EXP_SUB_LESS_EQ: "(All::(nat => bool) => bool)
+ (%a::nat.
+ (All::(nat => bool) => bool)
+ (%b::nat.
+ (op <=::nat => nat => bool)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ ((op -::nat => nat => nat) a b))
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ a)))"
by (import bits EXP_SUB_LESS_EQ)
lemma BITS_THM: "ALL (x::nat) (xa::nat) xb::nat.
- BITS x xa xb = xb div (2::nat) ^ xa mod (2::nat) ^ (Suc x - xa)"
+ BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)"
by (import bits BITS_THM)
-lemma BITSLT_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n < (2::nat) ^ (Suc h - l)"
+lemma BITSLT_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n < 2 ^ (Suc h - l)"
by (import bits BITSLT_THM)
-lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. (0::nat) < n --> m div n * n <= m"
+lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. 0 < n --> m div n * n <= m"
by (import bits DIV_MULT_LEM)
-lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat.
- n mod (2::nat) ^ x = n - n div (2::nat) ^ x * (2::nat) ^ x"
+lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat. n mod 2 ^ x = n - n div 2 ^ x * 2 ^ x"
by (import bits MOD_2EXP_LEM)
-lemma BITS2_THM: "ALL (h::nat) (l::nat) n::nat.
- BITS h l n = n mod (2::nat) ^ Suc h div (2::nat) ^ l"
+lemma BITS2_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n = n mod 2 ^ Suc h div 2 ^ l"
by (import bits BITS2_THM)
lemma BITS_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
@@ -179,71 +215,65 @@
by (import bits BITS_COMP_THM)
lemma BITS_DIV_THM: "ALL (h::nat) (l::nat) (x::nat) n::nat.
- BITS h l x div (2::nat) ^ n = BITS h (l + n) x"
+ BITS h l x div 2 ^ n = BITS h (l + n) x"
by (import bits BITS_DIV_THM)
-lemma BITS_LT_HIGH: "ALL (h::nat) (l::nat) n::nat.
- n < (2::nat) ^ Suc h --> BITS h l n = n div (2::nat) ^ l"
+lemma BITS_LT_HIGH: "ALL (h::nat) (l::nat) n::nat. n < 2 ^ Suc h --> BITS h l n = n div 2 ^ l"
by (import bits BITS_LT_HIGH)
-lemma BITS_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> BITS h l n = (0::nat)"
+lemma BITS_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> BITS h l n = 0"
by (import bits BITS_ZERO)
-lemma BITS_ZERO2: "ALL (h::nat) l::nat. BITS h l (0::nat) = (0::nat)"
+lemma BITS_ZERO2: "ALL (h::nat) l::nat. BITS h l 0 = 0"
by (import bits BITS_ZERO2)
-lemma BITS_ZERO3: "ALL (h::nat) x::nat. BITS h (0::nat) x = x mod (2::nat) ^ Suc h"
+lemma BITS_ZERO3: "ALL (h::nat) x::nat. BITS h 0 x = x mod 2 ^ Suc h"
by (import bits BITS_ZERO3)
lemma BITS_COMP_THM2: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n"
by (import bits BITS_COMP_THM2)
-lemma NOT_MOD2_LEM: "ALL n::nat. (n mod (2::nat) ~= (0::nat)) = (n mod (2::nat) = (1::nat))"
+lemma NOT_MOD2_LEM: "ALL n::nat. (n mod 2 ~= 0) = (n mod 2 = 1)"
by (import bits NOT_MOD2_LEM)
-lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a::type.
- (n mod (2::nat) ~= (1::nat)) = (n mod (2::nat) = (0::nat))"
+lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a::type. (n mod 2 ~= 1) = (n mod 2 = 0)"
by (import bits NOT_MOD2_LEM2)
-lemma EVEN_MOD2_LEM: "ALL n::nat. EVEN n = (n mod (2::nat) = (0::nat))"
+lemma EVEN_MOD2_LEM: "ALL n::nat. EVEN n = (n mod 2 = 0)"
by (import bits EVEN_MOD2_LEM)
-lemma ODD_MOD2_LEM: "ALL n::nat. ODD n = (n mod (2::nat) = (1::nat))"
+lemma ODD_MOD2_LEM: "ALL n::nat. ODD n = (n mod 2 = 1)"
by (import bits ODD_MOD2_LEM)
lemma LSB_ODD: "LSBn = ODD"
by (import bits LSB_ODD)
-lemma DIV_MULT_THM: "ALL (x::nat) n::nat.
- n div (2::nat) ^ x * (2::nat) ^ x = n - n mod (2::nat) ^ x"
+lemma DIV_MULT_THM: "ALL (x::nat) n::nat. n div 2 ^ x * 2 ^ x = n - n mod 2 ^ x"
by (import bits DIV_MULT_THM)
-lemma DIV_MULT_THM2: "ALL x::nat. (2::nat) * (x div (2::nat)) = x - x mod (2::nat)"
+lemma DIV_MULT_THM2: "ALL x::nat. 2 * (x div 2) = x - x mod 2"
by (import bits DIV_MULT_THM2)
-lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. (2::nat) ^ b = x * (2::nat) ^ a)"
+lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. 2 ^ b = x * 2 ^ a)"
by (import bits LESS_EQ_EXP_MULT)
lemma SLICE_LEM1: "ALL (a::nat) (x::nat) y::nat.
- a div (2::nat) ^ (x + y) * (2::nat) ^ (x + y) =
- a div (2::nat) ^ x * (2::nat) ^ x -
- a div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
+ a div 2 ^ (x + y) * 2 ^ (x + y) =
+ a div 2 ^ x * 2 ^ x - a div 2 ^ x mod 2 ^ y * 2 ^ x"
by (import bits SLICE_LEM1)
lemma SLICE_LEM2: "ALL (a::'a::type) (x::nat) y::nat.
- (n::nat) mod (2::nat) ^ (x + y) =
- n mod (2::nat) ^ x + n div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
+ (n::nat) mod 2 ^ (x + y) = n mod 2 ^ x + n div 2 ^ x mod 2 ^ y * 2 ^ x"
by (import bits SLICE_LEM2)
-lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat.
- l < h --> n mod (2::nat) ^ Suc l <= n mod (2::nat) ^ h"
+lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat. l < h --> n mod 2 ^ Suc l <= n mod 2 ^ h"
by (import bits SLICE_LEM3)
-lemma SLICE_THM: "ALL (n::nat) (h::nat) l::nat. SLICE h l n = BITS h l n * (2::nat) ^ l"
+lemma SLICE_THM: "ALL (n::nat) (h::nat) l::nat. SLICE h l n = BITS h l n * 2 ^ l"
by (import bits SLICE_THM)
-lemma SLICELT_THM: "ALL (h::nat) (l::nat) n::nat. SLICE h l n < (2::nat) ^ Suc h"
+lemma SLICELT_THM: "ALL (h::nat) (l::nat) n::nat. SLICE h l n < 2 ^ Suc h"
by (import bits SLICELT_THM)
lemma BITS_SLICE_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l (SLICE h l n) = BITS h l n"
@@ -253,44 +283,41 @@
h <= (h2::nat) --> BITS h2 l (SLICE h l n) = BITS h l n"
by (import bits BITS_SLICE_THM2)
-lemma MOD_2EXP_MONO: "ALL (n::nat) (h::nat) l::nat.
- l <= h --> n mod (2::nat) ^ l <= n mod (2::nat) ^ Suc h"
+lemma MOD_2EXP_MONO: "ALL (n::nat) (h::nat) l::nat. l <= h --> n mod 2 ^ l <= n mod 2 ^ Suc h"
by (import bits MOD_2EXP_MONO)
lemma SLICE_COMP_THM: "ALL (h::nat) (m::nat) (l::nat) n::nat.
Suc m <= h & l <= m --> SLICE h (Suc m) n + SLICE m l n = SLICE h l n"
by (import bits SLICE_COMP_THM)
-lemma SLICE_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> SLICE h l n = (0::nat)"
+lemma SLICE_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> SLICE h l n = 0"
by (import bits SLICE_ZERO)
lemma BIT_COMP_THM3: "ALL (h::nat) (m::nat) (l::nat) n::nat.
Suc m <= h & l <= m -->
- BITS h (Suc m) n * (2::nat) ^ (Suc m - l) + BITS m l n = BITS h l n"
+ BITS h (Suc m) n * 2 ^ (Suc m - l) + BITS m l n = BITS h l n"
by (import bits BIT_COMP_THM3)
-lemma NOT_BIT: "ALL (n::nat) a::nat. (~ bit n a) = (BITS n n a = (0::nat))"
+lemma NOT_BIT: "ALL (n::nat) a::nat. (~ bit n a) = (BITS n n a = 0)"
by (import bits NOT_BIT)
-lemma NOT_BITS: "ALL (n::nat) a::nat. (BITS n n a ~= (0::nat)) = (BITS n n a = (1::nat))"
+lemma NOT_BITS: "ALL (n::nat) a::nat. (BITS n n a ~= 0) = (BITS n n a = 1)"
by (import bits NOT_BITS)
-lemma NOT_BITS2: "ALL (n::nat) a::nat. (BITS n n a ~= (1::nat)) = (BITS n n a = (0::nat))"
+lemma NOT_BITS2: "ALL (n::nat) a::nat. (BITS n n a ~= 1) = (BITS n n a = 0)"
by (import bits NOT_BITS2)
lemma BIT_SLICE: "ALL (n::nat) (a::nat) b::nat.
(bit n a = bit n b) = (SLICE n n a = SLICE n n b)"
by (import bits BIT_SLICE)
-lemma BIT_SLICE_LEM: "ALL (y::nat) (x::nat) n::nat.
- SBIT (bit x n) (x + y) = SLICE x x n * (2::nat) ^ y"
+lemma BIT_SLICE_LEM: "ALL (y::nat) (x::nat) n::nat. SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y"
by (import bits BIT_SLICE_LEM)
lemma BIT_SLICE_THM: "ALL (x::nat) xa::nat. SBIT (bit x xa) x = SLICE x x xa"
by (import bits BIT_SLICE_THM)
-lemma SBIT_DIV: "ALL (b::bool) (m::nat) n::nat.
- n < m --> SBIT b (m - n) = SBIT b m div (2::nat) ^ n"
+lemma SBIT_DIV: "ALL (b::bool) (m::nat) n::nat. n < m --> SBIT b (m - n) = SBIT b m div 2 ^ n"
by (import bits SBIT_DIV)
lemma BITS_SUC: "ALL (h::nat) (l::nat) n::nat.
@@ -300,8 +327,7 @@
lemma BITS_SUC_THM: "ALL (h::nat) (l::nat) n::nat.
BITS (Suc h) l n =
- (if Suc h < l then 0::nat
- else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
+ (if Suc h < l then 0 else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
by (import bits BITS_SUC_THM)
lemma BIT_BITS_THM: "ALL (h::nat) (l::nat) (a::nat) b::nat.
@@ -310,12 +336,39 @@
by (import bits BIT_BITS_THM)
lemma BITWISE_LT_2EXP: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- BITWISE n oper a b < (2::nat) ^ n"
+ BITWISE n oper a b < 2 ^ n"
by (import bits BITWISE_LT_2EXP)
-lemma LESS_EXP_MULT2: "ALL (a::nat) b::nat.
- a < b -->
- (EX x::nat. (2::nat) ^ b = (2::nat) ^ (x + (1::nat)) * (2::nat) ^ a)"
+lemma LESS_EXP_MULT2: "(All::(nat => bool) => bool)
+ (%a::nat.
+ (All::(nat => bool) => bool)
+ (%b::nat.
+ (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b)
+ ((Ex::(nat => bool) => bool)
+ (%x::nat.
+ (op =::nat => nat => bool)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ b)
+ ((op *::nat => nat => nat)
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ ((op +::nat => nat => nat) x (1::nat)))
+ ((op ^::nat => nat => nat)
+ ((number_of::bin => nat)
+ ((op BIT::bin => bit => bin)
+ ((op BIT::bin => bit => bin) (Numeral.Pls::bin)
+ (bit.B1::bit))
+ (bit.B0::bit)))
+ a))))))"
by (import bits LESS_EXP_MULT2)
lemma BITWISE_THM: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
@@ -324,27 +377,21 @@
lemma BITWISE_COR: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
x < n -->
- oper (bit x a) (bit x b) -->
- BITWISE n oper a b div (2::nat) ^ x mod (2::nat) = (1::nat)"
+ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 1"
by (import bits BITWISE_COR)
lemma BITWISE_NOT_COR: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
x < n -->
- ~ oper (bit x a) (bit x b) -->
- BITWISE n oper a b div (2::nat) ^ x mod (2::nat) = (0::nat)"
+ ~ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 0"
by (import bits BITWISE_NOT_COR)
-lemma MOD_PLUS_RIGHT: "ALL n>0::nat. ALL (j::nat) k::nat. (j + k mod n) mod n = (j + k) mod n"
+lemma MOD_PLUS_RIGHT: "ALL n>0. ALL (j::nat) k::nat. (j + k mod n) mod n = (j + k) mod n"
by (import bits MOD_PLUS_RIGHT)
-lemma MOD_PLUS_1: "ALL n>0::nat.
- ALL x::nat. ((x + (1::nat)) mod n = (0::nat)) = (x mod n + (1::nat) = n)"
+lemma MOD_PLUS_1: "ALL n>0. ALL x::nat. ((x + 1) mod n = 0) = (x mod n + 1 = n)"
by (import bits MOD_PLUS_1)
-lemma MOD_ADD_1: "ALL n>0::nat.
- ALL x::nat.
- (x + (1::nat)) mod n ~= (0::nat) -->
- (x + (1::nat)) mod n = x mod n + (1::nat)"
+lemma MOD_ADD_1: "ALL n>0. ALL x::nat. (x + 1) mod n ~= 0 --> (x + 1) mod n = x mod n + 1"
by (import bits MOD_ADD_1)
;end_setup
@@ -379,18 +426,18 @@
MODw :: "nat => nat"
defs
- MODw_primdef: "MODw == %n::nat. n mod (2::nat) ^ WL"
+ MODw_primdef: "MODw == %n::nat. n mod 2 ^ WL"
-lemma MODw_def: "ALL n::nat. MODw n = n mod (2::nat) ^ WL"
+lemma MODw_def: "ALL n::nat. MODw n = n mod 2 ^ WL"
by (import word32 MODw_def)
consts
INw :: "nat => bool"
defs
- INw_primdef: "INw == %n::nat. n < (2::nat) ^ WL"
+ INw_primdef: "INw == %n::nat. n < 2 ^ WL"
-lemma INw_def: "ALL n::nat. INw n = (n < (2::nat) ^ WL)"
+lemma INw_def: "ALL n::nat. INw n = (n < 2 ^ WL)"
by (import word32 INw_def)
consts
@@ -429,7 +476,7 @@
lemma TOw_QT: "ALL a::nat. EQUIV (MODw a) a"
by (import word32 TOw_QT)
-lemma MODw_THM: "MODw = BITS HB (0::nat)"
+lemma MODw_THM: "MODw = BITS HB 0"
by (import word32 MODw_THM)
lemma MOD_ADD: "ALL (a::nat) b::nat. MODw (a + b) = MODw (MODw a + MODw b)"
@@ -442,16 +489,16 @@
AONE :: "nat"
defs
- AONE_primdef: "AONE == 1::nat"
+ AONE_primdef: "AONE == 1"
-lemma AONE_def: "AONE = (1::nat)"
+lemma AONE_def: "AONE = 1"
by (import word32 AONE_def)
-lemma ADD_QT: "(ALL n::nat. EQUIV ((0::nat) + n) n) &
+lemma ADD_QT: "(ALL n::nat. EQUIV (0 + n) n) &
(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n)))"
by (import word32 ADD_QT)
-lemma ADD_0_QT: "ALL a::nat. EQUIV (a + (0::nat)) a"
+lemma ADD_0_QT: "ALL a::nat. EQUIV (a + 0) a"
by (import word32 ADD_0_QT)
lemma ADD_COMM_QT: "ALL (a::nat) b::nat. EQUIV (a + b) (b + a)"
@@ -460,30 +507,29 @@
lemma ADD_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (a + (b + c)) (a + b + c)"
by (import word32 ADD_ASSOC_QT)
-lemma MULT_QT: "(ALL n::nat. EQUIV ((0::nat) * n) (0::nat)) &
+lemma MULT_QT: "(ALL n::nat. EQUIV (0 * n) 0) &
(ALL (m::nat) n::nat. EQUIV (Suc m * n) (m * n + n))"
by (import word32 MULT_QT)
lemma ADD1_QT: "ALL m::nat. EQUIV (Suc m) (m + AONE)"
by (import word32 ADD1_QT)
-lemma ADD_CLAUSES_QT: "(ALL m::nat. EQUIV ((0::nat) + m) m) &
-(ALL m::nat. EQUIV (m + (0::nat)) m) &
+lemma ADD_CLAUSES_QT: "(ALL m::nat. EQUIV (0 + m) m) &
+(ALL m::nat. EQUIV (m + 0) m) &
(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n))) &
(ALL (m::nat) n::nat. EQUIV (m + Suc n) (Suc (m + n)))"
by (import word32 ADD_CLAUSES_QT)
-lemma SUC_EQUIV_COMP: "ALL (a::nat) b::nat.
- EQUIV (Suc a) b --> EQUIV a (b + ((2::nat) ^ WL - (1::nat)))"
+lemma SUC_EQUIV_COMP: "ALL (a::nat) b::nat. EQUIV (Suc a) b --> EQUIV a (b + (2 ^ WL - 1))"
by (import word32 SUC_EQUIV_COMP)
lemma INV_SUC_EQ_QT: "ALL (m::nat) n::nat. EQUIV (Suc m) (Suc n) = EQUIV m n"
by (import word32 INV_SUC_EQ_QT)
-lemma ADD_INV_0_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m --> EQUIV n (0::nat)"
+lemma ADD_INV_0_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m --> EQUIV n 0"
by (import word32 ADD_INV_0_QT)
-lemma ADD_INV_0_EQ_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m = EQUIV n (0::nat)"
+lemma ADD_INV_0_EQ_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m = EQUIV n 0"
by (import word32 ADD_INV_0_EQ_QT)
lemma EQ_ADD_LCANCEL_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (m + n) (m + p) = EQUIV n p"
@@ -502,8 +548,8 @@
by (import word32 MULT_COMM_QT)
lemma MULT_CLAUSES_QT: "ALL (m::nat) n::nat.
- EQUIV ((0::nat) * m) (0::nat) &
- EQUIV (m * (0::nat)) (0::nat) &
+ EQUIV (0 * m) 0 &
+ EQUIV (m * 0) 0 &
EQUIV (AONE * m) m &
EQUIV (m * AONE) m &
EQUIV (Suc m * n) (m * n + n) & EQUIV (m * Suc n) (m + m * n)"
@@ -522,21 +568,21 @@
ONE_COMP :: "nat => nat"
defs
- ONE_COMP_primdef: "ONE_COMP == %x::nat. (2::nat) ^ WL - (1::nat) - MODw x"
+ ONE_COMP_primdef: "ONE_COMP == %x::nat. 2 ^ WL - 1 - MODw x"
-lemma ONE_COMP_def: "ALL x::nat. ONE_COMP x = (2::nat) ^ WL - (1::nat) - MODw x"
+lemma ONE_COMP_def: "ALL x::nat. ONE_COMP x = 2 ^ WL - 1 - MODw x"
by (import word32 ONE_COMP_def)
consts
TWO_COMP :: "nat => nat"
defs
- TWO_COMP_primdef: "TWO_COMP == %x::nat. (2::nat) ^ WL - MODw x"
+ TWO_COMP_primdef: "TWO_COMP == %x::nat. 2 ^ WL - MODw x"
-lemma TWO_COMP_def: "ALL x::nat. TWO_COMP x = (2::nat) ^ WL - MODw x"
+lemma TWO_COMP_def: "ALL x::nat. TWO_COMP x = 2 ^ WL - MODw x"
by (import word32 TWO_COMP_def)
-lemma ADD_TWO_COMP_QT: "ALL a::nat. EQUIV (MODw a + TWO_COMP a) (0::nat)"
+lemma ADD_TWO_COMP_QT: "ALL a::nat. EQUIV (MODw a + TWO_COMP a) 0"
by (import word32 ADD_TWO_COMP_QT)
lemma TWO_COMP_ONE_COMP_QT: "ALL a::nat. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
@@ -557,8 +603,7 @@
((EQUIV::nat => nat => bool) x xa)))"
by (import word32 BIT_EQUIV_THM)
-lemma BITS_SUC2: "ALL (n::nat) a::nat.
- BITS (Suc n) (0::nat) a = SLICE (Suc n) (Suc n) a + BITS n (0::nat) a"
+lemma BITS_SUC2: "ALL (n::nat) a::nat. BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a"
by (import word32 BITS_SUC2)
lemma BITWISE_ONE_COMP_THM: "ALL (a::nat) b::nat. BITWISE WL (%(x::bool) y::bool. ~ x) a b = ONE_COMP a"
@@ -598,9 +643,9 @@
COMP0 :: "nat"
defs
- COMP0_primdef: "COMP0 == ONE_COMP (0::nat)"
+ COMP0_primdef: "COMP0 == ONE_COMP 0"
-lemma COMP0_def: "COMP0 = ONE_COMP (0::nat)"
+lemma COMP0_def: "COMP0 = ONE_COMP 0"
by (import word32 COMP0_def)
lemma BITWISE_THM2: "(All::(nat => bool) => bool)
@@ -655,7 +700,7 @@
lemma OR_COMP_QT: "ALL a::nat. EQUIV (OR a (ONE_COMP a)) COMP0"
by (import word32 OR_COMP_QT)
-lemma AND_COMP_QT: "ALL a::nat. EQUIV (AND a (ONE_COMP a)) (0::nat)"
+lemma AND_COMP_QT: "ALL a::nat. EQUIV (AND a (ONE_COMP a)) 0"
by (import word32 AND_COMP_QT)
lemma ONE_COMP_QT: "ALL a::nat. EQUIV (ONE_COMP (ONE_COMP a)) a"
@@ -683,16 +728,14 @@
by (import word32 MSB_WELLDEF)
lemma BITWISE_ISTEP: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- (0::nat) < n -->
- BITWISE n oper (a div (2::nat)) (b div (2::nat)) =
- BITWISE n oper a b div (2::nat) +
- SBIT (oper (bit n a) (bit n b)) (n - (1::nat))"
+ 0 < n -->
+ BITWISE n oper (a div 2) (b div 2) =
+ BITWISE n oper a b div 2 + SBIT (oper (bit n a) (bit n b)) (n - 1)"
by (import word32 BITWISE_ISTEP)
lemma BITWISE_EVAL: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
BITWISE (Suc n) oper a b =
- (2::nat) * BITWISE n oper (a div (2::nat)) (b div (2::nat)) +
- SBIT (oper (LSBn a) (LSBn b)) (0::nat)"
+ 2 * BITWISE n oper (a div 2) (b div 2) + SBIT (oper (LSBn a) (LSBn b)) 0"
by (import word32 BITWISE_EVAL)
lemma BITWISE_WELLDEF: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) (b::nat) (c::nat) d::nat.
