Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
labels_flag:= true;; | |
let dirac_delta = new_definition `dirac_delta (i:num) = | |
(\j. if (i=j) then (&.1) else (&.0))`;; | |
let min_num = new_definition | |
`min_num (X:num->bool) = @m. (m IN X) /\ (!n. (n IN X) ==> (m <= n))`;; | |
let min_least = prove_by_refinement ( | |
`!(X:num->bool) c. (X c) ==> (X (min_num X) /\ (min_num X <=| c))`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[min_num;IN]; | |
REPEAT GEN_TAC; | |
DISCH_TAC; | |
SUBGOAL_THEN `?n. (X:num->bool) n /\ (!m. m <| n ==> ~X m)` MP_TAC; | |
REWRITE_TAC[(GSYM (ISPEC `X:num->bool` num_WOP))]; | |
ASM_MESON_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
ASSUME_TAC (select_thm `\m. (X:num->bool) m /\ (!n. X n ==> m <=| n)` `n:num`); | |
ABBREV_TAC `r = @m. (X:num->bool) m /\ (!n. X n ==> m <=| n)`; | |
ASM_MESON_TAC[ ARITH_RULE `~(n' < n) ==> (n <=| n') `] | |
]);; | |
(* }}} *) | |
let max_real = new_definition(`max_real x y = | |
if (y <. x) then x else y`);; | |
let min_real = new_definition(`min_real x y = | |
if (x <. y) then x else y`);; | |
let deriv = new_definition(`deriv f x = @d. (f diffl d)(x)`);; | |
let deriv2 = new_definition(`deriv2 f = (deriv (deriv f))`);; | |
let square_le = prove_by_refinement( | |
`!x y. (&.0 <=. x) /\ (&.0 <=. y) /\ (x*.x <=. y*.y) ==> (x <=. y)`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
UNDISCH_FIND_TAC `( *. )` ; | |
ONCE_REWRITE_TAC[REAL_ARITH `(a <=. b) <=> (&.0 <= (b - a))`]; | |
REWRITE_TAC[GSYM REAL_DIFFSQ]; | |
DISCH_TAC; | |
DISJ_CASES_TAC (REAL_ARITH `&.0 < (y+x) \/ (y+x <=. (&.0))`); | |
MATCH_MP_TAC (SPEC `(y+x):real` REAL_LE_LCANCEL_IMP); | |
ASM_REWRITE_TAC [REAL_ARITH `x * (&.0) = (&.0)`]; | |
CLEAN_ASSUME_TAC (REAL_ARITH `(&.0 <= y) /\ (&.0 <=. x) /\ (y+x <= (&.0)) ==> ((x= &.0) /\ (y= &.0))`); | |
ASM_REWRITE_TAC[REAL_ARITH `&.0 <=. (&.0 -. (&.0))`]; | |
]);; | |
(* }}} *) | |
let max_num_sequence = prove_by_refinement( | |
`!(t:num->num). (?n. !m. (n <=| m) ==> (t m = 0)) ==> | |
(?M. !i. (t i <=| M))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
REWRITE_TAC[GSYM LEFT_FORALL_IMP_THM]; | |
GEN_TAC; | |
SPEC_TAC (`t:num->num`,`t:num->num`); | |
SPEC_TAC (`n:num`,`n:num`); | |
INDUCT_TAC; | |
GEN_TAC; | |
REWRITE_TAC[ARITH_RULE `0<=|m`]; | |
DISCH_TAC; | |
EXISTS_TAC `0`; | |
ASM_MESON_TAC[ARITH_RULE`(a=0) ==> (a <=|0)`]; | |
DISCH_ALL_TAC; | |
ABBREV_TAC `b = \m. (if (m=n) then 0 else (t (m:num)) )`; | |
FIRST_ASSUM (fun t-> ASSUME_TAC (SPEC `b:num->num` t)); | |
SUBGOAL_TAC `((b:num->num) (n) = 0) /\ (!m. ~(m=n) ==> (b m = t m))`; | |
EXPAND_TAC "b"; | |
CONJ_TAC; | |
COND_CASES_TAC; | |
REWRITE_TAC[]; | |
ASM_MESON_TAC[]; | |
GEN_TAC; | |
COND_CASES_TAC; | |
REWRITE_TAC[]; | |
REWRITE_TAC[]; | |
DISCH_ALL_TAC; | |
FIRST_ASSUM (fun t-> MP_TAC(SPEC `b:num->num` t)); | |