@@ -728,9 +771,9 @@
LSR_ONE :: "nat => nat"
defs
- LSR_ONE_primdef: "LSR_ONE == %a::nat. MODw a div (2::nat)"
+ LSR_ONE_primdef: "LSR_ONE == %a::nat. MODw a div 2"
-lemma LSR_ONE_def: "ALL a::nat. LSR_ONE a = MODw a div (2::nat)"
+lemma LSR_ONE_def: "ALL a::nat. LSR_ONE a = MODw a div 2"
by (import word32 LSR_ONE_def)
consts
@@ -772,7 +815,7 @@
lemma RRX_WELLDEF: "ALL (a::nat) (b::nat) c::bool. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)"
by (import word32 RRX_WELLDEF)
-lemma LSR_ONE: "LSR_ONE = BITS HB (1::nat)"
+lemma LSR_ONE: "LSR_ONE = BITS HB 1"
by (import word32 LSR_ONE)
typedef (open) word32 = "{x::nat => bool. EX xa::nat. x = EQUIV xa}"
@@ -793,9 +836,9 @@
w_0 :: "word32"
defs
- w_0_primdef: "w_0 == mk_word32 (EQUIV (0::nat))"
+ w_0_primdef: "w_0 == mk_word32 (EQUIV 0)"
-lemma w_0_def: "w_0 = mk_word32 (EQUIV (0::nat))"
+lemma w_0_def: "w_0 = mk_word32 (EQUIV 0)"
by (import word32 w_0_def)
consts
@@ -1081,10 +1124,10 @@
word_1comp (bitwise_or x xa) = bitwise_and (word_1comp x) (word_1comp xa)"
by (import word32 DE_MORGAN_THMw)
-lemma w_0: "w_0 = n2w (0::nat)"
+lemma w_0: "w_0 = n2w 0"
by (import word32 w_0)
-lemma w_1: "w_1 = n2w (1::nat)"
+lemma w_1: "w_1 = n2w 1"
by (import word32 w_1)
lemma w_T: "w_T =
@@ -1139,9 +1182,9 @@
constdefs
word_lsl :: "word32 => nat => word32"
- "word_lsl == %(a::word32) n::nat. word_mul a (n2w ((2::nat) ^ n))"
+ "word_lsl == %(a::word32) n::nat. word_mul a (n2w (2 ^ n))"
-lemma word_lsl: "ALL (a::word32) n::nat. word_lsl a n = word_mul a (n2w ((2::nat) ^ n))"
+lemma word_lsl: "ALL (a::word32) n::nat. word_lsl a n = word_mul a (n2w (2 ^ n))"
by (import word32 word_lsl)
constdefs
@@ -1320,14 +1363,13 @@
lemma LSL_LIMIT: "ALL (w::word32) n::nat. HB < n --> word_lsl w n = w_0"
by (import word32 LSL_LIMIT)
-lemma MOD_MOD_DIV: "ALL (a::nat) b::nat. INw (MODw a div (2::nat) ^ b)"
+lemma MOD_MOD_DIV: "ALL (a::nat) b::nat. INw (MODw a div 2 ^ b)"
by (import word32 MOD_MOD_DIV)
-lemma MOD_MOD_DIV_2EXP: "ALL (a::nat) n::nat.
- MODw (MODw a div (2::nat) ^ n) div (2::nat) = MODw a div (2::nat) ^ Suc n"
+lemma MOD_MOD_DIV_2EXP: "ALL (a::nat) n::nat. MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n"
by (import word32 MOD_MOD_DIV_2EXP)
-lemma LSR_EVAL: "ALL n::nat. word_lsr (n2w (a::nat)) n = n2w (MODw a div (2::nat) ^ n)"
+lemma LSR_EVAL: "ALL n::nat. word_lsr (n2w (a::nat)) n = n2w (MODw a div 2 ^ n)"
by (import word32 LSR_EVAL)
lemma LSR_THM: "ALL (x::nat) n::nat. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
@@ -1336,16 +1378,13 @@
lemma LSR_LIMIT: "ALL (x::nat) w::word32. HB < x --> word_lsr w x = w_0"
by (import word32 LSR_LIMIT)
-lemma LEFT_SHIFT_LESS: "ALL (n::nat) (m::nat) a::nat.
- a < (2::nat) ^ m -->
- (2::nat) ^ n + a * (2::nat) ^ n <= (2::nat) ^ (m + n)"
+lemma LEFT_SHIFT_LESS: "ALL (n::nat) (m::nat) a::nat. a < 2 ^ m --> 2 ^ n + a * 2 ^ n <= 2 ^ (m + n)"
by (import word32 LEFT_SHIFT_LESS)
lemma ROR_THM: "ALL (x::nat) n::nat.
word_ror (n2w n) x =
(let x'::nat = x mod WL
- in n2w (BITS HB x' n +
- BITS (x' - (1::nat)) (0::nat) n * (2::nat) ^ (WL - x')))"
+ in n2w (BITS HB x' n + BITS (x' - 1) 0 n * 2 ^ (WL - x')))"
by (import word32 ROR_THM)
lemma ROR_CYCLE: "ALL (x::nat) w::word32. word_ror w (x * WL) = w"
@@ -1354,7 +1393,7 @@
lemma ASR_THM: "ALL (x::nat) n::nat.
word_asr (n2w n) x =
(let x'::nat = min HB x; s::nat = BITS HB x' n
- in n2w (if MSBn n then (2::nat) ^ WL - (2::nat) ^ (WL - x') + s else s))"
+ in n2w (if MSBn n then 2 ^ WL - 2 ^ (WL - x') + s else s))"
by (import word32 ASR_THM)
lemma ASR_LIMIT: "ALL (x::nat) w::word32.
@@ -1366,10 +1405,9 @@
(ALL n::nat. word_lsr w_0 n = w_0) & (ALL n::nat. word_ror w_0 n = w_0)"
by (import word32 ZERO_SHIFT)
-lemma ZERO_SHIFT2: "(ALL a::word32. word_lsl a (0::nat) = a) &
-(ALL a::word32. word_asr a (0::nat) = a) &
-(ALL a::word32. word_lsr a (0::nat) = a) &
-(ALL a::word32. word_ror a (0::nat) = a)"
+lemma ZERO_SHIFT2: "(ALL a::word32. word_lsl a 0 = a) &
+(ALL a::word32. word_asr a 0 = a) &
+(ALL a::word32. word_lsr a 0 = a) & (ALL a::word32. word_ror a 0 = a)"
by (import word32 ZERO_SHIFT2)
lemma ASR_w_T: "ALL n::nat. word_asr w_T n = w_T"
@@ -1425,19 +1463,19 @@
lemma ONE_COMP_EVAL2: "ALL a::nat.
word_1comp (n2w a) =
- n2w ((2::nat) ^
+ n2w (2 ^
NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
- (1::nat) -
+ 1 -
MODw a)"
by (import word32 ONE_COMP_EVAL2)
lemma TWO_COMP_EVAL2: "ALL a::nat.
word_2comp (n2w a) =
n2w (MODw
- ((2::nat) ^
+ (2 ^
NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
@@ -1445,12 +1483,12 @@
MODw a))"
by (import word32 TWO_COMP_EVAL2)
-lemma LSR_ONE_EVAL2: "ALL a::nat. word_lsr1 (n2w a) = n2w (MODw a div (2::nat))"
+lemma LSR_ONE_EVAL2: "ALL a::nat. word_lsr1 (n2w a) = n2w (MODw a div 2)"
by (import word32 LSR_ONE_EVAL2)
lemma ASR_ONE_EVAL2: "ALL a::nat.
word_asr1 (n2w a) =
- n2w (MODw a div (2::nat) +
+ n2w (MODw a div 2 +
SBIT (MSBn a)
(NUMERAL
(NUMERAL_BIT1
@@ -1460,7 +1498,7 @@
lemma ROR_ONE_EVAL2: "ALL a::nat.
word_ror1 (n2w a) =
- n2w (MODw a div (2::nat) +
+ n2w (MODw a div 2 +
SBIT (LSBn a)
(NUMERAL
(NUMERAL_BIT1
@@ -1470,7 +1508,7 @@
lemma RRX_EVAL2: "ALL (c::bool) a::nat.
RRX c (n2w a) =
- n2w (MODw a div (2::nat) +
+ n2w (MODw a div 2 +
SBIT c
(NUMERAL
(NUMERAL_BIT1
@@ -1520,50 +1558,14 @@
(%(x::bool) y::bool. x ~= y) a b)"
by (import word32 EOR_EVAL2)
-lemma BITWISE_EVAL2: "(All::(nat => bool) => bool)
- (%n::nat.
- (All::((bool => bool => bool) => bool) => bool)
- (%oper::bool => bool => bool.
- (All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%y::nat.
- (op =::nat => nat => bool)
- ((BITWISE::nat
- => (bool => bool => bool)
- => nat => nat => nat)
- n oper x y)
- ((If::bool => nat => nat => nat)
- ((op =::nat => nat => bool) n (0::nat)) (0::nat)
- ((op +::nat => nat => nat)
- ((op *::nat => nat => nat)
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin)
- (Numeral.Pls::bin) (bit.B1::bit))
- (bit.B0::bit)))
- ((BITWISE::nat
-=> (bool => bool => bool) => nat => nat => nat)
- ((op -::nat => nat => nat) n (1::nat)) oper
- ((op div::nat => nat => nat) x
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin)
- (Numeral.Pls::bin) (bit.B1::bit))
- (bit.B0::bit))))
- ((op div::nat => nat => nat) y
- ((number_of::bin => nat)
- ((op BIT::bin => bit => bin)
- ((op BIT::bin => bit => bin)
- (Numeral.Pls::bin) (bit.B1::bit))
- (bit.B0::bit))))))
- ((If::bool => nat => nat => nat)
- (oper ((ODD::nat => bool) x)
- ((ODD::nat => bool) y))
- (1::nat) (0::nat))))))))"
+lemma BITWISE_EVAL2: "ALL (n::nat) (oper::bool => bool => bool) (x::nat) y::nat.
+ BITWISE n oper x y =
+ (if n = 0 then 0
+ else 2 * BITWISE (n - 1) oper (x div 2) (y div 2) +
+ (if oper (ODD x) (ODD y) then 1 else 0))"
by (import word32 BITWISE_EVAL2)
-lemma BITSwLT_THM: "ALL (h::nat) (l::nat) n::word32. BITSw h l n < (2::nat) ^ (Suc h - l)"
+lemma BITSwLT_THM: "ALL (h::nat) (l::nat) n::word32. BITSw h l n < 2 ^ (Suc h - l)"
by (import word32 BITSwLT_THM)
lemma BITSw_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::word32.
@@ -1572,29 +1574,29 @@
by (import word32 BITSw_COMP_THM)
lemma BITSw_DIV_THM: "ALL (h::nat) (l::nat) (n::nat) x::word32.
- BITSw h l x div (2::nat) ^ n = BITSw h (l + n) x"
+ BITSw h l x div 2 ^ n = BITSw h (l + n) x"
by (import word32 BITSw_DIV_THM)
-lemma BITw_THM: "ALL (b::nat) n::word32. BITw b n = (BITSw b b n = (1::nat))"
+lemma BITw_THM: "ALL (b::nat) n::word32. BITw b n = (BITSw b b n = 1)"
by (import word32 BITw_THM)
-lemma SLICEw_THM: "ALL (n::word32) (h::nat) l::nat. SLICEw h l n = BITSw h l n * (2::nat) ^ l"
+lemma SLICEw_THM: "ALL (n::word32) (h::nat) l::nat. SLICEw h l n = BITSw h l n * 2 ^ l"
by (import word32 SLICEw_THM)
lemma BITS_SLICEw_THM: "ALL (h::nat) (l::nat) n::word32. BITS h l (SLICEw h l n) = BITSw h l n"
by (import word32 BITS_SLICEw_THM)
-lemma SLICEw_ZERO_THM: "ALL (n::word32) h::nat. SLICEw h (0::nat) n = BITSw h (0::nat) n"
+lemma SLICEw_ZERO_THM: "ALL (n::word32) h::nat. SLICEw h 0 n = BITSw h 0 n"
by (import word32 SLICEw_ZERO_THM)
lemma SLICEw_COMP_THM: "ALL (h::nat) (m::nat) (l::nat) a::word32.
Suc m <= h & l <= m --> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a"
by (import word32 SLICEw_COMP_THM)
-lemma BITSw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> BITSw h l n = (0::nat)"
+lemma BITSw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> BITSw h l n = 0"
by (import word32 BITSw_ZERO)
-lemma SLICEw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> SLICEw h l n = (0::nat)"
+lemma SLICEw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> SLICEw h l n = 0"
by (import word32 SLICEw_ZERO)
;end_setup
--- a/src/HOL/Import/HOL/bool.imp Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOL/bool.imp Mon Sep 26 16:10:19 2005 +0200
@@ -41,7 +41,7 @@
"boolAxiom" > "HOL4Base.bool.boolAxiom"
"UNWIND_THM2" > "HOL.simp_thms_39"
"UNWIND_THM1" > "HOL.simp_thms_40"
- "UNWIND_FORALL_THM2" > "HOL4Base.bool.UNWIND_FORALL_THM2"
+ "UNWIND_FORALL_THM2" > "HOL.simp_thms_41"
"UNWIND_FORALL_THM1" > "HOL.simp_thms_42"
"UEXISTS_SIMP" > "HOL4Base.bool.UEXISTS_SIMP"
"UEXISTS_OR_THM" > "HOL4Base.bool.UEXISTS_OR_THM"
@@ -95,7 +95,7 @@
"NOT_AND" > "HOL4Base.bool.NOT_AND"
"MONO_OR" > "Inductive.basic_monos_3"
"MONO_NOT" > "HOL.rev_contrapos"
- "MONO_IMP" > "HOL4Base.bool.MONO_IMP"
+ "MONO_IMP" > "Set.imp_mono"
"MONO_EXISTS" > "Inductive.basic_monos_5"
"MONO_COND" > "HOL4Base.bool.MONO_COND"
"MONO_AND" > "Inductive.basic_monos_4"
@@ -108,11 +108,11 @@
"LEFT_OR_OVER_AND" > "HOL.disj_conj_distribL"
"LEFT_OR_EXISTS_THM" > "HOL.ex_simps_3"
"LEFT_FORALL_OR_THM" > "HOL.all_simps_3"
- "LEFT_FORALL_IMP_THM" > "HOL4Base.bool.LEFT_FORALL_IMP_THM"
+ "LEFT_FORALL_IMP_THM" > "HOL.imp_ex"
"LEFT_EXISTS_IMP_THM" > "HOL.imp_all"
"LEFT_EXISTS_AND_THM" > "HOL.ex_simps_1"
"LEFT_AND_OVER_OR" > "HOL.conj_disj_distribL"
- "LEFT_AND_FORALL_THM" > "HOL4Base.bool.LEFT_AND_FORALL_THM"
+ "LEFT_AND_FORALL_THM" > "HOL.all_simps_1"
"IN_def" > "HOL4Base.bool.IN_def"
"IN_DEF" > "HOL4Base.bool.IN_DEF"
"INFINITY_AX" > "HOL4Setup.INFINITY_AX"
@@ -146,7 +146,7 @@
"EQ_SYM" > "HOL.meta_eq_to_obj_eq"
"EQ_REFL" > "Presburger.fm_modd_pinf"
"EQ_IMP_THM" > "HOL.iff_conv_conj_imp"
- "EQ_EXT" > "HOL4Base.bool.EQ_EXT"
+ "EQ_EXT" > "HOL.meta_eq_to_obj_eq"
"EQ_EXPAND" > "HOL4Base.bool.EQ_EXPAND"
"EQ_CLAUSES" > "HOL4Base.bool.EQ_CLAUSES"
"DISJ_SYM" > "HOL.disj_comms_1"
--- a/src/HOL/Import/HOLLight/HOLLight.thy Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/HOLLight/HOLLight.thy Mon Sep 26 16:10:19 2005 +0200
@@ -96,7 +96,7 @@
by (import hollight EXCLUDED_MIDDLE)
constdefs
- COND :: "bool => 'A::type => 'A::type => 'A::type"
+ 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)"
@@ -174,14 +174,14 @@
by (import hollight th_cond)
constdefs
- LET_END :: "'A::type => 'A::type"
+ 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::type => bool) => 'A::type"
+ GABS :: "('A => bool) => 'A"
"(op ==::(('A::type => bool) => 'A::type)
=> (('A::type => bool) => 'A::type) => prop)
(GABS::('A::type => bool) => 'A::type)
@@ -194,7 +194,7 @@
by (import hollight DEF_GABS)
constdefs
- GEQ :: "'A::type => 'A::type => bool"
+ 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)"
@@ -209,7 +209,7 @@
by (import hollight PAIR_EXISTS_THM)
constdefs
- CURRY :: "('A::type * 'B::type => 'C::type) => 'A::type => 'B::type => 'C::type"
+ CURRY :: "('A * 'B => 'C) => 'A => 'B => 'C"
"CURRY ==
%(u::'A::type * 'B::type => 'C::type) (ua::'A::type) ub::'B::type.
u (ua, ub)"
@@ -220,7 +220,7 @@
by (import hollight DEF_CURRY)
constdefs
- UNCURRY :: "('A::type => 'B::type => 'C::type) => 'A::type * 'B::type => 'C::type"
+ 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)"
@@ -231,8 +231,7 @@
by (import hollight DEF_UNCURRY)
constdefs
- PASSOC :: "(('A::type * 'B::type) * 'C::type => 'D::type)
-=> 'A::type * 'B::type * 'C::type => 'D::type"
+ 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.
@@ -245,12 +244,11 @@
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::nat) = e & (ALL n::nat. fn (Suc n) = f (fn n) n)"
+ 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::nat) + x = x) &
-(ALL x::nat. x + (0::nat) = x) &
+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)
@@ -259,25 +257,25 @@
m + n + (p::nat) = m + (n + p) & m + (n + p) = n + (m + p)"
by (import hollight ADD_AC)
-lemma EQ_ADD_LCANCEL_0: "ALL (m::nat) n::nat. (m + n = m) = (n = (0::nat))"
+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::nat))"
+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::nat) = Suc (0::nat)"
+lemma ONE: "NUMERAL_BIT1 0 = Suc 0"
by (import hollight ONE)
-lemma TWO: "NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) = Suc (NUMERAL_BIT1 (0::nat))"
+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::nat)"
+lemma ADD1: "ALL x::nat. Suc x = x + NUMERAL_BIT1 0"
by (import hollight ADD1)
-lemma MULT_CLAUSES: "(ALL x::nat. (0::nat) * x = (0::nat)) &
-(ALL x::nat. x * (0::nat) = (0::nat)) &
-(ALL x::nat. NUMERAL_BIT1 (0::nat) * x = x) &
-(ALL x::nat. x * NUMERAL_BIT1 (0::nat) = x) &
+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)
@@ -286,40 +284,39 @@
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::nat)) * n = n + n"
+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::nat)) =
- (m = NUMERAL_BIT1 (0::nat) & n = NUMERAL_BIT1 (0::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::nat) = NUMERAL_BIT1 (0::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::nat) = NUMERAL_BIT1 (0::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::nat)) = (m = (0::nat) & n ~= (0::nat))"
+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::nat)) n = NUMERAL_BIT1 (0::nat)"
+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::nat)) = x"
+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::nat))) = x * x"
+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"
@@ -334,12 +331,12 @@
defs
"<=_def": "<= ==
SOME u::nat => nat => bool.
- (ALL m::nat. u m (0::nat) = (m = (0::nat))) &
+ (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::nat) = (m = (0::nat))) &
+ (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_)
@@ -349,12 +346,12 @@
defs
"<_def": "< ==
SOME u::nat => nat => bool.
- (ALL m::nat. u m (0::nat) = False) &
+ (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::nat) = False) &
+ (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_)
@@ -388,10 +385,10 @@
lemma LT_SUC: "ALL (x::nat) xa::nat. < (Suc x) (Suc xa) = < x xa"
by (import hollight LT_SUC)
-lemma LE_0: "All (<= (0::nat))"
+lemma LE_0: "All (<= 0)"
by (import hollight LE_0)
-lemma LT_0: "ALL x::nat. < (0::nat) (Suc x)"
+lemma LT_0: "ALL x::nat. < 0 (Suc x)"
by (import hollight LT_0)
lemma LE_REFL: "ALL n::nat. <= n n"
@@ -436,7 +433,7 @@
lemma LTE_CASES: "ALL (x::nat) xa::nat. < x xa | <= xa x"
by (import hollight LTE_CASES)
-lemma LT_NZ: "ALL n::nat. < (0::nat) n = (n ~= (0::nat))"
+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)"
@@ -469,10 +466,10 @@
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::nat) n"
+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::nat) x"
+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"
@@ -501,28 +498,26 @@
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::nat) (m * n) = (< (0::nat) m & < (0::nat) n)"
+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::nat) & < n p --> < (m * n) (m * p)"
+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::nat) | <= n p)"
+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::nat))"
+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::nat) & < n p)"
+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::nat))"
+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)"
@@ -558,22 +553,21 @@
EVEN :: "nat => bool"
"EVEN ==
SOME EVEN::nat => bool.
- EVEN (0::nat) = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
+ EVEN 0 = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
lemma DEF_EVEN: "EVEN =
(SOME EVEN::nat => bool.
- EVEN (0::nat) = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n)))"
+ 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::nat) = False & (ALL n::nat. ODD (Suc n) = (~ ODD n))"
+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::nat) = False & (ALL n::nat. ODD (Suc n) = (~ ODD n)))"
+ 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"
@@ -594,7 +588,7 @@
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::nat))"
+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)"
@@ -603,83 +597,76 @@
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::nat))"
+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::nat)) * n)"
+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::nat)) * x))"
+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::nat)) * m)) &
- (~ EVEN n -->
- (EX m::nat. n = Suc (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * m)))"
+ (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::nat)) * m)"
+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::nat)) * m))"
+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::nat))) k * m) =
- (n ~= (0::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::nat) - x = (0::nat) & x - (0::nat) = x"
+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::nat)) = <= m n"
+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::nat)"
+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::nat) = x"
+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::nat) = NUMERAL_BIT1 (0::nat) &
- (ALL n::nat. FACT (Suc n) = Suc n * FACT n)"
+ 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::nat) = NUMERAL_BIT1 (0::nat) &
- (ALL n::nat. FACT (Suc n) = Suc n * FACT n))"
+ 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::nat) (FACT n)"
+lemma FACT_LT: "ALL n::nat. < 0 (FACT n)"
by (import hollight FACT_LT)
-lemma FACT_LE: "ALL x::nat. <= (NUMERAL_BIT1 (0::nat)) (FACT x)"
+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::nat) --> (EX (q::nat) r::nat. m = q * n + r & < r n)"
+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::nat)) (x = (0::nat) & xa = (0::nat))
- (m = x * n + xa & < xa n)"
+ COND (n = 0) (x = 0 & xa = 0) (m = x * n + xa & < xa n)"
by (import hollight DIVMOD_EXIST_0)
constdefs
@@ -688,14 +675,14 @@
SOME q::nat => nat => nat.
EX r::nat => nat => nat.
ALL (m::nat) n::nat.
- COND (n = (0::nat)) (q m n = (0::nat) & r m n = (0::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::nat)) (q m n = (0::nat) & r m n = (0::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)
@@ -704,18 +691,17 @@
"MOD ==
SOME r::nat => nat => nat.
ALL (m::nat) n::nat.
- COND (n = (0::nat)) (DIV m n = (0::nat) & r m n = (0::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::nat)) (DIV m n = (0::nat) & r m n = (0::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::nat) --> m = DIV m n * n + MOD m n & < (MOD m n) n"
+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.
@@ -733,14 +719,13 @@
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::nat) --> MOD (x * xa) x = (0::nat)"
+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::nat) --> DIV (x * xa) x = xa"
+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::nat) --> DIV (DIV m n) p = DIV m (n * p)"
+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"
@@ -750,166 +735,150 @@
by (import hollight MOD_EQ)
lemma DIV_MOD: "ALL (m::nat) (n::nat) p::nat.
- n * p ~= (0::nat) --> MOD (DIV m n) p = DIV (MOD m (n * p)) n"
+ 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::nat)) = n"
+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::nat) (EXP xa x) = (xa ~= (0::nat) | x = (0::nat))"
+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::nat) --> <= (DIV m n) m"
+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::nat) --> DIV (0::nat) n = (0::nat)"
+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::nat) --> MOD (0::nat) n = (0::nat)"
+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::nat)"
+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::nat) --> MOD (MOD m (n * p)) n = MOD m n"
+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::nat) --> MOD (MOD m n) n = MOD m n"
+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::nat) --> DIV (x * xa) (x * xb) = DIV xa xb"
+ 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::nat) --> MOD (x * xa) (x * xb) = x * MOD xa xb"
+ 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::nat)) = (0::nat)"
+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::nat)) (m = (0::nat)) (MOD m n = (0::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::nat))) x & < m n |
- x = (0::nat) & m ~= (0::nat) & n = (0::nat))"
+ (<= (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::nat)) (m = (0::nat) --> n = (0::nat))
- (x = NUMERAL_BIT1 (0::nat) | <= m 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::nat) & <= m n --> <= (DIV m p) (DIV n p)"
+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::nat) & <= (m + p) n --> < (DIV m p) (DIV n p)"
+ 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::nat) & <= b (a * n) --> <= (DIV b a) n"
+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::nat) --> <= n (DIV b a) = <= (a * n) b"
+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::nat) --> <= (DIV b a) n = < b (a * (n + NUMERAL_BIT1 (0::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::nat) --> (DIV m n = (0::nat)) = < m n"
+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::nat) --> (MOD m n = (0::nat)) = (EX q::nat. m = q * n)"
+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::nat))) = (0::nat))"
+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::nat))) = NUMERAL_BIT1 (0::nat))"
+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::nat) --> MOD (m * MOD p n) n = MOD (m * p) n"
+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::nat) --> MOD (MOD x xa * xb) xa = MOD (x * xb) xa"
+ 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::nat) --> MOD (MOD x xa * MOD xb xa) xa = MOD (x * xb) xa"
+ 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::nat) --> MOD (EXP (MOD m n) p) n = MOD (EXP m p) n"
+ 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::nat) --> MOD (MOD a n + MOD b n) n = MOD (a + b) n"
+ 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::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::nat) --> DIV n n = NUMERAL_BIT1 (0::nat)"
+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::nat) --> <= (MOD m n) m"
+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::nat) & <= p m --> <= (DIV n m) (DIV n p)"
+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::nat) & < (b * c) ((a + NUMERAL_BIT1 (0::nat)) * d) -->
- <= (DIV c d) (DIV a b)"
+ 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::nat)) * (d::nat)) &
-< (a * d) ((c + NUMERAL_BIT1 (0::nat)) * b) -->
+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::nat)) & (a = b + x --> P x))"
+(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::nat) --> P (0::nat)) & (n = Suc m --> P m))"
+(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::nat) & P (0::nat) (0::nat) |
- n ~= (0::nat) & (ALL (q::nat) r::nat. m = q * n + r & < r n --> P q r))"
+(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::type
-=> 'q_9909::type => ('q_9910::type => 'q_9909::type => bool) => bool"
+ 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"
@@ -942,7 +911,7 @@
by (import hollight MINIMAL)
constdefs
- WF :: "('A::type => 'A::type => bool) => bool"
+ WF :: "('A => 'A => bool) => bool"
"WF ==
%u::'A::type => 'A::type => bool.
ALL P::'A::type => bool.
@@ -1081,7 +1050,7 @@
by (import hollight WF_REC_num)
consts
- measure :: "('q_11107::type => nat) => 'q_11107::type => 'q_11107::type => bool"
+ measure :: "('q_11107 => nat) => 'q_11107 => 'q_11107 => bool"
defs
measure_def: "hollight.measure ==
@@ -1128,73 +1097,216 @@
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: "(0::nat) = (0::nat) & NUMERAL_BIT0 (0::nat) = (0::nat)"
+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::nat) = NUMERAL_BIT1 (0::nat) &
+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::nat) = (0::nat) &
+Pred 0 = 0 &
(ALL x::nat.
- Pred (NUMERAL_BIT0 x) =
- COND (x = (0::nat)) (0::nat) (NUMERAL_BIT1 (Pred x))) &
+ 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: "(ALL (x::nat) xa::nat. x + xa = x + xa) &
-(0::nat) + (0::nat) = (0::nat) &
-(ALL x::nat. (0::nat) + NUMERAL_BIT0 x = NUMERAL_BIT0 x) &
-(ALL x::nat. (0::nat) + NUMERAL_BIT1 x = NUMERAL_BIT1 x) &
-(ALL x::nat. NUMERAL_BIT0 x + (0::nat) = NUMERAL_BIT0 x) &
-(ALL x::nat. NUMERAL_BIT1 x + (0::nat) = NUMERAL_BIT1 x) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT0 x + NUMERAL_BIT0 xa = NUMERAL_BIT0 (x + xa)) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT0 x + NUMERAL_BIT1 xa = NUMERAL_BIT1 (x + xa)) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT1 x + NUMERAL_BIT0 xa = NUMERAL_BIT1 (x + xa)) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT1 x + NUMERAL_BIT1 xa = NUMERAL_BIT0 (Suc (x + xa)))"
+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: "(ALL (x::nat) xa::nat. x * xa = x * xa) &
-(0::nat) * (0::nat) = (0::nat) &
-(ALL x::nat. (0::nat) * NUMERAL_BIT0 x = (0::nat)) &
-(ALL x::nat. (0::nat) * NUMERAL_BIT1 x = (0::nat)) &
-(ALL x::nat. NUMERAL_BIT0 x * (0::nat) = (0::nat)) &
-(ALL x::nat. NUMERAL_BIT1 x * (0::nat) = (0::nat)) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT0 x * NUMERAL_BIT0 xa =
- NUMERAL_BIT0 (NUMERAL_BIT0 (x * xa))) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT0 x * NUMERAL_BIT1 xa =
- NUMERAL_BIT0 x + NUMERAL_BIT0 (NUMERAL_BIT0 (x * xa))) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT1 x * NUMERAL_BIT0 xa =
- NUMERAL_BIT0 xa + NUMERAL_BIT0 (NUMERAL_BIT0 (x * xa))) &
-(ALL (x::nat) xa::nat.
- NUMERAL_BIT1 x * NUMERAL_BIT1 xa =
- NUMERAL_BIT1 x +
- (NUMERAL_BIT0 xa + NUMERAL_BIT0 (NUMERAL_BIT0 (x * xa))))"
+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::nat) (0::nat) = NUMERAL_BIT1 (0::nat) &
-(ALL m::nat. EXP (NUMERAL_BIT0 m) (0::nat) = NUMERAL_BIT1 (0::nat)) &
-(ALL m::nat. EXP (NUMERAL_BIT1 m) (0::nat) = NUMERAL_BIT1 (0::nat)) &
-(ALL n::nat.
- EXP (0::nat) (NUMERAL_BIT0 n) = EXP (0::nat) n * EXP (0::nat) n) &
+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::nat) (NUMERAL_BIT1 n) = (0::nat)) &
+(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)) &
@@ -1204,23 +1316,23 @@
by (import hollight ARITH_EXP)
lemma ARITH_EVEN: "(ALL x::nat. EVEN x = EVEN x) &
-EVEN (0::nat) = True &
+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::nat) = False &
+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::nat) (0::nat) = True &
-(ALL x::nat. <= (NUMERAL_BIT0 x) (0::nat) = (x = (0::nat))) &
-(ALL x::nat. <= (NUMERAL_BIT1 x) (0::nat) = False) &
-(ALL x::nat. <= (0::nat) (NUMERAL_BIT0 x) = True) &
-(ALL x::nat. <= (0::nat) (NUMERAL_BIT1 x) = True) &
+<= 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) &
@@ -1228,44 +1340,181 @@
by (import hollight ARITH_LE)
lemma ARITH_LT: "(ALL (x::nat) xa::nat. < x xa = < x xa) &
-< (0::nat) (0::nat) = False &
-(ALL x::nat. < (NUMERAL_BIT0 x) (0::nat) = False) &
-(ALL x::nat. < (NUMERAL_BIT1 x) (0::nat) = False) &
-(ALL x::nat. < (0::nat) (NUMERAL_BIT0 x) = < (0::nat) x) &
-(ALL x::nat. < (0::nat) (NUMERAL_BIT1 x) = True) &
+< 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: "(ALL (x::nat) xa::nat. (x = xa) = (x = xa)) &
-((0::nat) = (0::nat)) = True &
-(ALL x::nat. (NUMERAL_BIT0 x = (0::nat)) = (x = (0::nat))) &
-(ALL x::nat. (NUMERAL_BIT1 x = (0::nat)) = False) &
-(ALL x::nat. ((0::nat) = NUMERAL_BIT0 x) = ((0::nat) = x)) &
-(ALL x::nat. ((0::nat) = NUMERAL_BIT1 x) = False) &
-(ALL (x::nat) xa::nat. (NUMERAL_BIT0 x = NUMERAL_BIT0 xa) = (x = xa)) &
-(ALL (x::nat) xa::nat. (NUMERAL_BIT0 x = NUMERAL_BIT1 xa) = False) &
-(ALL (x::nat) xa::nat. (NUMERAL_BIT1 x = NUMERAL_BIT0 xa) = False) &
-(ALL (x::nat) xa::nat. (NUMERAL_BIT1 x = NUMERAL_BIT1 xa) = (x = xa))"
+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: "(ALL (x::nat) xa::nat. x - xa = x - xa) &
-(0::nat) - (0::nat) = (0::nat) &
-(ALL x::nat. (0::nat) - NUMERAL_BIT0 x = (0::nat)) &
-(ALL x::nat. (0::nat) - NUMERAL_BIT1 x = (0::nat)) &
-(ALL x::nat. NUMERAL_BIT0 x - (0::nat) = NUMERAL_BIT0 x) &
-(ALL x::nat. NUMERAL_BIT1 x - (0::nat) = NUMERAL_BIT1 x) &
-(ALL (m::nat) n::nat.
- NUMERAL_BIT0 m - NUMERAL_BIT0 n = NUMERAL_BIT0 (m - n)) &
-(ALL (m::nat) n::nat.
- NUMERAL_BIT0 m - NUMERAL_BIT1 n = Pred (NUMERAL_BIT0 (m - n))) &
-(ALL (m::nat) n::nat.
- NUMERAL_BIT1 m - NUMERAL_BIT0 n =
- COND (<= n m) (NUMERAL_BIT1 (m - n)) (0::nat)) &
-(ALL (m::nat) n::nat.
- NUMERAL_BIT1 m - NUMERAL_BIT1 n = NUMERAL_BIT0 (m - n))"
+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)"
@@ -1282,7 +1531,7 @@
(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::nat) = r1) &
+(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) =
@@ -1318,24 +1567,24 @@
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::nat))) &
+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::nat) = r1 &
-pwr x (NUMERAL_BIT1 (0::nat)) = x &
+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::nat) + x = 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::nat) * x = x) &
-(ALL x::nat. (0::nat) * x = (0::nat)) &
+(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::nat) = NUMERAL_BIT1 (0::nat)) &
+(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)
@@ -1343,7 +1592,7 @@
(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::nat) * x = (0::nat)) &
+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))"
@@ -1360,13 +1609,13 @@
NUMPAIR :: "nat => nat => nat"
"NUMPAIR ==
%(u::nat) ua::nat.
- EXP (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) u *
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * ua + NUMERAL_BIT1 (0::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::nat))) u *
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * ua + NUMERAL_BIT1 (0::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.
@@ -1405,13 +1654,13 @@
NUMSUM :: "bool => nat => nat"
"NUMSUM ==
%(u::bool) ua::nat.
- COND u (Suc (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * ua))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * ua)"
+ 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::nat)) * ua))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * ua))"
+ 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.
@@ -1443,7 +1692,7 @@
by (import hollight DEF_NUMRIGHT)
constdefs
- INJN :: "nat => nat => 'A::type => bool"
+ 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)"
@@ -1462,7 +1711,7 @@
by (import hollight INJN_INJ)
constdefs
- INJA :: "'A::type => nat => 'A::type => bool"
+ 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)"
@@ -1472,7 +1721,7 @@
by (import hollight INJA_INJ)
constdefs
- INJF :: "(nat => nat => 'A::type => bool) => nat => 'A::type => bool"
+ 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 =
@@ -1484,8 +1733,7 @@
by (import hollight INJF_INJ)
constdefs
- INJP :: "(nat => 'A::type => bool)
-=> (nat => 'A::type => bool) => nat => 'A::type => bool"
+ 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)"
@@ -1501,8 +1749,7 @@
by (import hollight INJP_INJ)
constdefs
- ZCONSTR :: "nat
-=> 'A::type => (nat => nat => 'A::type => bool) => nat => 'A::type => bool"
+ 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))"
@@ -1513,10 +1760,10 @@
by (import hollight DEF_ZCONSTR)
constdefs
- ZBOT :: "nat => 'A::type => bool"
- "ZBOT == INJP (INJN (0::nat)) (SOME z::nat => 'A::type => bool. True)"
-
-lemma DEF_ZBOT: "ZBOT = INJP (INJN (0::nat)) (SOME z::nat => 'A::type => bool. True)"
+ 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.
@@ -1524,7 +1771,7 @@
by (import hollight ZCONSTR_ZBOT)
constdefs
- ZRECSPACE :: "(nat => 'A::type => bool) => bool"
+ ZRECSPACE :: "(nat => 'A => bool) => bool"
"ZRECSPACE ==
%a::nat => 'A::type => bool.
ALL ZRECSPACE'::(nat => 'A::type => bool) => bool.
@@ -1559,29 +1806,74 @@
"_mk_rec" :: _ ("'_mk'_rec")
lemmas "TYDEF_recspace_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type recspace" and r=r ,
+ [where a="a :: 'A recspace" and r=r ,
OF type_definition_recspace]
constdefs
- BOTTOM :: "'A::type recspace"
- "BOTTOM == _mk_rec ZBOT"
-
-lemma DEF_BOTTOM: "BOTTOM = _mk_rec ZBOT"
+ 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::type => (nat => 'A::type recspace) => 'A::type recspace"
- "CONSTR ==
-%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
- _mk_rec (ZCONSTR u ua (%n::nat. _dest_rec (ub n)))"
-
-lemma DEF_CONSTR: "CONSTR =
-(%(u::nat) (ua::'A::type) ub::nat => 'A::type recspace.
- _mk_rec (ZCONSTR u ua (%n::nat. _dest_rec (ub n))))"
+ 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 (x::nat => 'A::type => bool) y::nat => 'A::type => bool.
- _mk_rec x = _mk_rec y --> ZRECSPACE x & ZRECSPACE y --> x = y"
+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.
@@ -1609,24 +1901,24 @@
by (import hollight CONSTR_REC)
constdefs
- FCONS :: "'A::type => (nat => 'A::type) => nat => 'A::type"
+ 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::nat) = a) &
+ (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::nat) = a) &
+ (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::nat)) (f o Suc)"
+lemma FCONS_UNDO: "ALL f::nat => 'A::type. f = FCONS (f 0) (f o Suc)"
by (import hollight FCONS_UNDO)
constdefs
- FNIL :: "nat => 'A::type"
+ FNIL :: "nat => 'A"
"FNIL == %u::nat. SOME x::'A::type. True"
lemma DEF_FNIL: "FNIL = (%u::nat. SOME x::'A::type. True)"
@@ -1695,35 +1987,77 @@
"_mk_sum" :: _ ("'_mk'_sum")
lemmas "TYDEF_sum_@intern" = typedef_hol2hollight
- [where a="a :: ('A::type, 'B::type) sum" and r=r ,
+ [where a="a :: ('A, 'B) sum" and r=r ,
OF type_definition_sum]
constdefs
- INL :: "'A::type => ('A::type, 'B::type) sum"
- "INL ==
-%a::'A::type.
- _mk_sum (CONSTR (0::nat) (a, SOME v::'B::type. True) (%n::nat. BOTTOM))"
-
-lemma DEF_INL: "INL =
-(%a::'A::type.
- _mk_sum (CONSTR (0::nat) (a, SOME v::'B::type. True) (%n::nat. BOTTOM)))"
+ 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::type => ('A::type, 'B::type) sum"
- "INR ==
-%a::'B::type.
- _mk_sum
- (CONSTR (Suc (0::nat)) (SOME v::'A::type. True, a) (%n::nat. BOTTOM))"
-
-lemma DEF_INR: "INR =
-(%a::'B::type.
- _mk_sum
- (CONSTR (Suc (0::nat)) (SOME v::'A::type. True, a) (%n::nat. BOTTOM)))"
+ 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::type, 'B::type) sum => 'A::type"
+ OUTL :: "('A, 'B) sum => 'A"
defs
OUTL_def: "hollight.OUTL ==
@@ -1736,7 +2070,7 @@
by (import hollight DEF_OUTL)
consts
- OUTR :: "('A::type, 'B::type) sum => 'B::type"
+ OUTR :: "('A, 'B) sum => 'B"
defs
OUTR_def: "hollight.OUTR ==
@@ -1790,26 +2124,55 @@
"_mk_option" :: _ ("'_mk'_option")
lemmas "TYDEF_option_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type hollight.option" and r=r ,
+ [where a="a :: 'A hollight.option" and r=r ,
OF type_definition_option]
constdefs
- NONE :: "'A::type hollight.option"
- "NONE ==
-_mk_option (CONSTR (0::nat) (SOME v::'A::type. True) (%n::nat. BOTTOM))"
-
-lemma DEF_NONE: "NONE =
-_mk_option (CONSTR (0::nat) (SOME v::'A::type. True) (%n::nat. BOTTOM))"
+ 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::type => 'A::type hollight.option" ("SOME")
+ SOME :: "'A => 'A hollight.option" ("SOME")
defs
- SOME_def: "SOME == %a::'A::type. _mk_option (CONSTR (Suc (0::nat)) a (%n::nat. BOTTOM))"
-
-lemma DEF_SOME: "SOME =
-(%a::'A::type. _mk_option (CONSTR (Suc (0::nat)) a (%n::nat. BOTTOM)))"
+ 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)
@@ -1859,27 +2222,61 @@
"_mk_list" :: _ ("'_mk'_list")
lemmas "TYDEF_list_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type hollight.list" and r=r ,
+ [where a="a :: 'A hollight.list" and r=r ,
OF type_definition_list]
constdefs
- NIL :: "'A::type hollight.list"
- "NIL == _mk_list (CONSTR (0::nat) (SOME v::'A::type. True) (%n::nat. BOTTOM))"
-
-lemma DEF_NIL: "NIL = _mk_list (CONSTR (0::nat) (SOME v::'A::type. True) (%n::nat. BOTTOM))"
+ 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::type => 'A::type hollight.list => 'A::type hollight.list"
- "CONS ==
-%(a0::'A::type) a1::'A::type hollight.list.
- _mk_list
- (CONSTR (Suc (0::nat)) a0 (FCONS (_dest_list a1) (%n::nat. BOTTOM)))"
-
-lemma DEF_CONS: "CONS =
-(%(a0::'A::type) a1::'A::type hollight.list.
- _mk_list
- (CONSTR (Suc (0::nat)) a0 (FCONS (_dest_list a1) (%n::nat. BOTTOM))))"
+ 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.
@@ -1888,12 +2285,11 @@
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::nat) = e & (ALL n::nat. fn (Suc n) = f n (fn n))"
+ 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::type => 'B::type) => ('B::type => 'A::type) => bool"
+ 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)"
@@ -1922,14 +2318,12 @@
typedef (open) N_2 = "{a::bool recspace.
ALL u::bool recspace => bool.
(ALL a::bool recspace.
- a =
- CONSTR (NUMERAL (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM) |
+ a = CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM) |
a =
- CONSTR (Suc (NUMERAL (0::nat))) (SOME x::bool. True)
- (%n::nat. BOTTOM) -->
+ 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::nat)) (SOME x::bool. True) (%n::nat. BOTTOM)"])
+ apply (rule light_ex_imp_nonempty[where t="CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM)"])
by (import hollight TYDEF_2)
syntax
@@ -1946,20 +2340,36 @@
"_10288" :: "N_2" ("'_10288")
defs
- "_10288_def": "_10288 == _mk_2 (CONSTR (0::nat) (SOME x::bool. True) (%n::nat. BOTTOM))"
-
-lemma DEF__10288: "_10288 = _mk_2 (CONSTR (0::nat) (SOME x::bool. True) (%n::nat. BOTTOM))"
+ "_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": "_10289 ==
-_mk_2 (CONSTR (Suc (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM))"
-
-lemma DEF__10289: "_10289 =
-_mk_2 (CONSTR (Suc (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM))"
+ "_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
@@ -1979,17 +2389,15 @@
typedef (open) N_3 = "{a::bool recspace.