SUBGOAL_TAC `!m. (n<=|m) ==> (b m =0)`; | |
GEN_TAC; | |
ASM_CASES_TAC `m = (n:num)`; | |
ASM_REWRITE_TAC[]; | |
SUBGOAL_TAC ( `(n <=| m) /\ (~(m = n)) ==> (SUC n <=| m)`); | |
ARITH_TAC; | |
ASM_REWRITE_TAC[]; | |
DISCH_ALL_TAC; | |
ASM_MESON_TAC[]; (* good *) | |
DISCH_THEN (fun t-> REWRITE_TAC[t]); | |
DISCH_THEN CHOOSE_TAC; | |
EXISTS_TAC `(M:num) + (t:num->num) n`; | |
GEN_TAC; | |
ASM_CASES_TAC `(i:num) = n`; | |
ASM_REWRITE_TAC[]; | |
ARITH_TAC; | |
MATCH_MP_TAC (ARITH_RULE `x <=| M ==> (x <=| M+ u)`); | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let REAL_INV_LT = prove_by_refinement( | |
`!x y z. (&.0 <. x) ==> ((inv(x)*y < z) <=> (y <. x*z))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_TAC; | |
REWRITE_TAC[REAL_ARITH `inv x * y = y* inv x`]; | |
REWRITE_TAC[GSYM real_div]; | |
ASM_SIMP_TAC[REAL_LT_LDIV_EQ]; | |
REAL_ARITH_TAC; | |
]);; | |
(* }}} *) | |
let REAL_MUL_NN = prove_by_refinement( | |
`!x y. (&.0 <= x*y) <=> | |
((&.0 <= x /\ (&.0 <=. y)) \/ ((x <= &.0) /\ (y <= &.0) ))`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
SUBGOAL_TAC `! x y. ((&.0 < x) ==> ((&.0 <= x*y) <=> ((&.0 <= x /\ (&.0 <=. y)) \/ ((x <= &.0) /\ (y <= &.0) ))))`; | |
DISCH_ALL_TAC; | |
ASM_SIMP_TAC[REAL_ARITH `((&.0 <. x) ==> (&.0 <=. x))`;REAL_ARITH `(&.0 <. x) ==> ~(x <=. &.0)`]; | |
EQ_TAC; | |
ASM_MESON_TAC[REAL_PROP_NN_LCANCEL]; | |
ASM_MESON_TAC[REAL_LE_MUL;REAL_LT_IMP_LE]; | |
DISCH_TAC; | |
DISJ_CASES_TAC (REAL_ARITH `(&.0 < x) \/ (x = &.0) \/ (x < &.0)`); | |
ASM_MESON_TAC[]; | |
UND 1 THEN DISCH_THEN DISJ_CASES_TAC; | |
ASM_REWRITE_TAC[]; | |
REAL_ARITH_TAC; | |
ASM_SIMP_TAC[REAL_ARITH `((x <. &.0) ==> ~(&.0 <=. x))`;REAL_ARITH `(x <. &.0) ==> (x <=. &.0)`]; | |
USE 0 (SPECL [`--. (x:real)`;`--. (y:real)`]); | |
UND 0; | |
REDUCE_TAC; | |
ASM_REWRITE_TAC[]; | |
ASM_SIMP_TAC[REAL_ARITH `((x <. &.0) ==> ~(&.0 <=. x))`;REAL_ARITH `(x <. &.0) ==> (x <=. &.0)`]; | |
]);; | |
(* }}} *) | |
let ABS_SQUARE = prove_by_refinement( | |
`!t u. abs(t) <. u ==> t*t <. u*u`, | |
(* {{{ proof *) | |
[ | |
REP_GEN_TAC; | |
CONV_TAC (SUBS_CONV[SPEC `t:real` (REWRITE_RULE[POW_2] (GSYM REAL_POW2_ABS))]); | |
ASSUME_TAC REAL_ABS_POS; | |
USE 0 (SPEC `t:real`); | |
ABBREV_TAC `(b:real) = (abs t)`; | |
KILL 1; | |
DISCH_ALL_TAC; | |
MATCH_MP_TAC REAL_PROP_LT_LRMUL; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let ABS_SQUARE_LE = prove_by_refinement( | |
`!t u. abs(t) <=. u ==> t*t <=. u*u`, | |
(* {{{ proof *) | |
[ | |
REP_GEN_TAC; | |
CONV_TAC (SUBS_CONV[SPEC `t:real` (REWRITE_RULE[POW_2] (GSYM REAL_POW2_ABS))]); | |
ASSUME_TAC REAL_ABS_POS; | |
USE 0 (SPEC `t:real`); | |
ABBREV_TAC `(b:real) = (abs t)`; | |
KILL 1; | |
DISCH_ALL_TAC; | |
MATCH_MP_TAC REAL_PROP_LE_LRMUL; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let twopow_eps = prove_by_refinement( | |