ALL u::bool recspace => bool.
(ALL a::bool recspace.
- a =
- CONSTR (NUMERAL (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM) |
+ a = CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM) |
a =
- CONSTR (Suc (NUMERAL (0::nat))) (SOME x::bool. True)
- (%n::nat. BOTTOM) |
+ CONSTR (Suc (NUMERAL 0)) (SOME x::bool. True) (%n::nat. BOTTOM) |
a =
- CONSTR (Suc (Suc (NUMERAL (0::nat)))) (SOME x::bool. True)
+ 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::nat)) (SOME x::bool. True) (%n::nat. BOTTOM)"])
+ apply (rule light_ex_imp_nonempty[where t="CONSTR (NUMERAL 0) (SOME x::bool. True) (%n::nat. BOTTOM)"])
by (import hollight TYDEF_3)
syntax
@@ -2006,31 +2414,55 @@
"_10312" :: "N_3" ("'_10312")
defs
- "_10312_def": "_10312 == _mk_3 (CONSTR (0::nat) (SOME x::bool. True) (%n::nat. BOTTOM))"
-
-lemma DEF__10312: "_10312 = _mk_3 (CONSTR (0::nat) (SOME x::bool. True) (%n::nat. BOTTOM))"
+ "_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": "_10313 ==
-_mk_3 (CONSTR (Suc (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM))"
-
-lemma DEF__10313: "_10313 =
-_mk_3 (CONSTR (Suc (0::nat)) (SOME x::bool. True) (%n::nat. BOTTOM))"
+ "_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": "_10314 ==
-_mk_3 (CONSTR (Suc (Suc (0::nat))) (SOME x::bool. True) (%n::nat. BOTTOM))"
-
-lemma DEF__10314: "_10314 =
-_mk_3 (CONSTR (Suc (Suc (0::nat))) (SOME x::bool. True) (%n::nat. BOTTOM))"
+ "_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
@@ -2062,7 +2494,7 @@
by (import hollight list_INDUCT)
constdefs
- HD :: "'A::type hollight.list => 'A::type"
+ 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"
@@ -2073,7 +2505,7 @@
by (import hollight DEF_HD)
constdefs
- TL :: "'A::type hollight.list => 'A::type hollight.list"
+ 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"
@@ -2084,7 +2516,7 @@
by (import hollight DEF_TL)
constdefs
- APPEND :: "'A::type hollight.list => 'A::type hollight.list => 'A::type hollight.list"
+ 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.
@@ -2102,7 +2534,7 @@
by (import hollight DEF_APPEND)
constdefs
- REVERSE :: "'A::type hollight.list => 'A::type hollight.list"
+ REVERSE :: "'A hollight.list => 'A hollight.list"
"REVERSE ==
SOME REVERSE::'A::type hollight.list => 'A::type hollight.list.
REVERSE NIL = NIL &
@@ -2117,22 +2549,22 @@
by (import hollight DEF_REVERSE)
constdefs
- LENGTH :: "'A::type hollight.list => nat"
+ LENGTH :: "'A hollight.list => nat"
"LENGTH ==
SOME LENGTH::'A::type hollight.list => nat.
- LENGTH NIL = (0::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::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::type => 'B::type) => 'A::type hollight.list => 'B::type hollight.list"
+ MAP :: "('A => 'B) => 'A hollight.list => 'B hollight.list"
"MAP ==
SOME MAP::('A::type => 'B::type)
=> 'A::type hollight.list => 'B::type hollight.list.
@@ -2149,7 +2581,7 @@
by (import hollight DEF_MAP)
constdefs
- LAST :: "'A::type hollight.list => 'A::type"
+ LAST :: "'A hollight.list => 'A"
"LAST ==
SOME LAST::'A::type hollight.list => 'A::type.
ALL (h::'A::type) t::'A::type hollight.list.
@@ -2162,22 +2594,22 @@
by (import hollight DEF_LAST)
constdefs
- REPLICATE :: "nat => 'q_16809::type => 'q_16809::type hollight.list"
+ 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::nat) x = NIL) &
+ (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::nat) x = NIL) &
+ (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::type hollight.list => bool"
+ NULL :: "'q_16824 hollight.list => bool"
"NULL ==
SOME NULL::'q_16824::type hollight.list => bool.
NULL NIL = True &
@@ -2192,7 +2624,7 @@
by (import hollight DEF_NULL)
constdefs
- ALL_list :: "('q_16844::type => bool) => 'q_16844::type hollight.list => bool"
+ 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) &
@@ -2207,7 +2639,7 @@
by (import hollight DEF_ALL)
consts
- EX :: "('q_16865::type => bool) => 'q_16865::type hollight.list => bool" ("EX")
+ EX :: "('q_16865 => bool) => 'q_16865 hollight.list => bool" ("EX")
defs
EX_def: "EX ==
@@ -2224,8 +2656,8 @@
by (import hollight DEF_EX)
constdefs
- ITLIST :: "('q_16888::type => 'q_16887::type => 'q_16887::type)
-=> 'q_16888::type hollight.list => 'q_16887::type => 'q_16887::type"
+ 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
@@ -2250,7 +2682,7 @@
by (import hollight DEF_ITLIST)
constdefs
- MEM :: "'q_16913::type => 'q_16913::type hollight.list => bool"
+ 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) &
@@ -2267,8 +2699,8 @@
by (import hollight DEF_MEM)
constdefs
- ALL2 :: "('q_16946::type => 'q_16953::type => bool)
-=> 'q_16946::type hollight.list => 'q_16953::type hollight.list => bool"
+ 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
@@ -2301,9 +2733,9 @@
by (import hollight ALL2)
constdefs
- 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"
+ 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
@@ -2336,23 +2768,22 @@
by (import hollight MAP2)
constdefs
- EL :: "nat => 'q_17106::type hollight.list => 'q_17106::type"
+ 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::nat) l = HD l) &
+ (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::nat) l = HD l) &
+ (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::type => bool)
-=> 'q_17131::type hollight.list => 'q_17131::type hollight.list"
+ FILTER :: "('q_17131 => bool) => 'q_17131 hollight.list => 'q_17131 hollight.list"
"FILTER ==
SOME FILTER::('q_17131::type => bool)
=> 'q_17131::type hollight.list
@@ -2374,8 +2805,7 @@
by (import hollight DEF_FILTER)
constdefs
- ASSOC :: "'q_17160::type
-=> ('q_17160::type * 'q_17154::type) hollight.list => 'q_17154::type"
+ 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
@@ -2394,9 +2824,8 @@
by (import hollight DEF_ASSOC)
constdefs
- 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"
+ 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)
@@ -2443,9 +2872,8 @@
by (import hollight ITLIST2)
consts
- ZIP :: "'q_17256::type hollight.list
-=> 'q_17264::type hollight.list
- => ('q_17256::type * 'q_17264::type) hollight.list"
+ ZIP :: "'q_17256 hollight.list
+=> 'q_17264 hollight.list => ('q_17256 * 'q_17264) hollight.list"
defs
ZIP_def: "hollight.ZIP ==
@@ -2544,7 +2972,7 @@
LENGTH (MAP f l) = LENGTH l"
by (import hollight LENGTH_MAP)
-lemma LENGTH_EQ_NIL: "ALL l::'A::type hollight.list. (LENGTH l = (0::nat)) = (l = NIL)"
+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.
@@ -2748,13 +3176,13 @@
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::nat)"
+lemma DIST_REFL: "ALL x::nat. dist (x, x) = 0"
by (import hollight DIST_REFL)
-lemma DIST_LZERO: "ALL x::nat. dist (0::nat, x) = x"
+lemma DIST_LZERO: "ALL x::nat. dist (0, x) = x"
by (import hollight DIST_LZERO)
-lemma DIST_RZERO: "ALL x::nat. dist (x, 0::nat) = x"
+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)"
@@ -2778,7 +3206,7 @@
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::nat)) = (x = xa)"
+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)) =
@@ -2815,7 +3243,7 @@
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::nat))"
+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.
@@ -2824,8 +3252,7 @@
by (import hollight BOUNDS_DIVIDED)
lemma BOUNDS_NOTZERO: "ALL (P::nat => nat => nat) (A::nat) B::nat.
- P (0::nat) (0::nat) = (0::nat) &
- (ALL (m::nat) n::nat. <= (P m n) (A * (m + n) + B)) -->
+ 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)
@@ -2847,11 +3274,11 @@
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::nat)"
+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::nat)"])
+ apply (rule light_ex_imp_nonempty[where t="%n::nat. NUMERAL 0"])
by (import hollight TYDEF_nadd)
syntax
@@ -2978,8 +3405,7 @@
lemma NADD_LE_TOTAL_LEMMA: "ALL (x::nadd) y::nadd.
~ nadd_le x y -->
- (ALL B::nat.
- EX n::nat. n ~= (0::nat) & < (dest_nadd y n + B) (dest_nadd x n))"
+ (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"
@@ -3015,7 +3441,7 @@
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::nat)) x) x"
+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.
@@ -3054,7 +3480,7 @@
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::nat))) x) x"
+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.
@@ -3075,7 +3501,7 @@
(nadd_of_num (x * xa))"
by (import hollight NADD_OF_NUM_MUL)
-lemma NADD_LE_0: "All (nadd_le (nadd_of_num (0::nat)))"
+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"
@@ -3103,13 +3529,13 @@
by (import hollight NADD_RDISTRIB)
lemma NADD_ARCH_MULT: "ALL (x::nadd) k::nat.
- ~ nadd_eq x (nadd_of_num (0::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::nat))"
+ nadd_eq x (nadd_of_num 0)"
by (import hollight NADD_ARCH_ZERO)
lemma NADD_ARCH_LEMMA: "ALL (x::nadd) (y::nadd) z::nadd.
@@ -3131,12 +3557,12 @@
by (import hollight NADD_UBOUND)
lemma NADD_NONZERO: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
- (EX N::nat. ALL n::nat. <= N n --> dest_nadd x n ~= (0::nat))"
+ ~ 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::nat)) -->
+ ~ 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)
@@ -3148,17 +3574,17 @@
by (import hollight DEF_nadd_rinv)
lemma NADD_MUL_LINV_LEMMA0: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ 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::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::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX N::nat.
ALL n::nat.
<= N n -->
@@ -3166,7 +3592,7 @@
by (import hollight NADD_MUL_LINV_LEMMA2)
lemma NADD_MUL_LINV_LEMMA3: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX N::nat.
ALL (m::nat) n::nat.
<= N n -->
@@ -3177,7 +3603,7 @@
by (import hollight NADD_MUL_LINV_LEMMA3)
lemma NADD_MUL_LINV_LEMMA4: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX N::nat.
ALL (m::nat) n::nat.
<= N m & <= N n -->
@@ -3188,7 +3614,7 @@
by (import hollight NADD_MUL_LINV_LEMMA4)
lemma NADD_MUL_LINV_LEMMA5: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX (B::nat) N::nat.
ALL (m::nat) n::nat.
<= N m & <= N n -->
@@ -3198,7 +3624,7 @@
by (import hollight NADD_MUL_LINV_LEMMA5)
lemma NADD_MUL_LINV_LEMMA6: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX (B::nat) N::nat.
ALL (m::nat) n::nat.
<= N m & <= N n -->
@@ -3207,7 +3633,7 @@
by (import hollight NADD_MUL_LINV_LEMMA6)
lemma NADD_MUL_LINV_LEMMA7: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(EX (B::nat) N::nat.
ALL (m::nat) n::nat.
<= N m & <= N n -->
@@ -3215,7 +3641,7 @@
by (import hollight NADD_MUL_LINV_LEMMA7)
lemma NADD_MUL_LINV_LEMMA7a: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ nadd_eq x (nadd_of_num 0) -->
(ALL N::nat.
EX (A::nat) B::nat.
ALL (m::nat) n::nat.
@@ -3224,7 +3650,7 @@
by (import hollight NADD_MUL_LINV_LEMMA7a)
lemma NADD_MUL_LINV_LEMMA8: "ALL x::nadd.
- ~ nadd_eq x (nadd_of_num (0::nat)) -->
+ ~ 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)))"
@@ -3234,26 +3660,25 @@
nadd_inv :: "nadd => nadd"
"nadd_inv ==
%u::nadd.
- COND (nadd_eq u (nadd_of_num (0::nat))) (nadd_of_num (0::nat))
- (mk_nadd (nadd_rinv u))"
+ 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::nat))) (nadd_of_num (0::nat))
+ 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::nat))) (%n::nat. 0::nat) (nadd_rinv 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::nat)) -->
- nadd_eq (nadd_mul (nadd_inv x) x) (nadd_of_num (NUMERAL_BIT1 (0::nat)))"
+ ~ 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::nat))) (nadd_of_num (0::nat))"
+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)"
@@ -3361,7 +3786,7 @@
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::nat)) = x"
+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.
@@ -3369,10 +3794,10 @@
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::nat)) m = hreal_of_num (0::nat)"
+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::nat)) = hreal_of_num (0::nat)"
+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 &
@@ -3390,9 +3815,9 @@
constdefs
treal_of_num :: "nat => hreal * hreal"
- "treal_of_num == %u::nat. (hreal_of_num u, hreal_of_num (0::nat))"
-
-lemma DEF_treal_of_num: "treal_of_num = (%u::nat. (hreal_of_num u, hreal_of_num (0::nat)))"
+ "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
@@ -3441,20 +3866,20 @@
treal_inv :: "hreal * hreal => hreal * hreal"
"treal_inv ==
%u::hreal * hreal.
- COND (fst u = snd u) (hreal_of_num (0::nat), hreal_of_num (0::nat))
+ 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::nat))
- (hreal_of_num (0::nat),
+ 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::nat), hreal_of_num (0::nat))
+ 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::nat))
- (hreal_of_num (0::nat),
+ 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)
@@ -3512,11 +3937,10 @@
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::nat)) x) x"
+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::nat))"
+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.
@@ -3528,7 +3952,7 @@
by (import hollight TREAL_MUL_ASSOC)
lemma TREAL_MUL_LID: "ALL x::hreal * hreal.
- treal_eq (treal_mul (treal_of_num (NUMERAL_BIT1 (0::nat))) x) x"
+ 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.
@@ -3555,18 +3979,16 @@
by (import hollight TREAL_LE_LADD_IMP)
lemma TREAL_LE_MUL: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_le (treal_of_num (0::nat)) x &
- treal_le (treal_of_num (0::nat)) y -->
- treal_le (treal_of_num (0::nat)) (treal_mul x y)"
+ 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::nat))) (treal_of_num (0::nat))"
+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::nat)) -->
- treal_eq (treal_mul (treal_inv x) x)
- (treal_of_num (NUMERAL_BIT1 (0::nat)))"
+ ~ 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)"
@@ -3749,25 +4171,23 @@
constdefs
real_abs :: "hollight.real => hollight.real"
"real_abs ==
-%u::hollight.real. COND (real_le (real_of_num (0::nat)) u) u (real_neg u)"
+%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::nat)) u) u (real_neg u))"
+(%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::nat) = real_of_num (NUMERAL_BIT1 (0::nat))) &
+ (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::nat) = real_of_num (NUMERAL_BIT1 (0::nat))) &
+ (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)
@@ -3795,20 +4215,19 @@
lemma REAL_HREAL_LEMMA1: "EX x::hreal => hollight.real.
(ALL xa::hollight.real.
- real_le (real_of_num (0::nat)) xa = (EX y::hreal. xa = x y)) &
+ 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::nat)) xa --> r (x xa) = xa) &
- (ALL x::hreal. real_le (real_of_num (0::nat)) (r x)) &
+ (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::nat)) x) &
+ (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) &
@@ -3830,7 +4249,7 @@
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::nat)"
+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.
@@ -3852,11 +4271,11 @@
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::nat)) = x"
+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::nat)) (real_add x y)"
+ 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.
@@ -3864,7 +4283,7 @@
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::nat))"
+ 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)"
@@ -3896,8 +4315,8 @@
by (import hollight REAL_LT_TRANS)
lemma REAL_LE_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_of_num (0::nat)) x & real_le (real_of_num (0::nat)) y -->
- real_le (real_of_num (0::nat)) (real_add x y)"
+ 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)"
@@ -3907,46 +4326,40 @@
by (import hollight REAL_LT_REFL)
lemma REAL_LET_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_le (real_of_num (0::nat)) x & real_lt (real_of_num (0::nat)) y -->
- real_lt (real_of_num (0::nat)) (real_add x y)"
+ 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::nat)) =
- (x = real_of_num (0::nat) | y = real_of_num (0::nat))"
+ (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::nat))) = real_mul x x"
+ 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::nat)) x = 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::nat))) x = x) &
-(ALL x::hollight.real.
- real_mul (real_of_num (0::nat)) x = real_of_num (0::nat)) &
+(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::nat) = real_of_num (NUMERAL_BIT1 (0::nat))) &
+(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::nat)))) x) &
+ 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::nat)))) 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"
@@ -3961,7 +4374,7 @@
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::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)
@@ -3983,16 +4396,16 @@
by (import hollight REAL_LT_LADD_IMP)
lemma REAL_LT_MUL: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num (0::nat)) x & real_lt (real_of_num (0::nat)) y -->
- real_lt (real_of_num (0::nat)) (real_mul x y)"
+ 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::nat))"
+ (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::nat))"
+ (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.
@@ -4009,7 +4422,7 @@
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::nat)) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+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)
@@ -4018,34 +4431,33 @@
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::nat)) (real_add x xa)"
+ 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::nat))"
+ 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::nat)) = (x = real_of_num (0::nat))"
+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::nat)) y"
+ 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::nat)) x"
+ 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::nat)) y"
+ 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::nat)) x"
+ 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)
@@ -4086,15 +4498,13 @@
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::nat)) = (x = real_of_num (0::nat))"
+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::nat)) = real_of_num (0::nat)"
+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::nat))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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.
@@ -4111,7 +4521,7 @@
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::nat)) (real_abs x)"
+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.
@@ -4119,7 +4529,7 @@
by (import hollight REAL_ABS_SUB)
lemma REAL_ABS_NZ: "ALL x::hollight.real.
- (x ~= real_of_num (0::nat)) = real_lt (real_of_num (0::nat)) (real_abs x)"
+ (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"
@@ -4128,11 +4538,11 @@
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::nat)) x"
+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::nat)) d &
+ (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)
@@ -4142,12 +4552,11 @@
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::nat)"
+ 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::nat) | real_lt (real_of_num (0::nat)) (real_abs x)"
+ 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.
@@ -4156,12 +4565,12 @@
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::nat)) x"
+ 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::nat))"
+ 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.
@@ -4178,11 +4587,11 @@
y::hollight.real.
real_lt x0 y0 &
real_lt
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
+ (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::nat))))
+ (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_abs (real_sub y y0)))
(real_sub y0 x0) -->
real_lt x y"
@@ -4271,13 +4680,11 @@
by (import hollight REAL_ABS_MUL)
lemma REAL_POW_LE: "ALL (x::hollight.real) n::nat.
- real_le (real_of_num (0::nat)) x -->
- real_le (real_of_num (0::nat)) (real_pow x n)"
+ 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::nat)) x -->
- real_lt (real_of_num (0::nat)) (real_pow x n)"
+ 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.
@@ -4285,89 +4692,82 @@
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::nat)) x & real_le xa xb -->
+ 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::nat)) z -->
+ 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::nat)) x & real_lt xa xb -->
+ 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::nat)) z -->
+ 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::nat) | y = z)"
+ (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::nat))"
+ (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::nat)) --> real_inv y = x"
+ 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::nat)) --> real_inv x = xa"
+ 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::nat)) = (x = real_of_num (0::nat))"
+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::nat)) x -->
- real_lt (real_of_num (0::nat)) (real_inv x)"
+ 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::nat)) (real_inv x) =
- real_lt (real_of_num (0::nat)) x"
+ 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::nat)) (real_inv x) =
- real_le (real_of_num (0::nat)) x"
+ 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::nat)) x -->
- real_le (real_of_num (0::nat)) (real_inv x)"
+ 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::nat))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat))) = x"
+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::nat) -->
- real_div x x = real_of_num (NUMERAL_BIT1 (0::nat))"
+ 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::nat) --> real_mul (real_div x xa) xa = x"
+ 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::nat) --> real_mul xa (real_div x xa) = x"
+ 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)"
@@ -4402,190 +4802,179 @@
by (import hollight REAL_POW_ADD)
lemma REAL_POW_NZ: "ALL (x::hollight.real) n::nat.
- x ~= real_of_num (0::nat) --> real_pow x n ~= real_of_num (0::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::nat) & <= m n -->
+ 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::nat)) x --> x ~= real_of_num (0::nat)"
+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::nat)) x &
- real_lt (real_mul x y) (real_mul x z) -->
+ 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::nat)) xb &
- real_lt (real_mul x xb) (real_mul xa xb) -->
+ 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::nat)) x &
- real_le (real_mul x y) (real_mul x z) -->
+ 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::nat)) xb &
- real_le (real_mul x xb) (real_mul xa xb) -->
+ 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::nat)) z -->
+ 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::nat)) z -->
+ 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::nat)) z -->
+ 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::nat)) xb -->
+ 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::nat)) xb -->
+ 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::nat)) xb -->
+ 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::nat)) xb -->
+ 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::nat)) xb -->
+ 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::nat)) xb -->
+ 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::nat)))) x =
- real_add x x"
+ 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::nat)) =
- (x = real_of_num (0::nat) & n ~= (0::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::nat)) w &
- real_le w x & real_le (real_of_num (0::nat)) y & real_le y z -->
+ 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::nat)) w &
- real_lt w x & real_le (real_of_num (0::nat)) y & real_lt y z -->
+ 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::nat)) (real_mul x x) =
- (x ~= real_of_num (0::nat))"
+ 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::nat))) x -->
- real_le (real_inv x) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ 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::nat))) x -->
- real_le (real_of_num (NUMERAL_BIT1 (0::nat))) (real_pow x n)"
+ 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::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_le (real_pow x n) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ 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::nat)) = x"
+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::nat))) n =
- real_of_num (NUMERAL_BIT1 (0::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::nat)) x & real_lt x y -->
+ 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::nat)) x & real_le x y -->
+ 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::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_le (real_of_num (NUMERAL_BIT1 (0::nat))) (real_inv x)"
+ 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::nat) & xa ~= real_of_num (0::nat) -->
+ 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::nat)) d -->
- (EX x::hollight.real. real_lt (real_of_num (0::nat)) x & real_lt x d)"
+ 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::nat)) d1 & real_lt (real_of_num (0::nat)) d2 -->
+ real_lt (real_of_num 0) d1 & real_lt (real_of_num 0) d2 -->
(EX e::hollight.real.
- real_lt (real_of_num (0::nat)) e & real_lt e d1 & real_lt e d2)"
+ 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::nat)) x & real_le x y -->
+ 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::nat))) x & <= m n -->
+ 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::nat) & real_le (real_of_num (0::nat)) x & real_lt x y -->
+ 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::nat))) x & < m n -->
+ 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)
@@ -4594,54 +4983,54 @@
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::nat) & real_mul x z = real_mul y z --> x = y"
+ 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::nat) & real_mul xb x = real_mul xb xa --> x = xa"
+ 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::nat)) x & real_lt (real_of_num (0::nat)) xa -->
- real_lt (real_of_num (0::nat)) (real_div x xa)"
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
- real_le (real_of_num (0::nat)) (real_div x xa)"
+ 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::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::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::nat))
- (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) x)"
+ 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::nat)))
- (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) n)"
+ 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::nat))) =
- real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))"
+ 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::nat))))
- (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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.
@@ -4674,39 +5063,39 @@
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::nat) &
-(y2::hollight.real) ~= real_of_num (0::nat) -->
+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::nat)) (y1::hollight.real) &
-real_lt (real_of_num (0::nat)) (y2::hollight.real) -->
+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::nat)) (y1::hollight.real) &
-real_lt (real_of_num (0::nat)) (y2::hollight.real) -->
+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::nat)) (y1::hollight.real) &
-real_lt (real_of_num (0::nat)) (y2::hollight.real) -->
+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::nat)) (y1::hollight.real) &
-real_lt (real_of_num (0::nat)) (y2::hollight.real) -->
+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)
@@ -4723,7 +5112,7 @@
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::nat))"])
+ apply (rule light_ex_imp_nonempty[where t="real_of_num (NUMERAL 0)"])
by (import hollight TYDEF_int)
syntax
@@ -4902,29 +5291,29 @@
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::nat)))) y"
+ 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::nat))))"
+ 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::nat)) i --> P i)"
+(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::nat))) =
- (int_abs x = int_of_num (NUMERAL_BIT1 (0::nat)) &
- int_abs y = int_of_num (NUMERAL_BIT1 (0::nat)))"
+ (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::nat) = int_of_num (NUMERAL_BIT1 (0::nat)) &
+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::nat)) x) x (int_neg x)"
+ 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"
@@ -4937,7 +5326,7 @@
by (import hollight INT_LT)
lemma INT_ARCH: "ALL (x::hollight.int) d::hollight.int.
- d ~= int_of_num (0::nat) --> (EX c::hollight.int. int_lt x (int_mul c d))"
+ d ~= int_of_num 0 --> (EX c::hollight.int. int_lt x (int_mul c d))"
by (import hollight INT_ARCH)
constdefs
@@ -4952,7 +5341,7 @@
by (import hollight DEF_mod_int)
constdefs
- IN :: "'A::type => ('A::type => bool) => bool"
+ 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)"
@@ -4963,14 +5352,14 @@
by (import hollight EXTENSION)
constdefs
- GSPEC :: "('A::type => bool) => 'A::type => bool"
+ 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::type => bool => 'q_36941::type => bool"
+ 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)"
@@ -4993,14 +5382,14 @@
by (import hollight IN_ELIM_THM)
constdefs
- EMPTY :: "'A::type => bool"
+ EMPTY :: "'A => bool"
"EMPTY == %x::'A::type. False"
lemma DEF_EMPTY: "EMPTY = (%x::'A::type. False)"
by (import hollight DEF_EMPTY)
constdefs
- INSERT :: "'A::type => ('A::type => bool) => 'A::type => bool"
+ 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 =
@@ -5008,7 +5397,7 @@
by (import hollight DEF_INSERT)
consts
- UNIV :: "'A::type => bool"
+ UNIV :: "'A => bool"
defs
UNIV_def: "hollight.UNIV == %x::'A::type. True"
@@ -5017,7 +5406,7 @@
by (import hollight DEF_UNIV)
consts
- UNION :: "('A::type => bool) => ('A::type => bool) => 'A::type => bool"
+ UNION :: "('A => bool) => ('A => bool) => 'A => bool"
defs
UNION_def: "hollight.UNION ==
@@ -5030,7 +5419,7 @@
by (import hollight DEF_UNION)
constdefs
- UNIONS :: "(('A::type => bool) => bool) => 'A::type => bool"
+ UNIONS :: "(('A => bool) => bool) => 'A => bool"
"UNIONS ==
%u::('A::type => bool) => bool.
GSPEC
@@ -5047,7 +5436,7 @@
by (import hollight DEF_UNIONS)
consts
- INTER :: "('A::type => bool) => ('A::type => bool) => 'A::type => bool"
+ INTER :: "('A => bool) => ('A => bool) => 'A => bool"
defs
INTER_def: "hollight.INTER ==
@@ -5060,7 +5449,7 @@
by (import hollight DEF_INTER)
constdefs
- INTERS :: "(('A::type => bool) => bool) => 'A::type => bool"
+ INTERS :: "(('A => bool) => bool) => 'A => bool"
"INTERS ==
%u::('A::type => bool) => bool.
GSPEC
@@ -5077,7 +5466,7 @@
by (import hollight DEF_INTERS)
constdefs
- DIFF :: "('A::type => bool) => ('A::type => bool) => 'A::type => bool"
+ 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)"
@@ -5093,7 +5482,7 @@
by (import hollight INSERT)
constdefs
- DELETE :: "('A::type => bool) => 'A::type => 'A::type => bool"
+ 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)"
@@ -5104,7 +5493,7 @@
by (import hollight DEF_DELETE)
constdefs
- SUBSET :: "('A::type => bool) => ('A::type => bool) => bool"
+ 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"
@@ -5115,7 +5504,7 @@
by (import hollight DEF_SUBSET)
constdefs
- PSUBSET :: "('A::type => bool) => ('A::type => bool) => bool"
+ PSUBSET :: "('A => bool) => ('A => bool) => bool"
"PSUBSET ==
%(u::'A::type => bool) ua::'A::type => bool. SUBSET u ua & u ~= ua"
@@ -5124,7 +5513,7 @@
by (import hollight DEF_PSUBSET)
constdefs
- DISJOINT :: "('A::type => bool) => ('A::type => bool) => bool"
+ DISJOINT :: "('A => bool) => ('A => bool) => bool"
"DISJOINT ==
%(u::'A::type => bool) ua::'A::type => bool. hollight.INTER u ua = EMPTY"
@@ -5133,14 +5522,14 @@
by (import hollight DEF_DISJOINT)
constdefs
- SING :: "('A::type => bool) => bool"
+ 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::type => bool) => bool"
+ FINITE :: "('A => bool) => bool"
"FINITE ==
%a::'A::type => bool.
ALL FINITE'::('A::type => bool) => bool.
@@ -5163,14 +5552,14 @@
by (import hollight DEF_FINITE)
constdefs
- INFINITE :: "('A::type => bool) => bool"
+ 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::type => 'B::type) => ('A::type => bool) => 'B::type => bool"
+ IMAGE :: "('A => 'B) => ('A => bool) => 'B => bool"
"IMAGE ==
%(u::'A::type => 'B::type) ua::'A::type => bool.
GSPEC
@@ -5185,7 +5574,7 @@
by (import hollight DEF_IMAGE)
constdefs
- INJ :: "('A::type => 'B::type) => ('A::type => bool) => ('B::type => bool) => bool"
+ 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) &
@@ -5199,7 +5588,7 @@
by (import hollight DEF_INJ)
constdefs
- SURJ :: "('A::type => 'B::type) => ('A::type => bool) => ('B::type => bool) => bool"
+ 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) &
@@ -5212,7 +5601,7 @@
by (import hollight DEF_SURJ)
constdefs
- BIJ :: "('A::type => 'B::type) => ('A::type => bool) => ('B::type => bool) => bool"
+ 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"
@@ -5223,21 +5612,21 @@
by (import hollight DEF_BIJ)
constdefs
- CHOICE :: "('A::type => bool) => 'A::type"
+ 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::type => bool) => 'A::type => bool"
+ 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::type => bool) => ('q_37575::type => bool) => bool"
+ 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.
@@ -5252,7 +5641,7 @@
by (import hollight DEF_CARD_GE)
constdefs
- CARD_LE :: "('q_37587::type => bool) => ('q_37586::type => bool) => bool"
+ 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"
@@ -5261,7 +5650,7 @@
by (import hollight DEF_CARD_LE)
constdefs
- CARD_EQ :: "('q_37597::type => bool) => ('q_37598::type => bool) => bool"
+ 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"
@@ -5272,7 +5661,7 @@
by (import hollight DEF_CARD_EQ)
constdefs
- CARD_GT :: "('q_37612::type => bool) => ('q_37613::type => bool) => bool"
+ 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"
@@ -5283,7 +5672,7 @@
by (import hollight DEF_CARD_GT)
constdefs
- CARD_LT :: "('q_37628::type => bool) => ('q_37629::type => bool) => bool"
+ 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"
@@ -5294,7 +5683,7 @@
by (import hollight DEF_CARD_LT)
constdefs
- COUNTABLE :: "('q_37642::type => bool) => bool"
+ COUNTABLE :: "('q_37642 => bool) => bool"
"(op ==::(('q_37642::type => bool) => bool)
=> (('q_37642::type => bool) => bool) => prop)
(COUNTABLE::('q_37642::type => bool) => bool)
@@ -5884,9 +6273,8 @@
by (import hollight FINITE_DIFF)
constdefs
- FINREC :: "('q_41615::type => 'q_41614::type => 'q_41614::type)
-=> 'q_41614::type
- => ('q_41615::type => bool) => 'q_41614::type => nat => bool"
+ 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
@@ -5894,7 +6282,7 @@
=> '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::nat) = (s = EMPTY & a = b)) &
+ 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.
@@ -5909,7 +6297,7 @@
=> '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::nat) = (s = EMPTY & a = b)) &
+ 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.
@@ -5920,7 +6308,7 @@
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::nat)) =
+ 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)
@@ -5973,8 +6361,8 @@
by (import hollight SET_RECURSION_LEMMA)
constdefs
- ITSET :: "('q_42316::type => 'q_42315::type => 'q_42315::type)
-=> ('q_42316::type => bool) => 'q_42315::type => 'q_42315::type"
+ 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.
@@ -6045,12 +6433,10 @@
by (import hollight FINITE_RESTRICT)
constdefs
- CARD :: "('q_42709::type => bool) => nat"
- "CARD ==
-%u::'q_42709::type => bool. ITSET (%x::'q_42709::type. Suc) u (0::nat)"
-
-lemma DEF_CARD: "CARD =
-(%u::'q_42709::type => bool. ITSET (%x::'q_42709::type. Suc) u (0::nat))"
+ 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)
@@ -6081,8 +6467,7 @@
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::nat)) (CARD 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)
@@ -6092,7 +6477,7 @@
by (import hollight CARD_UNION_EQ)
constdefs
- HAS_SIZE :: "('q_42990::type => bool) => nat => bool"
+ 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)"
@@ -6101,8 +6486,7 @@
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::nat) = (s = EMPTY)"
+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.
@@ -6130,7 +6514,7 @@
(xb * xc)"
by (import hollight HAS_SIZE_UNIONS)
-lemma HAS_SIZE_CLAUSES: "HAS_SIZE (s::'q_43395::type => bool) (0::nat) = (s = EMPTY) &
+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)"
@@ -6148,8 +6532,7 @@
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::nat)) = (s = EMPTY)"
+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.
@@ -6293,7 +6676,7 @@
(GSPEC
(%u::'A::type => bool.
EX t::'A::type => bool. SETSPEC u (SUBSET t s) t))
- (EXP (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) n)"
+ (EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) n)"
by (import hollight HAS_SIZE_POWERSET)
lemma CARD_POWERSET: "ALL s::'A::type => bool.
@@ -6302,7 +6685,7 @@
(GSPEC
(%u::'A::type => bool.
EX t::'A::type => bool. SETSPEC u (SUBSET t s) t)) =
- EXP (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) (CARD s)"
+ EXP (NUMERAL_BIT0 (NUMERAL_BIT1 0)) (CARD s)"
by (import hollight CARD_POWERSET)
lemma FINITE_POWERSET: "ALL s::'A::type => bool.
@@ -6336,7 +6719,7 @@
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::nat))"
+ (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))"
@@ -6344,7 +6727,7 @@
lemma CARD_NUMSEG_LE: "ALL x::nat.
CARD (GSPEC (%u::nat. EX m::nat. SETSPEC u (<= m x) m)) =
- x + NUMERAL_BIT1 (0::nat)"
+ 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)"
@@ -6364,7 +6747,7 @@
by (import hollight HAS_SIZE_INDEX)
constdefs
- set_of_list :: "'q_45759::type hollight.list => 'q_45759::type => bool"
+ 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 &
@@ -6379,7 +6762,7 @@
by (import hollight DEF_set_of_list)
constdefs
- list_of_set :: "('q_45777::type => bool) => 'q_45777::type hollight.list"
+ 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.
@@ -6419,8 +6802,7 @@
by (import hollight SET_OF_LIST_APPEND)
constdefs
- pairwise :: "('q_45989::type => 'q_45989::type => bool)
-=> ('q_45989::type => bool) => bool"
+ 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.
@@ -6433,8 +6815,7 @@
by (import hollight DEF_pairwise)
constdefs
- PAIRWISE :: "('q_46011::type => 'q_46011::type => bool)
-=> 'q_46011::type hollight.list => bool"
+ 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.
@@ -6477,7 +6858,7 @@
mk_finite_image :: _
lemmas "TYDEF_finite_image_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type finite_image" and r=r ,
+ [where a="a :: 'A finite_image" and r=r ,
OF type_definition_finite_image]
lemma FINITE_IMAGE_IMAGE: "(op =::('A::type finite_image => bool)
@@ -6497,7 +6878,7 @@
by (import hollight FINITE_IMAGE_IMAGE)
constdefs
- dimindex :: "('A::type => bool) => nat"
+ dimindex :: "('A => bool) => nat"
"(op ==::(('A::type => bool) => nat) => (('A::type => bool) => nat) => prop)
(dimindex::('A::type => bool) => nat)
(%u::'A::type => bool.
@@ -6536,10 +6917,10 @@
(hollight.UNIV::'A::type finite_image => bool)"
by (import hollight FINITE_FINITE_IMAGE)
-lemma DIMINDEX_NONZERO: "ALL s::'A::type => bool. dimindex s ~= (0::nat)"
+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::nat)) (dimindex x)"
+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.
@@ -6547,7 +6928,7 @@
by (import hollight DIMINDEX_FINITE_IMAGE)
constdefs
- finite_index :: "nat => 'A::type"
+ 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)
@@ -6669,11 +7050,11 @@
mk_cart :: _
lemmas "TYDEF_cart_@intern" = typedef_hol2hollight
- [where a="a :: ('A::type, 'B::type) cart" and r=r ,
+ [where a="a :: ('A, 'B) cart" and r=r ,
OF type_definition_cart]
consts
- "$" :: "('q_46418::type, 'q_46425::type) cart => nat => 'q_46418::type" ("$")
+ "$" :: "('q_46418, 'q_46425) cart => nat => 'q_46418" ("$")
defs
"$_def": "$ ==
@@ -6709,7 +7090,7 @@
by (import hollight CART_EQ)
constdefs
- lambda :: "(nat => 'A::type) => ('A::type, 'B::type) cart"
+ 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)
@@ -6806,13 +7187,11 @@
mk_finite_sum :: _
lemmas "TYDEF_finite_sum_@intern" = typedef_hol2hollight
- [where a="a :: ('A::type, 'B::type) finite_sum" and r=r ,
+ [where a="a :: ('A, 'B) finite_sum" and r=r ,
OF type_definition_finite_sum]
constdefs
- pastecart :: "('A::type, 'M::type) cart
-=> ('A::type, 'N::type) cart
- => ('A::type, ('M::type, 'N::type) finite_sum) cart"
+ 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)
@@ -6863,8 +7242,7 @@
by (import hollight DEF_pastecart)
constdefs
- fstcart :: "('A::type, ('M::type, 'N::type) finite_sum) cart
-=> ('A::type, 'M::type) cart"
+ fstcart :: "('A, ('M, 'N) finite_sum) cart => ('A, 'M) cart"
"fstcart ==
%u::('A::type, ('M::type, 'N::type) finite_sum) cart. lambda ($ u)"
@@ -6873,8 +7251,7 @@
by (import hollight DEF_fstcart)
constdefs
- sndcart :: "('A::type, ('M::type, 'N::type) finite_sum) cart
-=> ('A::type, 'N::type) cart"
+ 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
@@ -7057,24 +7434,20 @@
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::nat)) n) =
- dotdot m n"
+ 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::nat))) (dotdot p n) =
- dotdot m n"
+ 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::nat)) xa) = dotdot x xa"
+ <= 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::nat))) = dotdot x xa"
+ <= 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.
@@ -7090,24 +7463,22 @@
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::nat)"
+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::nat) - m"
+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::nat) - x)"
+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::nat)) x) = x"
+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::nat)) x) x"
+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::nat) = COND (m = (0::nat)) (INSERT (0::nat) EMPTY) EMPTY) &
+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))"
@@ -7117,10 +7488,10 @@
FINITE s =
(EX f::nat => 'A::type.
(ALL (i::nat) j::nat.
- IN i (dotdot (NUMERAL_BIT1 (0::nat)) (CARD s)) &
- IN j (dotdot (NUMERAL_BIT1 (0::nat)) (CARD s)) & f i = f j -->
+ 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::nat)) (CARD s)))"
+ s = IMAGE f (dotdot (NUMERAL_BIT1 0) (CARD s)))"
by (import hollight FINITE_INDEX_NUMSEG)
lemma FINITE_INDEX_NUMBERS: "ALL s::'A::type => bool.
@@ -7136,10 +7507,9 @@
by (import hollight DISJOINT_NUMSEG)
lemma NUMSEG_ADD_SPLIT: "ALL (x::nat) (xa::nat) xb::nat.
- <= x (xa + NUMERAL_BIT1 (0::nat)) -->
+ <= x (xa + NUMERAL_BIT1 0) -->
dotdot x (xa + xb) =
- hollight.UNION (dotdot x xa)
- (dotdot (xa + NUMERAL_BIT1 (0::nat)) (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.
@@ -7151,7 +7521,7 @@
by (import hollight SUBSET_NUMSEG)
constdefs
- neutral :: "('q_48776::type => 'q_48776::type => 'q_48776::type) => 'q_48776::type"
+ 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"
@@ -7162,7 +7532,7 @@
by (import hollight DEF_neutral)
constdefs
- monoidal :: "('A::type => 'A::type => 'A::type) => bool"
+ 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) &
@@ -7179,8 +7549,7 @@
by (import hollight DEF_monoidal)
constdefs
- support :: "('B::type => 'B::type => 'B::type)
-=> ('A::type => 'B::type) => ('A::type => bool) => 'A::type => bool"
+ 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.
@@ -7197,8 +7566,8 @@
by (import hollight DEF_support)
constdefs
- iterate :: "('q_48881::type => 'q_48881::type => 'q_48881::type)
-=> ('A::type => bool) => ('A::type => 'q_48881::type) => 'q_48881::type"
+ 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.
@@ -7406,7 +7775,7 @@
by (import hollight ITERATE_IMAGE)
constdefs
- nsum :: "('q_50348::type => bool) => ('q_50348::type => nat) => nat"
+ 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)
@@ -7424,10 +7793,11 @@
(op +::nat => nat => nat))"
by (import hollight DEF_nsum)
-lemma NEUTRAL_ADD: "neutral op + = (0::nat)"
+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::nat)"
+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)"
@@ -7436,7 +7806,7 @@
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::nat)) &
+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 -->
@@ -7498,28 +7868,27 @@
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::nat)) --> nsum xa x = (0::nat)"
+ (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::nat) = (0::nat)"
+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::nat) (x xb)) -->
- <= (0::nat) (nsum xa x)"
+ 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::nat) (f xa)) & <= (nsum x f) b -->
+ (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::nat) (x xb)) &
- nsum xa x = (0::nat) -->
- (ALL xb::'q_51076::type. IN xb xa --> x xb = (0::nat))"
+ (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.
@@ -7527,8 +7896,7 @@
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::nat)) =
- COND (IN xa x) b (0::nat)"
+ 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)
@@ -7548,23 +7916,23 @@
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::nat)) -->
+ 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::nat)) -->
+ 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::nat)) -->
+ 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::nat)) = nsum s f"
+ 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.
@@ -7610,7 +7978,7 @@
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::nat)) -->
+ (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)
@@ -7621,7 +7989,7 @@
(%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::nat))"
+ 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)
@@ -7645,7 +8013,7 @@
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::nat))"
+ 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)
@@ -7696,8 +8064,7 @@
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::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)
@@ -7726,35 +8093,32 @@
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::nat) - xa) * x"
+ 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::nat)) -->
- nsum (dotdot m n) x = (0::nat)"
+ (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::nat)"
+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::nat) (xb p)) -->
- <= (0::nat) (nsum (dotdot x xa) xb)"
+ (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::nat) (f p)) &
- nsum (dotdot m n) f = (0::nat) -->
- (ALL p::nat. <= m p & <= p n --> f p = (0::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::nat)) (f::nat => nat) =
- COND (x = (0::nat)) (f (0::nat)) (0::nat)) &
+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))
@@ -7767,10 +8131,9 @@
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::nat)) -->
+ <= xa (xb + NUMERAL_BIT1 0) -->
nsum (dotdot xa (xb + xc)) x =
- nsum (dotdot xa xb) x +
- nsum (dotdot (xb + NUMERAL_BIT1 (0::nat)) (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.
@@ -7780,24 +8143,21 @@
lemma NSUM_OFFSET_0: "ALL (x::nat => nat) (xa::nat) xb::nat.
<= xa xb -->
- nsum (dotdot xa xb) x =
- nsum (dotdot (0::nat) (xb - xa)) (%i::nat. x (i + xa))"
+ 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::nat)) xb) x"
+ 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::nat) n & <= m n -->
- nsum (dotdot m n) f = nsum (dotdot m (n - NUMERAL_BIT1 (0::nat))) f + f n"
+ < 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::type => bool)
-=> ('q_53311::type => hollight.real) => hollight.real"
+ sum :: "('q_53311 => bool) => ('q_53311 => hollight.real) => hollight.real"
defs
sum_def: "(op ==::(('q_53311::type => bool)
@@ -7825,10 +8185,10 @@
(real_add::hollight.real => hollight.real => hollight.real))"
by (import hollight DEF_sum)
-lemma NEUTRAL_REAL_ADD: "neutral real_add = real_of_num (0::nat)"
+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::nat))"
+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"
@@ -7838,7 +8198,7 @@
by (import hollight MONOIDAL_REAL_MUL)
lemma SUM_CLAUSES: "(ALL x::'q_53353::type => hollight.real.
- hollight.sum EMPTY x = real_of_num (0::nat)) &
+ 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 -->
@@ -7876,24 +8236,22 @@
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::nat)) (x xb)) -->
- real_le (real_of_num (0::nat)) (hollight.sum xa x)"
+ (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::nat)) (f xa)) &
+ (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::nat)) (x xb)) &
- hollight.sum xa x = real_of_num (0::nat) -->
- (ALL xb::'q_54187::type. IN xb xa --> x xb = real_of_num (0::nat))"
+ (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.
@@ -7902,9 +8260,8 @@
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::nat))) =
- COND (IN xa x) b (real_of_num (0::nat))"
+ (%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)
@@ -7919,28 +8276,28 @@
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::nat)) -->
+ (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::nat)) -->
+ (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::nat)) -->
+ (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::nat))) =
+ (%x::'q_54580::type. COND (IN x s) (f x) (real_of_num 0)) =
hollight.sum s f"
by (import hollight SUM_RESTRICT)
@@ -7996,7 +8353,7 @@
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::nat)) -->
+ (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)
@@ -8007,7 +8364,7 @@
(%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::nat)))"
+ 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)
@@ -8033,7 +8390,7 @@
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::nat)))"
+ 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)
@@ -8098,10 +8455,8 @@
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::nat))) &
- (ALL x::'A::type.
- IN x (DIFF v u) --> real_le (real_of_num (0::nat)) (f x)) -->
+ (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)
@@ -8109,8 +8464,7 @@
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::nat)) (f x)) -->
+ (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)
@@ -8151,37 +8505,35 @@
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::nat) - xa)) 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::nat)) -->
- hollight.sum (dotdot m n) x = real_of_num (0::nat)"
+ (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::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::nat)) (xb p)) -->
- real_le (real_of_num (0::nat)) (hollight.sum (dotdot x xa) xb)"
+ (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::nat)) (f p)) &
- hollight.sum (dotdot m n) f = real_of_num (0::nat) -->
- (ALL p::nat. <= m p & <= p n --> f p = real_of_num (0::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::nat)) (f::nat => hollight.real) =
- COND (x = (0::nat)) (f (0::nat)) (real_of_num (0::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))
@@ -8196,28 +8548,28 @@
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::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::nat)) (xb + xc)) 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::nat) (xb - xa)) (%i::nat. x (i + xa))"
+ 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::nat)) 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::nat) n & <= m n -->
+ < 0 n & <= m n -->
hollight.sum (dotdot m n) f =
- real_add (hollight.sum (dotdot m (n - NUMERAL_BIT1 (0::nat))) f) (f n)"
+ 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.
@@ -8226,9 +8578,8 @@
by (import hollight REAL_OF_NUM_SUM_NUMSEG)
constdefs
- 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"
+ 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
@@ -8346,12 +8697,10 @@
by (import hollight CASEWISE_WORKS)
constdefs
- admissible :: "('q_57089::type => 'q_57082::type => bool)
-=> (('q_57089::type => 'q_57085::type) => 'q_57095::type => bool)
- => ('q_57095::type => 'q_57082::type)
- => (('q_57089::type => 'q_57085::type)
- => 'q_57095::type => 'q_57090::type)
- => bool"
+ 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)
@@ -8378,10 +8727,9 @@
by (import hollight DEF_admissible)
constdefs
- tailadmissible :: "('A::type => 'A::type => bool)
-=> (('A::type => 'B::type) => 'P::type => bool)
- => ('P::type => 'A::type)
- => (('A::type => 'B::type) => 'P::type => 'B::type) => bool"
+ 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)
@@ -8416,12 +8764,10 @@
by (import hollight DEF_tailadmissible)
constdefs
- superadmissible :: "('q_57239::type => 'q_57239::type => bool)
-=> (('q_57239::type => 'q_57241::type) => 'q_57247::type => bool)
- => ('q_57247::type => 'q_57239::type)
- => (('q_57239::type => 'q_57241::type)
- => 'q_57247::type => 'q_57241::type)
- => bool"
+ 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)
@@ -8695,19 +9041,19 @@
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::nat) - m = (0::nat)) &
-(ALL (m::nat) n::nat. Suc m - n = COND (< m n) (0::nat) (Suc (m - n)))"
+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::nat))) = x"
+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::nat) -->
- real_mul x (real_inv x) = real_of_num (NUMERAL_BIT1 (0::nat))"
+ 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.
@@ -8723,31 +9069,29 @@
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::nat))"
+ (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::nat))"
+ (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::nat)) = (x = real_neg y)"
+ (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::nat)) = (y = real_neg x)"
+ (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::nat)) x = real_of_num (0::nat)"
+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::nat)) = real_of_num (0::nat)"
+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.
@@ -8807,45 +9151,39 @@
by (import hollight REAL_LE_TRANS)
lemma REAL_NEG_LT0: "ALL x::hollight.real.
- real_lt (real_neg x) (real_of_num (0::nat)) =
- real_lt (real_of_num (0::nat)) x"
+ 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::nat)) (real_neg x) =
- real_lt x (real_of_num (0::nat))"
+ 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::nat)) =
- real_le (real_of_num (0::nat)) x"
+ 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::nat)) (real_neg x) =
- real_le x (real_of_num (0::nat))"
+ 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::nat) |
- real_lt (real_of_num (0::nat)) x |
- real_lt (real_of_num (0::nat)) (real_neg x)"
+ 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::nat)) x |
- real_le (real_of_num (0::nat)) (real_neg x)"
+ 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::nat)) x & real_le (real_of_num (0::nat)) y -->
- real_le (real_of_num (0::nat)) (real_mul x y)"
+ 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::nat)) (real_mul x x)"
+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::nat)) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+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.
@@ -8862,8 +9200,8 @@
by (import hollight REAL_LT_ADD2)
lemma REAL_LT_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num (0::nat)) x & real_lt (real_of_num (0::nat)) y -->
- real_lt (real_of_num (0::nat)) (real_add x y)"
+ 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.
@@ -8875,8 +9213,7 @@
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::nat))))"
+ 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"
@@ -8885,31 +9222,27 @@
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::nat)"
+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::nat)) = (x = y)"
+ (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::nat)) (real_add x x) =
- real_le (real_of_num (0::nat)) x"
+ 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::nat)) x"
+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::nat))"
+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::nat)) = (x = real_of_num (0::nat))"
+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::nat)) = real_of_num (0::nat)"
+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.
@@ -8917,19 +9250,19 @@
by (import hollight REAL_NEG_SUB)
lemma REAL_SUB_LT: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num (0::nat)) (real_sub x y) = real_lt y x"
+ 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::nat)) (real_sub x y) = real_le y x"
+ 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::nat) | y = z)"
+ (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::nat) | x = y)"
+ (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.
@@ -8944,80 +9277,73 @@
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::nat)))) x"
+ 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::nat) --> real_inv x ~= real_of_num (0::nat)"
+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::nat) --> real_inv (real_inv x) = x"
+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::nat)) x -->
- real_lt (real_of_num (0::nat)) (real_inv x)"
+ 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::nat)) x -->
- real_lt (real_of_num (0::nat)) (real_mul x y) =
- real_lt (real_of_num (0::nat)) y"
+ real_lt (real_of_num 0) x -->
+ real_lt (real_of_num 0) (real_mul x y) = real_lt (real_of_num 0) y"
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::nat)) y -->
- real_lt (real_of_num (0::nat)) (real_mul x y) =
- real_lt (real_of_num (0::nat)) x"
+ 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::nat)) x -->
+ 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::nat)) z -->
+ 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::nat)) z -->
+ 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::nat)) x -->
+ 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::nat)) --> x = real_inv y"
+ 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::nat)) --> y = real_inv x"
+ 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::nat) -->
- real_neg (real_inv x) = real_inv (real_neg x)"
+ 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::nat))) x"
+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::nat)))"
+ 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::nat)) (real_of_num n)"
+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"
@@ -9040,71 +9366,64 @@
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::nat))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat)) x = real_of_num (0::nat)"
+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::nat)) =
- real_lt (real_of_num (0::nat)) (real_of_num n)"
+ (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::nat) --> real_lt (real_of_num (0::nat)) (real_of_num n)"
+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::nat)) z -->
- real_lt (real_of_num (0::nat)) (real_div y z) =
- real_lt (real_of_num (0::nat)) y"
+ 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::nat)) z -->
+ 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::nat) -->
- real_lt (real_of_num (0::nat)) (real_div d (real_of_num n)) =
- real_lt (real_of_num (0::nat)) d"
+ 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::nat)) x -->
- real_lt xa (real_mul (real_of_num x) xa) =
- real_lt (real_of_num (0::nat)) xa"
+ < (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::nat)) n -->
- real_lt (real_div d (real_of_num n)) d = real_lt (real_of_num (0::nat)) d"
+ < (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::nat))
- (real_div d (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) =
- real_lt (real_of_num (0::nat)) d"
+ 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::nat)))))
- d =
- real_lt (real_of_num (0::nat)) d"
+ 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::nat)))) x"
+ 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::nat)))))
- (real_div x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) =
+ 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)
@@ -9140,10 +9459,10 @@
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::nat)) x = real_neg x"
+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::nat)) = x"
+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)
@@ -9152,14 +9471,14 @@
by (import hollight REAL_LTE_ADD2)
lemma REAL_LTE_ADD: "ALL (x::hollight.real) y::hollight.real.
- real_lt (real_of_num (0::nat)) x & real_le (real_of_num (0::nat)) y -->
- real_lt (real_of_num (0::nat)) (real_add x y)"
+ 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::nat)) x1 &
- real_le (real_of_num (0::nat)) y1 & real_lt x1 x2 & real_lt y1 y2 -->
+ 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)
@@ -9180,22 +9499,22 @@
by (import hollight REAL_SUB_TRIANGLE)
lemma REAL_INV_MUL_WEAK: "ALL (x::hollight.real) xa::hollight.real.
- x ~= real_of_num (0::nat) & xa ~= real_of_num (0::nat) -->
+ 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::nat)) x -->
+ 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::nat)) z -->
+ 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::nat) & y ~= real_of_num (0::nat) -->
+ 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)
@@ -9208,77 +9527,74 @@
by (import hollight REAL_MEAN)
lemma REAL_EQ_LMUL2: "ALL (x::hollight.real) (y::hollight.real) z::hollight.real.
- x ~= real_of_num (0::nat) --> (y = z) = (real_mul x y = real_mul x z)"
+ 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::nat)) x1 &
- real_le (real_of_num (0::nat)) y1 & real_le x1 x2 & real_le y1 y2 -->
+ 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::nat)) x & real_le y (real_mul z x) -->
+ 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::nat)) x & real_le (real_mul y x) z -->
+ 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::nat)) x & real_lt x y -->
- real_lt (real_div x y) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ 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::nat)) x & real_le y z -->
+ 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::nat)) x & real_le xa xb -->
+ 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::nat)) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_lt (real_of_num (NUMERAL_BIT1 (0::nat))) (real_inv x)"
+ 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::nat)) x --> x ~= real_of_num (0::nat)"
+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::nat) & real_mul x z = real_mul y z --> x = y"
+ 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::nat) & real_mul x xa = real_mul x xb --> xa = xb"
+ 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::nat)"
+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::nat)) (real_mul x x) =
- (x ~= real_of_num (0::nat))"
+ 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::nat)) =
- (x = real_of_num (0::nat) & y = real_of_num (0::nat))"
+ (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::nat) & z ~= real_of_num (0::nat) -->
+ 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)
@@ -9286,27 +9602,23 @@
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::nat)))))"
+ (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::nat)))))
+ (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::nat)) = (x = real_of_num (0::nat))"
+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::nat)) = real_of_num (0::nat)"
+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::nat))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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"
@@ -9316,7 +9628,7 @@
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::nat)) (real_abs x)"
+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.
@@ -9334,12 +9646,11 @@
by (import hollight ABS_SUB)
lemma ABS_NZ: "ALL x::hollight.real.
- (x ~= real_of_num (0::nat)) = real_lt (real_of_num (0::nat)) (real_abs x)"
+ (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::nat) -->
- real_abs (real_inv x) = real_inv (real_abs x)"
+ 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"
@@ -9348,14 +9659,14 @@
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::nat)) x"
+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::nat)) d &
+ (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)
@@ -9365,12 +9676,11 @@
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::nat)"
+ 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::nat) | real_lt (real_of_num (0::nat)) (real_abs x)"
+ 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.
@@ -9379,16 +9689,16 @@
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::nat)) x"
+ 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::nat))"
+ real_lt x (real_of_num 0)"
by (import hollight ABS_SIGN2)
lemma ABS_DIV: "ALL y::hollight.real.
- y ~= real_of_num (0::nat) -->
+ 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)
@@ -9412,10 +9722,10 @@
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::nat))))) &
+ (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::nat))))) -->
+ (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) -->
real_lt x y"
by (import hollight ABS_BETWEEN2)
@@ -9423,16 +9733,15 @@
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::nat)) (Suc n) = real_of_num (0::nat)"
+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::nat) --> real_pow c n ~= real_of_num (0::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::nat) -->
- real_inv (real_pow c x) = real_pow (real_inv c) x"
+ 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.
@@ -9440,37 +9749,35 @@
by (import hollight POW_ABS)
lemma POW_PLUS1: "ALL (e::hollight.real) x::nat.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
real_le
- (real_add (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_mul (real_of_num x) e))
- (real_pow (real_add (real_of_num (NUMERAL_BIT1 (0::nat))) e) x)"
+ (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::nat)) = x"
+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::nat))) = real_mul x x"
+ 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::nat)) x -->
- real_le (real_of_num (0::nat)) (real_pow x xa)"
+ 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::nat)) x & real_le x y -->
+ 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::nat)))) n) =
- real_of_num (NUMERAL_BIT1 (0::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.
@@ -9478,46 +9785,45 @@
by (import hollight POW_MUL)
lemma REAL_LE_SQUARE_POW: "ALL x::hollight.real.
- real_le (real_of_num (0::nat))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat)))) =
- real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))"
+ 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::nat))) x -->
- real_le (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat))) x -->
- real_lt (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat)) x -->
- real_lt (real_of_num (0::nat)) (real_pow x (Suc n))"
+ 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::nat)))
- (real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) n)"
+ 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::nat)))) 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::nat))))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n) =
- real_of_num (NUMERAL_BIT1 (0::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.
@@ -9588,22 +9894,21 @@
by (import hollight REAL_ARCH_SIMPLE)
lemma REAL_ARCH: "ALL x::hollight.real.
- real_lt (real_of_num (0::nat)) x -->
+ 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::nat)) y -->
+ real_lt (real_of_num 0) y -->
(ALL x::hollight.real.
- real_le (real_of_num (0::nat)) x -->
+ 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::nat) f = real_of_num (0::nat)) &
+ (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)
@@ -9612,20 +9917,18 @@
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::nat) f = real_of_num (0::nat)) &
+ (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::nat) f = real_of_num (0::nat)) &
+ (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::nat) (f::nat => hollight.real) = real_of_num (0::nat) &
+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)
@@ -9636,16 +9939,16 @@
by (import hollight PSUM_SUM)
lemma PSUM_SUM_NUMSEG: "ALL (f::nat => hollight.real) (m::nat) n::nat.
- ~ (m = (0::nat) & n = (0::nat)) -->
- psum (m, n) f = hollight.sum (dotdot m (m + n - NUMERAL_BIT1 (0::nat))) f"
+ ~ (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::nat, n) f) (psum (n, p) f) = psum (0::nat, n + p) f"
+ 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::nat, m + n) f) (psum (0::nat, m) f)"
+ 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.
@@ -9664,13 +9967,13 @@
by (import hollight SUM_EQ)
lemma SUM_POS: "ALL f::nat => hollight.real.
- (ALL n::nat. real_le (real_of_num (0::nat)) (f n)) -->
- (ALL (m::nat) n::nat. real_le (real_of_num (0::nat)) (psum (m, n) f))"
+ (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::nat)) (f n)) -->
- real_le (real_of_num (0::nat)) (psum (m, n) f)"
+ (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.
@@ -9684,8 +9987,8 @@
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::nat)) -->
- (ALL (m::nat) n::nat. >= m N --> psum (m, n) f = real_of_num (0::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.
@@ -9712,8 +10015,8 @@
by (import hollight SUM_SUBST)
lemma SUM_NSUB: "ALL (n::nat) (f::nat => hollight.real) c::hollight.real.
- real_sub (psum (0::nat, n) f) (real_mul (real_of_num n) c) =
- psum (0::nat, n) (%p::nat. real_sub (f p) c)"
+ 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.
@@ -9722,29 +10025,27 @@
by (import hollight SUM_BOUND)
lemma SUM_GROUP: "ALL (n::nat) (k::nat) f::nat => hollight.real.
- psum (0::nat, n) (%m::nat. psum (m * k, k) f) = psum (0::nat, n * k) f"
+ 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::nat)) f = f n"
+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::nat))) f =
- real_add (f n) (f (n + NUMERAL_BIT1 (0::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::nat, n) (%m::nat. f (m + k)) =
- real_sub (psum (0::nat, n + k) f) (psum (0::nat, k) f)"
+ 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::nat)) = real_of_num (0::nat)"
+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.
@@ -9753,14 +10054,14 @@
by (import hollight SUM_CANCEL)
lemma SUM_HORNER: "ALL (f::nat => hollight.real) (n::nat) x::hollight.real.
- psum (0::nat, Suc n) (%i::nat. real_mul (f i) (real_pow x i)) =
- real_add (f (0::nat))
+ psum (0, Suc n) (%i::nat. real_mul (f i) (real_pow x i)) =
+ real_add (f 0)
(real_mul x
- (psum (0::nat, n) (%i::nat. real_mul (f (Suc i)) (real_pow x i))))"
+ (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::nat, n) (%m::nat. c) = real_mul (real_of_num n) c"
+ 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.
@@ -9774,15 +10075,15 @@
lemma SUM_EQ_0: "(ALL r::nat.
<= (m::nat) r & < r (m + (n::nat)) -->
- (f::nat => hollight.real) r = real_of_num (0::nat)) -->
-psum (m, n) f = real_of_num (0::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::nat)) -->
+ (f::nat => hollight.real) r = real_of_num 0) -->
psum (m, p) f = psum (m, n) f"
by (import hollight SUM_MORETERMS_EQ)
@@ -9796,7 +10097,7 @@
by (import hollight SUM_DIFFERENCES_EQ)
constdefs
- re_Union :: "(('A::type => bool) => bool) => 'A::type => bool"
+ 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"
@@ -9807,7 +10108,7 @@
by (import hollight DEF_re_Union)
constdefs
- re_union :: "('A::type => bool) => ('A::type => bool) => 'A::type => bool"
+ 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"
@@ -9816,7 +10117,7 @@
by (import hollight DEF_re_union)
constdefs
- re_intersect :: "('A::type => bool) => ('A::type => bool) => 'A::type => bool"
+ 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"
@@ -9825,21 +10126,21 @@
by (import hollight DEF_re_intersect)
constdefs
- re_null :: "'A::type => bool"
+ 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::type => bool"
+ 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::type => bool) => ('A::type => bool) => bool"
+ 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"
@@ -9848,7 +10149,7 @@
by (import hollight DEF_re_subset)
constdefs
- re_compl :: "('A::type => bool) => 'A::type => bool"
+ 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)"
@@ -9869,7 +10170,7 @@
by (import hollight SUBSETA_TRANS)
constdefs
- istopology :: "(('A::type => bool) => bool) => bool"
+ istopology :: "(('A => bool) => bool) => bool"
"istopology ==
%u::('A::type => bool) => bool.
u re_null &
@@ -9901,7 +10202,7 @@
topology :: _
lemmas "TYDEF_topology_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type topology" and r=r ,
+ [where a="a :: 'A topology" and r=r ,
OF type_definition_topology]
lemma TOPOLOGY: "ALL L::'A::type topology.
@@ -9918,7 +10219,7 @@
by (import hollight TOPOLOGY_UNION)
constdefs
- neigh :: "'A::type topology => ('A::type => bool) * 'A::type => bool"
+ 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)"
@@ -9952,7 +10253,7 @@
by (import hollight OPEN_NEIGH)
constdefs
- closed :: "'A::type topology => ('A::type => bool) => bool"
+ closed :: "'A topology => ('A => bool) => bool"
"closed == %(u::'A::type topology) ua::'A::type => bool. open u (re_compl ua)"
lemma DEF_closed: "closed =
@@ -9960,7 +10261,7 @@
by (import hollight DEF_closed)
constdefs
- limpt :: "'A::type topology => 'A::type => ('A::type => bool) => bool"
+ limpt :: "'A topology => 'A => ('A => bool) => bool"
"limpt ==
%(u::'A::type topology) (ua::'A::type) ub::'A::type => bool.
ALL N::'A::type => bool.
@@ -9977,18 +10278,16 @@
by (import hollight CLOSED_LIMPT)
constdefs
- ismet :: "('A::type * 'A::type => hollight.real) => bool"
+ 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::nat)) = (x = y)) &
+ (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::nat)) = (x = y)) &
+ (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)
@@ -10008,21 +10307,21 @@
metric :: _
lemmas "TYDEF_metric_@intern" = typedef_hol2hollight
- [where a="a :: 'A::type metric" and r=r ,
+ [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::nat)) = (x = y)"
+ (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::nat)"
+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::nat)) (mdist m (x, y))"
+ 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.
@@ -10034,11 +10333,11 @@
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::nat)) (mdist m (x, y))"
+ x ~= y --> real_lt (real_of_num 0) (mdist m (x, y))"
by (import hollight METRIC_NZ)
constdefs
- mtop :: "'A::type metric => 'A::type topology"
+ mtop :: "'A metric => 'A topology"
"mtop ==
%u::'A::type metric.
topology
@@ -10046,7 +10345,7 @@
ALL x::'A::type.
S x -->
(EX e::hollight.real.
- real_lt (real_of_num (0::nat)) e &
+ real_lt (real_of_num 0) e &
(ALL y::'A::type. real_lt (mdist u (x, y)) e --> S y)))"
lemma DEF_mtop: "mtop =
@@ -10056,7 +10355,7 @@
ALL x::'A::type.
S x -->
(EX e::hollight.real.
- real_lt (real_of_num (0::nat)) e &
+ 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)
@@ -10066,7 +10365,7 @@
ALL x::'A::type.
S x -->
(EX e::hollight.real.
- real_lt (real_of_num (0::nat)) e &
+ 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)
@@ -10075,12 +10374,12 @@
(ALL x::'A::type.
S x -->
(EX e::hollight.real.
- real_lt (real_of_num (0::nat)) e &
+ 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::type metric => 'A::type * hollight.real => 'A::type => bool"
+ 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)"
@@ -10091,17 +10390,17 @@
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::nat)) e --> open (mtop m) (ball m (x, e))"
+ 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::nat)) e --> neigh (mtop m) (ball m (x, e), x)"
+ 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::nat)) e -->
+ 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)
@@ -10142,19 +10441,19 @@
by (import hollight MR1_SUB)
lemma MR1_ADD_LE: "ALL (x::hollight.real) d::hollight.real.
- real_le (real_of_num (0::nat)) d --> mdist mr1 (x, real_add x d) = d"
+ 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::nat)) d --> mdist mr1 (x, real_sub x d) = d"
+ 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::nat)) d --> mdist mr1 (x, real_add x d) = d"
+ 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::nat)) d --> mdist mr1 (x, real_sub x d) = d"
+ 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.
@@ -10165,7 +10464,7 @@
by (import hollight MR1_LIMPT)
constdefs
- dorder :: "('A::type => 'A::type => bool) => bool"
+ dorder :: "('A => 'A => bool) => bool"
"dorder ==
%u::'A::type => 'A::type => bool.
ALL (x::'A::type) y::'A::type.
@@ -10180,8 +10479,7 @@
by (import hollight DEF_dorder)
constdefs
- tends :: "('B::type => 'A::type)
-=> 'A::type => 'A::type topology * ('B::type => 'B::type => bool) => bool"
+ 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).
@@ -10200,8 +10498,7 @@
by (import hollight DEF_tends)
constdefs
- bounded :: "'A::type metric * ('B::type => 'B::type => bool)
-=> ('B::type => 'A::type) => bool"
+ bounded :: "'A metric * ('B => 'B => bool) => ('B => 'A) => bool"
"bounded ==
%(u::'A::type metric * ('B::type => 'B::type => bool))
ua::'B::type => 'A::type.
@@ -10218,15 +10515,15 @@
by (import hollight DEF_bounded)
constdefs
- tendsto :: "'A::type metric * 'A::type => 'A::type => 'A::type => bool"
+ 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::nat)) (mdist (fst u) (snd u, ua)) &
+ 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::nat)) (mdist (fst u) (snd u, ua)) &
+ 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)
@@ -10248,7 +10545,7 @@
(x::'B::type => 'A::type) x0::'A::type.
tends x x0 (mtop d, g) =
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ 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)))"
@@ -10264,7 +10561,7 @@
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::nat)) xa -->
+ 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)
@@ -10274,11 +10571,11 @@
limpt (mtop m1) x0 re_universe -->
tends f y0 (mtop m2, tendsto (m1, x0)) =
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL x::'A::type.
- real_lt (real_of_num (0::nat)) (mdist m1 (x, x0)) &
+ 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)
@@ -10288,11 +10585,11 @@
limpt (mtop m1) x0 re_universe -->
tends f y0 (mtop m2, tendsto (m1, x0)) =
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL x::'A::type.
- real_lt (real_of_num (0::nat)) (mdist m1 (x, x0)) &
+ 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)
@@ -10306,8 +10603,7 @@
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::nat))
- (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)
@@ -10316,39 +10612,38 @@
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::nat) -->
+ 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::nat)))"
+ 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::nat) -->
+ 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::nat)) (mtop mr1, g) &
- tends y (real_of_num (0::nat)) (mtop mr1, g) -->
- tends (%n::'A::type. real_add (x n) (y n)) (real_of_num (0::nat))
+ 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::nat)) (mtop mr1, g) -->
- tends (%n::'A::type. real_mul (x n) (y n)) (real_of_num (0::nat))
+ 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::nat)) (mtop mr1, g) -->
- tends (%n::'A::type. real_mul k (x n)) (real_of_num (0::nat))
- (mtop mr1, g)"
+ 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)
@@ -10383,7 +10678,7 @@
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::nat) -->
+ 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)
@@ -10391,7 +10686,7 @@
(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::nat) -->
+ 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)
@@ -10403,7 +10698,7 @@
lemma NET_SUM: "ALL g::'q_71813::type => 'q_71813::type => bool.
dorder g &
- tends (%x::'q_71813::type. real_of_num (0::nat)) (real_of_num (0::nat))
+ tends (%x::'q_71813::type. real_of_num 0) (real_of_num 0)
(mtop mr1, g) -->
(ALL (x::nat) n::nat.
(ALL r::nat.
@@ -10436,7 +10731,7 @@
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::nat)) e -->
+ 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)
@@ -10462,7 +10757,7 @@
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::nat) -->
+ 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)
@@ -10474,8 +10769,7 @@
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::nat) -->
+ 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)
@@ -10485,7 +10779,7 @@
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::nat))"
+ 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)
@@ -10512,7 +10806,7 @@
"cauchy ==
%u::nat => hollight.real.
ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (m::nat) n::nat.
>= m N & >= n N -->
@@ -10521,7 +10815,7 @@
lemma DEF_cauchy: "cauchy =
(%u::nat => hollight.real.
ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX N::nat.
ALL (m::nat) n::nat.
>= m N & >= n N -->
@@ -10630,11 +10924,11 @@
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::nat)) &
+ 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::nat))"
+ 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.
@@ -10642,8 +10936,8 @@
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::nat)) =
- tends_num_real f (real_of_num (0::nat))"
+ 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.
@@ -10654,17 +10948,17 @@
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::nat))"
+ 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::nat))) -->
- tends_num_real (real_pow (real_abs c)) (real_of_num (0::nat))"
+ 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::nat))) -->
- tends_num_real (real_pow c) (real_of_num (0::nat))"
+ 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.
@@ -10681,7 +10975,7 @@
(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::nat)) -->
+ 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)"
@@ -10692,7 +10986,7 @@
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::nat)) d &
+ 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))) -->
@@ -10701,11 +10995,9 @@
constdefs
sums :: "(nat => hollight.real) => hollight.real => bool"
- "sums ==
-%u::nat => hollight.real. tends_num_real (%n::nat. psum (0::nat, n) u)"
-
-lemma DEF_sums: "sums =
-(%u::nat => hollight.real. tends_num_real (%n::nat. psum (0::nat, n) u))"
+ "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
@@ -10736,25 +11028,21 @@
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::nat)) -->
- sums f (psum (0::nat, n) f)"
+ (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::nat)) (f m)) -->
- real_le (psum (0::nat, n) f) (suminf f)"
+ 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::nat)) (f m)) -->
- real_lt (psum (0::nat, n) f) (suminf f)"
+ 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::nat) k -->
- sums (%n::nat. psum (n * k, k) f) (suminf f)"
+ 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.
@@ -10762,8 +11050,7 @@
sums
(%n::nat.
psum
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n,
- NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))
+ (NUMERAL_BIT0 (NUMERAL_BIT1 0) * n, NUMERAL_BIT0 (NUMERAL_BIT1 0))
f)
(suminf f)"
by (import hollight SER_PAIR)
@@ -10771,23 +11058,22 @@
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::nat, k) f)))"
+ 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::nat, k) f) (suminf (%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::nat))
- (real_add (f (n + NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * d))
+ real_lt (real_of_num 0)
+ (real_add (f (n + NUMERAL_BIT0 (NUMERAL_BIT1 0) * d))
(f (n +
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * d +
- NUMERAL_BIT1 (0::nat)))))) -->
- real_lt (psum (0::nat, n) f) (suminf f)"
+ (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)
@@ -10817,14 +11103,13 @@
lemma SER_CAUCHY: "ALL f::nat => hollight.real.
summable f =
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ 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::nat))"
+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.
@@ -10859,28 +11144,25 @@
by (import hollight SER_ABS)
lemma GP_FINITE: "ALL x::hollight.real.
- x ~= real_of_num (NUMERAL_BIT1 (0::nat)) -->
+ x ~= real_of_num (NUMERAL_BIT1 0) -->
(ALL n::nat.
- psum (0::nat, n) (real_pow x) =
- real_div
- (real_sub (real_pow x n) (real_of_num (NUMERAL_BIT1 (0::nat))))
- (real_sub x (real_of_num (NUMERAL_BIT1 (0::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::nat))) -->
- sums (real_pow x)
- (real_inv (real_sub (real_of_num (NUMERAL_BIT1 (0::nat))) x))"
+ 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::nat)) -->
- real_le (real_abs x) (real_mul c (real_abs y)) -->
- x = real_of_num (0::nat)"
+ 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::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)))) -->
@@ -10889,7 +11171,7 @@
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::nat, n) f))) b"
+ real_le (real_abs (real_sub l (psum (0, n) f))) b"
by (import hollight SEQ_TRUNCATION)
constdefs
@@ -10907,11 +11189,11 @@
x0::hollight.real.
tends_real_real f y0 x0 =
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL x::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs (real_sub x x0)) &
+ 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)
@@ -10939,7 +11221,7 @@
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::nat) -->
+ 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)
@@ -10953,15 +11235,14 @@
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::nat) -->
+ 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::nat)) 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)
@@ -10988,8 +11269,8 @@
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::nat)) x0 &
+ 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)
@@ -11000,13 +11281,13 @@
%(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::nat))"
+ (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::nat)))"
+ ua (real_of_num 0))"
by (import hollight DEF_diffl)
constdefs
@@ -11014,12 +11295,12 @@
"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::nat))"
+ (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::nat)))"
+ (real_of_num 0))"
by (import hollight DEF_contl)
constdefs
@@ -11069,8 +11350,7 @@
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::nat) -->
+ 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)
@@ -11082,8 +11362,7 @@
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::nat) -->
+ 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)
@@ -11108,8 +11387,7 @@
(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::nat)))"
+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)
@@ -11155,37 +11433,33 @@
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::nat))))"
+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::nat)))) x"
+ (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::nat) -->
+ x ~= real_of_num 0 -->
diffl real_inv
- (real_neg
- (real_pow (real_inv x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
- x"
+ (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::nat) -->
+ 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::nat))))))
+ (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::nat) -->
+ 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::nat)))))
+ (real_pow (g x) (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x"
by (import hollight DIFF_DIV)
@@ -11254,54 +11528,54 @@
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::nat)) l -->
+ diffl f l x & real_lt (real_of_num 0) l -->
(EX d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL h::hollight.real.
- real_lt (real_of_num (0::nat)) h & real_lt h d -->
+ 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::nat)) -->
+ diffl f l x & real_lt l (real_of_num 0) -->
(EX d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL h::hollight.real.
- real_lt (real_of_num (0::nat)) h & real_lt h d -->
+ 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::nat)) d &
+ 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::nat)"
+ 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::nat)) d &
+ 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::nat)"
+ 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::nat)) d &
+ 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::nat)"
+ 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::nat)) xa &
+ 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))"
@@ -11310,7 +11584,7 @@
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::nat)) xa &
+ 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))"
@@ -11323,7 +11597,7 @@
(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::nat)) z)"
+ 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.
@@ -11358,7 +11632,7 @@
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::nat)) x) -->
+ 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)
@@ -11366,19 +11640,19 @@
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::nat)) x) -->
+ 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::nat)) x) -->
+ 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::nat))) --> x xa = x xb"
+ 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.
@@ -11388,7 +11662,7 @@
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::nat)) d &
+ 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.
@@ -11399,7 +11673,7 @@
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::nat)) d &
+ 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.
@@ -11410,13 +11684,13 @@
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::nat)) d &
+ 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::nat)) e &
+ 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.
@@ -11425,7 +11699,7 @@
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::nat)) d &
+ 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.
@@ -11435,23 +11709,23 @@
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::nat)) d &
+ 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::nat) -->
+ 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::nat)) d &
+ 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::nat) -->
+ diffl f l x & l ~= real_of_num 0 -->
diffl g (real_inv l) (f x)"
by (import hollight DIFF_INVERSE_LT)
@@ -11459,10 +11733,9 @@
(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::nat)) &
- real_lt (f' b) (real_of_num (0::nat)) -->
- (EX z::hollight.real.
- real_lt a z & real_lt z b & f' z = real_of_num (0::nat))"
+ 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)
@@ -11486,9 +11759,9 @@
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::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX (N::nat) d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ 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))) &
@@ -11499,60 +11772,60 @@
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::nat)) d &
+ 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::nat)) d &
+ real_lt (real_of_num 0) d &
(ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ 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::nat, n) (s x)) (f x))) e)))"
+ 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::nat)) d &
+ 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::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX (N::nat) d::hollight.real.
- real_lt (real_of_num (0::nat)) d &
+ 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::nat, n) (s x)) (f x))) e)))"
+ 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::nat, Suc n)
+ psum (0, Suc n)
(%p::nat. real_mul (real_pow x p) (real_pow y (Suc n - p))) =
real_mul y
- (psum (0::nat, Suc n)
+ (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::nat, Suc n)
+ (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::nat, Suc n)
+ psum (0, Suc n)
(%xa::nat. real_mul (real_pow x xa) (real_pow y (n - xa))) =
- psum (0::nat, Suc n)
+ psum (0, Suc n)
(%xa::nat. real_mul (real_pow x (n - xa)) (real_pow y xa))"
by (import hollight POWREV)
@@ -11584,25 +11857,24 @@
by (import hollight DIFFS_NEG)
lemma DIFFS_LEMMA: "ALL (n::nat) (c::nat => hollight.real) x::hollight.real.
- psum (0::nat, n) (%n::nat. real_mul (diffs c n) (real_pow x n)) =
+ psum (0, n) (%n::nat. real_mul (diffs c n) (real_pow x n)) =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%n::nat.
real_mul (real_of_num n)
- (real_mul (c n) (real_pow x (n - NUMERAL_BIT1 (0::nat))))))
+ (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::nat)))))"
+ (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::nat, n)
+ psum (0, n)
(%n::nat.
real_mul (real_of_num n)
- (real_mul (c n) (real_pow x (n - NUMERAL_BIT1 (0::nat))))) =
- real_sub
- (psum (0::nat, n) (%n::nat. real_mul (diffs c n) (real_pow x 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::nat)))))"
+ (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.
@@ -11610,73 +11882,70 @@
sums
(%n::nat.
real_mul (real_of_num n)
- (real_mul (c n) (real_pow x (n - NUMERAL_BIT1 (0::nat)))))
+ (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::nat, m)
+ 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::nat, 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::nat) -->
+ 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::nat)))) =
+ (real_mul (real_of_num n) (real_pow z (n - NUMERAL_BIT1 0))) =
real_mul h
- (psum (0::nat, n - NUMERAL_BIT1 (0::nat))
+ (psum (0, n - NUMERAL_BIT1 0)
(%p::nat.
real_mul (real_pow z p)
- (psum (0::nat, n - NUMERAL_BIT1 (0::nat) - 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::nat)) - p - q))))))"
+ (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::nat) &
+ 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::nat))))))
+ (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::nat)))
- (real_mul (real_pow K (n - NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
+ (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::nat)) k &
+ real_lt (real_of_num 0) k &
(ALL h::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs h) &
- real_lt (real_abs h) k -->
+ 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::nat)) (real_of_num (0::nat))"
+ 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::nat)) k &
+ real_lt (real_of_num 0) k &
summable f &
(ALL h::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs h) &
- real_lt (real_abs h) k -->
+ 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::nat))
- (real_of_num (0::nat))"
+ 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.
@@ -11690,9 +11959,9 @@
by (import hollight TERMDIFF)
lemma SEQ_NPOW: "ALL x::hollight.real.
- real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat))"
+ (real_of_num 0)"
by (import hollight SEQ_NPOW)
lemma TERMDIFF_CONVERGES: "ALL K::hollight.real.
@@ -11715,52 +11984,48 @@
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::nat)) n))
- (a (0::nat))"
+ 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::nat)) s &
+ 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::nat)) (real_of_num (0::nat))"
+ 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::nat)) s &
+ real_lt (real_of_num 0) s &
(ALL x::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs x) &
- real_lt (real_abs x) s -->
+ 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::nat)) (real_of_num (0::nat))"
+ 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::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX x::hollight.real.
P x &
- real_lt (real_of_num (0::nat)) (real_abs x) &
- real_lt (real_abs x) e)) &
+ 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::nat)) (real_abs x) & P x -->
+ 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::nat) = b (0::nat)"
+ 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::nat)) e -->
+ real_lt (real_of_num 0) e -->
(EX x::hollight.real.
P x &
- real_lt (real_of_num (0::nat)) (real_abs x) &
- real_lt (real_abs x) e)) &
+ 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) &
@@ -11769,35 +12034,32 @@
by (import hollight POWSER_EQUAL)
lemma MULT_DIV_2: "ALL n::nat.
- DIV (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n)
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::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::nat))) =
- Suc (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat) --> x = real_of_num (0::nat)"
+ 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::nat)) = (x = real_of_num (0::nat))"
+ (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::nat)) x & real_lt x y -->
+ 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::nat)) x &
- real_le (real_of_num (0::nat)) y &
- real_pow x (Suc n) = real_pow y (Suc n) -->
+ 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)
@@ -11821,11 +12083,10 @@
suminf
(%n::nat.
real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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))"
@@ -11834,11 +12095,10 @@
suminf
(%n::nat.
real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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)
@@ -11852,10 +12112,10 @@
real_mul
(COND (EVEN n)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
+ (real_of_num 0))
(real_pow u n))"
lemma DEF_cos: "cos =
@@ -11865,10 +12125,10 @@
real_mul
(COND (EVEN n)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
+ (real_of_num 0))
(real_pow u n)))"
by (import hollight DEF_cos)
@@ -11881,11 +12141,10 @@
sums
(%n::nat.
real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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)"
@@ -11897,10 +12156,10 @@
real_mul
(COND (EVEN n)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
+ (real_of_num 0))
(real_pow x n))
(cos x)"
by (import hollight COS_CONVERGES)
@@ -11911,36 +12170,34 @@
lemma SIN_FDIFF: "diffs
(%n::nat.
- COND (EVEN n) (real_of_num (0::nat))
+ COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))"
+ (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::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) =
+ (real_of_num 0)) =
(%n::nat.
real_neg
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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)
@@ -11950,11 +12207,10 @@
(%n::nat.
real_neg
(real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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)
@@ -11970,17 +12226,16 @@
lemma DIFF_COMPOSITE: "(diffl (f::hollight.real => hollight.real) (l::hollight.real)
(x::hollight.real) &
- f x ~= real_of_num (0::nat) -->
+ 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::nat))))))
+ (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::nat) -->
+ 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::nat)))))
+ (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) &
@@ -11992,8 +12247,7 @@
(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::nat))))
+ (real_mul (real_mul (real_of_num n) (real_pow (g x) (n - NUMERAL_BIT1 0)))
m)
x) &
(diffl g m x -->
@@ -12004,17 +12258,17 @@
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::nat)) = real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat)) x -->
- real_le (real_add (real_of_num (NUMERAL_BIT1 (0::nat))) x) (exp x)"
+ 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::nat)) x -->
- real_lt (real_of_num (NUMERAL_BIT1 (0::nat))) (exp x)"
+ 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.
@@ -12022,11 +12276,11 @@
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::nat))"
+ 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::nat))"
+ 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)"
@@ -12036,13 +12290,13 @@
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::nat)) (exp x)"
+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::nat)"
+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::nat)) (exp x)"
+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.
@@ -12069,24 +12323,22 @@
by (import hollight REAL_EXP_INJ)
lemma REAL_EXP_TOTAL_LEMMA: "ALL y::hollight.real.
- real_le (real_of_num (NUMERAL_BIT1 (0::nat))) y -->
+ real_le (real_of_num (NUMERAL_BIT1 0)) y -->
(EX x::hollight.real.
- real_le (real_of_num (0::nat)) x &
- real_le x (real_sub y (real_of_num (NUMERAL_BIT1 (0::nat)))) &
- exp x = y)"
+ 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::nat)) y --> (EX x::hollight.real. exp x = y)"
+ 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::nat)) x &
- real_le x
- (real_inv (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) -->
+ 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::nat)))
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) 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
@@ -12099,67 +12351,66 @@
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::nat)) x"
+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::nat)) x & real_lt (real_of_num (0::nat)) y -->
+ 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::nat)) x & real_lt (real_of_num (0::nat)) y -->
+ 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::nat))) = real_of_num (0::nat)"
+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::nat)) x --> ln (real_inv x) = real_neg (ln x)"
+ 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::nat)) x &
- real_lt (real_of_num (0::nat)) (y::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::nat)) x & real_lt (real_of_num (0::nat)) y -->
+ 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::nat)) x & real_lt (real_of_num (0::nat)) y -->
+ 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::nat)) x -->
+ 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::nat)) x -->
- real_le (ln (real_add (real_of_num (NUMERAL_BIT1 (0::nat))) x)) x"
+ 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::nat)) x --> real_lt (ln x) x"
+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::nat))) x -->
- real_le (real_of_num (0::nat)) (ln x)"
+ 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::nat))) x -->
- real_lt (real_of_num (0::nat)) (ln x)"
+ 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::nat)) x --> diffl ln (real_inv x) x"
+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
@@ -12167,15 +12418,13 @@
"root ==
%(u::nat) ua::hollight.real.
SOME ub::hollight.real.
- (real_lt (real_of_num (0::nat)) ua -->
- real_lt (real_of_num (0::nat)) ub) &
+ (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::nat)) ua -->
- real_lt (real_of_num (0::nat)) ub) &
+ (real_lt (real_of_num 0) ua --> real_lt (real_of_num 0) ub) &
real_pow ub u = ua)"
by (import hollight DEF_root)
@@ -12184,243 +12433,233 @@
"sqrt ==
%u::hollight.real.
SOME y::hollight.real.
- real_le (real_of_num (0::nat)) y &
- real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) = u"
+ 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::nat)) y &
- real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) = u)"
+ 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::nat))) x"
+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::nat)) x -->
+ 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::nat)) x -->
+ 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::nat)) = real_of_num (0::nat)"
+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::nat))) =
- real_of_num (NUMERAL_BIT1 (0::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::nat)) x -->
- real_pow (root (Suc n) x) (Suc n) = x"
+ 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::nat)) x -->
- root (Suc n) (real_pow x (Suc n)) = x"
+ 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::nat)) x -->
- real_le (real_of_num (0::nat)) (root (Suc n) x)"
+ 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::nat)) x &
- real_le (real_of_num (0::nat)) y & real_pow y (Suc n) = x -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) y -->
+ 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::nat)) x -->
+ 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::nat)) xa & real_le (real_of_num (0::nat)) xb -->
+ 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::nat)) x & real_lt x y -->
+ 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::nat)) x & real_le x y -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) y -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) y -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)) = real_of_num (0::nat)"
+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::nat))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat)) x -->
- real_lt (real_of_num (0::nat)) (sqrt x)"
+ 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::nat)) x -->
- real_le (real_of_num (0::nat)) (sqrt x)"
+ 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::nat))) = x) =
- real_le (real_of_num (0::nat)) x"
+ (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::nat)) x -->
- real_pow (sqrt x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) = x"
+ 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::nat)) (x::hollight.real) -->
-sqrt (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) = x"
+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::nat)) x &
- real_le (real_of_num (0::nat)) xa &
- real_pow xa (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))) = x -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)) x -->
- sqrt (real_inv x) = real_inv (sqrt x)"
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)) x & real_lt x xa -->
- real_lt (sqrt x) (sqrt xa)"
+ 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::nat)) x & real_le x xa -->
- real_le (sqrt x) (sqrt xa)"
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)) x & real_le (real_of_num (0::nat)) xa -->
+ 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::nat)))) n) =
- real_pow (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
- (DIV n (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat)) x --> real_div x (sqrt x) = sqrt x"
+ 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::nat)))) = real_abs x"
+ 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::nat)) x -->
- (sqrt x = real_of_num (0::nat)) = (x = real_of_num (0::nat))"
+ 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::nat)) x &
- real_le (real_of_num (0::nat)) y &
- real_le x (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) -->
+ 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::nat))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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::nat)))) y -->
+ 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::nat)) = real_of_num (0::nat)"
+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::nat)) = real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat))))
- (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) =
- real_of_num (NUMERAL_BIT1 (0::nat))"
+ 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::nat)))"
+ 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::nat)))) (sin x) &
- real_le (sin x) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ 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::nat)))"
+ 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::nat)))) (cos x) &
- real_le (cos x) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ 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.
@@ -12428,21 +12667,21 @@
(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::nat))))
+ (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::nat)))) =
- real_of_num (0::nat)"
+ (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::nat))))
+ (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(real_pow (real_sub (cos (real_neg x)) (cos x))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) =
- real_of_num (0::nat)"
+ (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.
@@ -12462,15 +12701,15 @@
by (import hollight COS_NEG)
lemma SIN_DOUBLE: "ALL x::hollight.real.
- sin (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) x) =
- real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
+ 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::nat)))) x) =
- real_sub (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
- (real_pow (sin x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+ 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"
@@ -12480,103 +12719,94 @@
sums
(%n::nat.
real_mul
- (real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat)))) n)
+ (real_div (real_pow (real_neg (real_of_num (NUMERAL_BIT1 0))) n)
(real_of_num
- (FACT
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n +
- NUMERAL_BIT1 (0::nat)))))
- (real_pow x
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n +
- NUMERAL_BIT1 (0::nat))))
+ (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::nat)) x &
- real_lt x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) -->
- real_lt (real_of_num (0::nat)) (sin x)"
+ 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::nat)))) n)
- (real_of_num (FACT (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n))))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)) * n)))
+ (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::nat)))))
- (real_of_num (0::nat))"
+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::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) &
- cos x = real_of_num (0::nat)"
+ 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::nat))))
+real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(SOME x::hollight.real.
- real_le (real_of_num (0::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) &
- cos x = real_of_num (0::nat))"
+ 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::nat))))
+real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))
(SOME x::hollight.real.
- real_le (real_of_num (0::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) &
- cos x = real_of_num (0::nat))"
+ 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::nat)))) =
+lemma PI2: "real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) =
(SOME x::hollight.real.
- real_le (real_of_num (0::nat)) x &
- real_le x (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) &
- cos x = real_of_num (0::nat))"
+ 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::nat))))) =
-real_of_num (0::nat)"
+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::nat))
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) &
-real_lt (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
- (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))"
+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::nat)) pi"
+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::nat))))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+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::nat)))"
+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::nat)"
+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::nat)))))
+ 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::nat)))))
+ sin (real_sub (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x)"
by (import hollight COS_SIN)
@@ -12588,152 +12818,139 @@
lemma SIN_PERIODIC: "ALL x::hollight.real.
sin (real_add x
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
- pi)) =
+ (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::nat))))
- pi)) =
+ (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::nat)))) n"
+ 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::nat)"
+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::nat)) x &
- real_lt x
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) -->
- real_lt (real_of_num (0::nat)) (sin x)"
+ 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::nat)) x &
- real_lt x
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) -->
- real_lt (real_of_num (0::nat)) (cos x)"
+ 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::nat))))))
+ (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::nat))))) -->
- real_lt (real_of_num (0::nat)) (cos 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::nat)) x & real_lt x pi -->
- real_lt (real_of_num (0::nat)) (sin x)"
+ 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::nat)) x & real_le x pi -->
- real_le (real_of_num (0::nat)) (sin x)"
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)) x & real_le x pi & cos x = y)"
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat))))))
+ (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::nat))))) &
+ (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::nat)) x & cos x = real_of_num (0::nat) -->
+ 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::nat))))))"
+ (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::nat)) x & sin x = real_of_num (0::nat) -->
+ 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::nat))))))"
+ (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::nat)) =
+ (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::nat)))))) |
+ (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::nat))))))))"
+ (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::nat)) =
+ (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::nat)))))) |
+ (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::nat))))))))"
+ (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::nat)) =
+ (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::nat))) =
+ (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::nat))))
- pi)) |
+ (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::nat))))
- pi))))"
+ (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi))))"
by (import hollight COS_ONE_2PI)
constdefs
@@ -12743,13 +12960,13 @@
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::nat)) = real_of_num (0::nat)"
+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::nat)"
+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::nat)"
+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)"
@@ -12757,8 +12974,7 @@
lemma TAN_PERIODIC: "ALL x::hollight.real.
tan (real_add x
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
- pi)) =
+ (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))) pi)) =
tan x"
by (import hollight TAN_PERIODIC)
@@ -12770,64 +12986,58 @@
by (import hollight TAN_PERIODIC_NPI)
lemma TAN_ADD: "ALL (x::hollight.real) y::hollight.real.
- cos x ~= real_of_num (0::nat) &
- cos y ~= real_of_num (0::nat) &
- cos (real_add x y) ~= real_of_num (0::nat) -->
+ 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::nat)))
- (real_mul (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::nat) &
- cos (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) x) ~=
- real_of_num (0::nat) -->
- tan (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) x) =
- real_div
- (real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) (tan x))
- (real_sub (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow (tan x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))"
+ 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::nat)) x &
- real_lt x
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) -->
- real_lt (real_of_num (0::nat)) (tan x)"
+ 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::nat) -->
- diffl tan
- (real_inv (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) x"
+ 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::nat) -->
+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::nat)))))
+ (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::nat)) y -->
+ real_lt (real_of_num 0) y -->
(EX x::hollight.real.
- real_lt (real_of_num (0::nat)) x &
+ real_lt (real_of_num 0) x &
real_lt x
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) &
+ (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::nat)) y -->
+ real_le (real_of_num 0) y -->
(EX x::hollight.real.
- real_le (real_of_num (0::nat)) x &
+ real_le (real_of_num 0) x &
real_lt x
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))) &
+ (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = y)"
by (import hollight TAN_TOTAL_POS)
@@ -12835,27 +13045,25 @@
EX! x::hollight.real.
real_lt
(real_neg
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))))) &
+ (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::nat)))) =
-real_mul (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))
- (real_div pi
- (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))"
+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::nat)))))) =
-real_of_num (NUMERAL_BIT1 (0::nat))"
+ (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::nat)))))
+ tan (real_sub (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0))))
x) =
real_inv (tan x)"
by (import hollight TAN_COT)
@@ -12863,14 +13071,13 @@
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::nat)))))) -->
- real_lt (real_abs (tan x)) (real_of_num (NUMERAL_BIT1 (0::nat)))"
+ (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::nat))))) -->
+ (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)
@@ -12881,10 +13088,10 @@
SOME x::hollight.real.
real_le
(real_neg
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))))) &
+ (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin x = u"
lemma DEF_asn: "asn =
@@ -12892,11 +13099,10 @@
SOME x::hollight.real.
real_le
(real_neg
- (real_div pi
- (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))))) &
+ (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
sin x = u)"
by (import hollight DEF_asn)
@@ -12905,12 +13111,12 @@
"acs ==
%u::hollight.real.
SOME x::hollight.real.
- real_le (real_of_num (0::nat)) x & real_le x pi & cos x = u"
+ 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::nat)) x & real_le x pi & cos x = u)"
+ real_le (real_of_num 0) x & real_le x pi & cos x = u)"
by (import hollight DEF_acs)
constdefs
@@ -12920,10 +13126,10 @@
SOME x::hollight.real.
real_lt
(real_neg
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))))) &
+ (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan x = u"
lemma DEF_atn: "atn =
@@ -12931,100 +13137,92 @@
SOME x::hollight.real.
real_lt
(real_neg
- (real_div pi
- (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))))) &
+ (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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat))))))
+ (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::nat))))) &
+ (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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat))))))
+ (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::nat)))))"
+ (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::nat)))) y &
- real_lt y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat))))))
+ (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::nat)))))"
+ (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::nat))))))
+ (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::nat))))) -->
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_le (real_of_num (0::nat)) (acs y) &
- real_le (acs y) pi & cos (acs y) = y"
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))) y &
- real_le y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_le (real_of_num (0::nat)) (acs y) & real_le (acs y) pi"
+ 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::nat)))) y &
- real_lt y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- real_lt (real_of_num (0::nat)) (acs y) & real_lt (acs y) pi"
+ 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::nat)) x & real_le x pi --> acs (cos x) = x"
+ 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::nat))))))
+ (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::nat))))) &
+ (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 0)))) &
tan (atn y) = y"
by (import hollight ATN)
@@ -13033,49 +13231,45 @@
lemma ATN_BOUNDS: "ALL x::hollight.real.
real_lt
- (real_neg
- (real_div pi (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat)))))"
+ (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::nat))))))
+ (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::nat))))) -->
+ 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::nat)) = real_of_num (0::nat)"
+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::nat))) =
-real_div pi
- (real_of_num (NUMERAL_BIT0 (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))"
+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::nat)"
+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::nat) -->
- real_add (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow (tan x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) =
- real_pow (real_inv (cos x)) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))"
+ 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::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (real_add (real_of_num (NUMERAL_BIT1 0))
+ (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 0)))))
x"
by (import hollight DIFF_ATN)
@@ -13084,8 +13278,8 @@
diffl (%x::hollight.real. atn (g x))
(real_mul
(real_inv
- (real_add (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow (g x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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)
@@ -13105,139 +13299,136 @@
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::nat)) (atn (x::hollight.real)) =
-real_lt (real_of_num (0::nat)) x"
+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::nat)) (atn (x::hollight.real)) =
-real_le (real_of_num (0::nat)) x"
+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::nat))) -->
+ 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::nat))))))"
+ (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::nat)))) x -->
+ 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::nat)))))))
+ (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::nat))) -->
+ 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::nat))))))"
+ (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::nat))) -->
+ 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::nat))))))"
+ (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::nat)) (cos x) -->
+ real_le (real_of_num 0) (cos x) -->
cos x =
sqrt
- (real_sub (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow (sin x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))"
+ (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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- cos (asn x) ~= real_of_num (0::nat)"
+ 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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))))
+ (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::nat)) (sin x) -->
+ real_le (real_of_num 0) (sin x) -->
sin x =
sqrt
- (real_sub (real_of_num (NUMERAL_BIT1 (0::nat)))
- (real_pow (cos x) (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))"
+ (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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
- sin (acs x) ~= real_of_num (0::nat)"
+ 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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))))
+ (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::nat))))
- (real_pow y (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))) =
- real_of_num (NUMERAL_BIT1 (0::nat)) -->
+ 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::nat)))) x &
- real_lt x y & real_lt y (real_of_num (NUMERAL_BIT1 (0::nat))) -->
+ 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::nat)) = (x = (0::nat))"
+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::nat) = fst u &
+ 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::nat) = fst u &
+ 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)))"
@@ -13281,11 +13472,11 @@
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::nat)) (ua x)"
+ 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::nat)) (ua x))"
+ ALL x::hollight.real. u x --> real_lt (real_of_num 0) (ua x))"
by (import hollight DEF_gauge)
constdefs
@@ -13312,13 +13503,13 @@
"rsum ==
%(u::(nat => hollight.real) * (nat => hollight.real))
ua::hollight.real => hollight.real.
- psum (0::nat, dsize (fst u))
+ 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::nat, dsize (fst u))
+ 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)
@@ -13330,7 +13521,7 @@
%(u::hollight.real * hollight.real) (ua::hollight.real => hollight.real)
ub::hollight.real.
ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ 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 &
@@ -13342,7 +13533,7 @@
(%(u::hollight.real * hollight.real) (ua::hollight.real => hollight.real)
ub::hollight.real.
ALL e::hollight.real.
- real_lt (real_of_num (0::nat)) e -->
+ 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 &
@@ -13352,25 +13543,24 @@
by (import hollight DEF_defint)
lemma DIVISION_0: "ALL (a::hollight.real) b::hollight.real.
- a = b --> dsize (%n::nat. COND (n = (0::nat)) a b) = (0::nat)"
+ 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::nat)) a b) = NUMERAL_BIT1 (0::nat)"
+ 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::nat)) a b)"
+ 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::nat) = a"
+ 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::nat) = a &
+ (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)
@@ -13386,7 +13576,7 @@
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::nat)) (D (Suc n)))"
+ (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.
@@ -13399,7 +13589,7 @@
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::nat))"
+ 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)
@@ -13407,8 +13597,7 @@
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::nat) --> real_lt xa (x (Suc xc))"
+ 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)
@@ -13478,13 +13667,13 @@
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::nat))"
+ 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::nat)) e -->
+ 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.
@@ -13508,22 +13697,21 @@
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::nat)) h &
- < (0::nat) n &
- diff (0::nat) = f &
+ 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::nat)) t & real_le t h -->
+ < 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::nat)) t &
+ real_lt (real_of_num 0) t &
real_lt t h &
f h =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_div (diff m (real_of_num (0::nat)))
- (real_of_num (FACT m)))
+ (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)))"
@@ -13531,22 +13719,21 @@
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::nat)) &
- < (0::nat) n &
- diff (0::nat) = f &
+ 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::nat)) -->
+ < 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::nat)) &
+ real_lt t (real_of_num 0) &
f h =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_div (diff m (real_of_num (0::nat)))
- (real_of_num (FACT m)))
+ (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)))"
@@ -13554,7 +13741,7 @@
lemma MCLAURIN_BI_LE: "ALL (f::hollight.real => hollight.real)
(diff::nat => hollight.real => hollight.real) (x::hollight.real) n::nat.
- diff (0::nat) = f &
+ 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) -->
@@ -13562,11 +13749,10 @@
real_le (real_abs xa) (real_abs x) &
f x =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_div (diff m (real_of_num (0::nat)))
- (real_of_num (FACT m)))
+ (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)))"
@@ -13574,19 +13760,19 @@
lemma MCLAURIN_ALL_LT: "ALL (f::hollight.real => hollight.real)
diff::nat => hollight.real => hollight.real.
- diff (0::nat) = f &
+ 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::nat) & < (0::nat) n -->
+ x ~= real_of_num 0 & < 0 n -->
(EX t::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs t) &
+ real_lt (real_of_num 0) (real_abs t) &
real_lt (real_abs t) (real_abs x) &
f x =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_div (diff m (real_of_num (0::nat)))
+ (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)))
@@ -13594,29 +13780,27 @@
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::nat) & < (0::nat) n -->
- psum (0::nat, n)
+ x = real_of_num 0 & < 0 n -->
+ psum (0, n)
(%m::nat.
- real_mul
- (real_div (diff m (real_of_num (0::nat))) (real_of_num (FACT m)))
+ real_mul (real_div (diff m (real_of_num 0)) (real_of_num (FACT m)))
(real_pow x m)) =
- diff (0::nat) (real_of_num (0::nat))"
+ 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::nat) = f &
+ 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::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_div (diff m (real_of_num (0::nat)))
- (real_of_num (FACT m)))
+ (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)))"
@@ -13626,13 +13810,13 @@
by (import hollight MCLAURIN_EXP_LEMMA)
lemma MCLAURIN_EXP_LT: "ALL (x::hollight.real) n::nat.
- x ~= real_of_num (0::nat) & < (0::nat) n -->
+ x ~= real_of_num 0 & < 0 n -->
(EX t::hollight.real.
- real_lt (real_of_num (0::nat)) (real_abs t) &
+ real_lt (real_of_num 0) (real_abs t) &
real_lt (real_abs t) (real_abs x) &
exp x =
real_add
- (psum (0::nat, n)
+ (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)
@@ -13642,67 +13826,61 @@
real_le (real_abs t) (real_abs x) &
exp x =
real_add
- (psum (0::nat, n)
+ (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::nat)) (g x) -->
+ 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::nat)) x & < (0::nat) n -->
+ real_lt (real_of_num 0) x & < 0 n -->
(EX t::hollight.real.
- real_lt (real_of_num (0::nat)) t &
+ real_lt (real_of_num 0) t &
real_lt t x &
- ln (real_add (real_of_num (NUMERAL_BIT1 (0::nat))) x) =
+ ln (real_add (real_of_num (NUMERAL_BIT1 0)) x) =
real_add
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (Suc m))
+ (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::nat))))
- (Suc n))
+ (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::nat))) t)
- 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::nat)) x &
- real_lt x (real_of_num (NUMERAL_BIT1 (0::nat))) & < (0::nat) n -->
+ 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::nat)) t &
+ real_lt (real_of_num 0) t &
real_lt t x &
- real_neg (ln (real_sub (real_of_num (NUMERAL_BIT1 (0::nat))) x)) =
+ real_neg (ln (real_sub (real_of_num (NUMERAL_BIT1 0)) x)) =
real_add
- (psum (0::nat, n)
- (%m::nat. real_div (real_pow x m) (real_of_num m)))
+ (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::nat))) t)
- 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::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (COND (EVEN m) (real_of_num (0::nat))
+ (COND (EVEN m) (real_of_num 0)
(real_div
- (real_pow
- (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (m - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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))"
@@ -13712,143 +13890,138 @@
real_le
(real_abs
(real_sub (cos x)
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
(COND (EVEN m)
(real_div
- (real_pow
- (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV m (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
+ (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::nat))) -->
+ real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
summable
(%n::nat.
real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) -->
+ 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::nat))
+ COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) -->
+ 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::nat))
+ COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))"
+ (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::nat))) -->
+ 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::nat))
+ COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow
- (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) -->
+ 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::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat)))
- (real_pow x (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat))))))
+ (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::nat))) -->
+ real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
sums
(%n::nat.
real_mul
- (COND (EVEN n) (real_of_num (0::nat))
+ (COND (EVEN n) (real_of_num 0)
(real_div
- (real_pow (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (n - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) -->
+ real_lt (real_abs x) (real_of_num (NUMERAL_BIT1 0)) -->
real_le
(real_abs
(real_sub (atn x)
- (psum (0::nat, n)
+ (psum (0, n)
(%m::nat.
real_mul
- (COND (EVEN m) (real_of_num (0::nat))
+ (COND (EVEN m) (real_of_num 0)
(real_div
- (real_pow
- (real_neg (real_of_num (NUMERAL_BIT1 (0::nat))))
- (DIV (m - NUMERAL_BIT1 (0::nat))
- (NUMERAL_BIT0 (NUMERAL_BIT1 (0::nat)))))
+ (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::nat))) (real_abs x)))"
+ (real_sub (real_of_num (NUMERAL_BIT1 0)) (real_abs x)))"
by (import hollight MCLAURIN_ATN)
;end_setup
--- a/src/HOL/Import/import_package.ML Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/import_package.ML Mon Sep 26 16:10:19 2005 +0200
@@ -53,8 +53,9 @@
val thm = equal_elim rew thm
val prew = ProofKernel.rewrite_hol4_term prem thy
val prem' = #2 (Logic.dest_equals (prop_of prew))
+ val _ = message ("Import proved " ^ (string_of_thm thm))
val thm = Drule.disambiguate_frees thm
- val _ = message ("Import proved " ^ (string_of_thm thm))
+ val _ = message ("Disambiguate: " ^ (string_of_thm thm))
in
case Shuffler.set_prop thy prem' [("",thm)] of
SOME (_,thm) =>
--- a/src/HOL/Import/proof_kernel.ML Mon Sep 26 15:56:28 2005 +0200
+++ b/src/HOL/Import/proof_kernel.ML Mon Sep 26 16:10:19 2005 +0200
@@ -182,6 +182,22 @@
fun quotename c =
if Syntax.is_identifier c andalso not (OuterSyntax.is_keyword c) then c else quote c
+fun simple_smart_string_of_cterm ct =
+ let
+ val {sign,t,T,...} = rep_cterm ct
+ (* Hack to avoid parse errors with Trueprop *)
+ val ct = (cterm_of sign (HOLogic.dest_Trueprop t)
+ handle TERM _ => ct)
+ in
+ quote(
+ Library.setmp print_mode [] (
+ Library.setmp show_brackets false (
+ Library.setmp show_all_types true (
+ Library.setmp Syntax.ambiguity_is_error false (
+ Library.setmp show_sorts true string_of_cterm))))
+ ct)
+ end
+
fun smart_string_of_cterm ct =
let
val {sign,t,T,...} = rep_cterm ct
@@ -189,29 +205,24 @@
val ct = (cterm_of sign (HOLogic.dest_Trueprop t)
handle TERM _ => ct)
fun match cu = t aconv (term_of cu)
- fun G 0 = I
- | G 1 = Library.setmp show_types true
- | G 2 = Library.setmp show_all_types true
- | G _ = error ("ProofKernel.smart_string_of_cterm internal error: " ^ (G 2 string_of_cterm ct))
- fun F sh_br n =
+ fun G 0 = Library.setmp show_types true (Library.setmp show_sorts true)
+ | G 1 = Library.setmp show_all_types true (G 0)
+ | G _ = error ("ProofKernel.smart_string_of_cterm internal error")
+ fun F n =
let
- val str = Library.setmp show_brackets sh_br (G n string_of_cterm) ct
+ val str = Library.setmp show_brackets false (G n string_of_cterm) ct
val cu = transform_error (read_cterm sign) (str,T)
in
if match cu
then quote str
- else F false (n+1)
+ else F (n+1)
end
- handle ERROR_MESSAGE mesg =>
- if String.isPrefix "Ambiguous" mesg andalso
- not sh_br
- then F true n
- else F false (n+1)
+ handle ERROR_MESSAGE mesg => F (n+1)
in
- transform_error (Library.setmp print_mode [] (Library.setmp Syntax.ambiguity_is_error true (F false))) 0
+ transform_error (Library.setmp print_mode [] (Library.setmp Syntax.ambiguity_is_error true F)) 0
end
- handle ERROR_MESSAGE mesg => quote (string_of_cterm ct)
-
+ handle ERROR_MESSAGE mesg => simple_smart_string_of_cterm ct
+
val smart_string_of_thm = smart_string_of_cterm o cprop_of
fun prth th = writeln ((Library.setmp print_mode [] string_of_thm) th)