`!R e. ?n. (&.0 <. R)/\ (&.0 <. e) ==> R*(twopow(--: (&:n))) <. e`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
REWRITE_TAC[TWOPOW_NEG]; (* cs6b *) | |
ASSUME_TAC (prove(`!n. &.0 < &.2 pow n`,REDUCE_TAC THEN ARITH_TAC)); | |
ONCE_REWRITE_TAC[REAL_MUL_AC]; | |
ASM_SIMP_TAC[REAL_INV_LT]; | |
ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ]; | |
CONV_TAC (quant_right_CONV "n"); | |
DISCH_ALL_TAC; | |
ASSUME_TAC (SPEC `R/e` REAL_ARCH_SIMPLE); | |
CHO 3; | |
EXISTS_TAC `n:num`; | |
UND 3; | |
MESON_TAC[POW_2_LT;REAL_LET_TRANS]; | |
]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* finite products, in imitation of finite sums *) | |
(* ------------------------------------------------------------------ *) | |
let prod_EXISTS = prove_by_refinement( | |
`?prod. (!f n. prod(n,0) f = &1) /\ | |
(!f m n. prod(n,SUC m) f = prod(n,m) f * f(n + m))`, | |
(* {{{ proof *) | |
[ | |
(CHOOSE_TAC o prove_recursive_functions_exist num_RECURSION) `(!f n. sm n 0 f = &1) /\ (!f m n. sm n (SUC m) f = sm n m f * f(n + m))` ; | |
EXISTS_TAC `\(n,m) f. (sm:num->num->(num->real)->real) n m f`; | |
CONV_TAC(DEPTH_CONV GEN_BETA_CONV) THEN ASM_REWRITE_TAC[] | |
]);; | |
(* }}} *) | |
let prod_DEF = new_specification ["prod"] prod_EXISTS;; | |
let prod = prove | |
(`!n m. (prod(n,0) f = &1) /\ | |
(prod(n,SUC m) f = prod(n,m) f * f(n + m))`, | |
(* {{{ proof *) | |
REWRITE_TAC[prod_DEF]);; | |
(* }}} *) | |
let PROD_TWO = prove_by_refinement( | |
`!f n p. prod(0,n) f * prod(n,p) f = prod(0,n + p) f`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[prod; REAL_MUL_RID; MULT_CLAUSES;ADD_0]; | |
REWRITE_TAC[ARITH_RULE `n+| (SUC p) = (SUC (n+|p))`;prod;ARITH_RULE `0+|n = n`]; | |
ASM_REWRITE_TAC[REAL_MUL_ASSOC]; | |
]);; | |
(* }}} *) | |
let ABS_PROD = prove_by_refinement( | |
`!f m n. abs(prod(m,n) f) = prod(m,n) (\n. abs(f n))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC; | |
REWRITE_TAC[prod]; | |
REAL_ARITH_TAC; | |
ASM_REWRITE_TAC[prod;ABS_MUL] | |
]);; | |
(* }}} *) | |
let PROD_EQ = prove_by_refinement | |
(`!f g m n. (!r. m <= r /\ r < (n + m) ==> (f(r) = g(r))) | |
==> (prod(m,n) f = prod(m,n) g)`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[prod]; | |
REWRITE_TAC[prod]; | |
DISCH_THEN (fun th -> MP_TAC th THEN (MP_TAC (SPEC `m+|n` th))); | |
REWRITE_TAC[ARITH_RULE `(m<=| (m+|n))/\ (m +| n <| (SUC n +| m))`]; | |
DISCH_ALL_TAC; | |
ASM_REWRITE_TAC[]; | |
AP_THM_TAC THEN AP_TERM_TAC; | |
FIRST_X_ASSUM MATCH_MP_TAC; | |
GEN_TAC THEN DISCH_TAC; | |
FIRST_X_ASSUM MATCH_MP_TAC; | |
ASM_MESON_TAC[ARITH_RULE `r <| (n+| m) ==> (r <| (SUC n +| m))`] | |
]);; | |
(* }}} *) | |
let PROD_POS = prove_by_refinement | |
(`!f. (!n. &0 <= f(n)) ==> !m n. &0 <= prod(m,n) f`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[prod]; | |
REAL_ARITH_TAC; | |
ASM_MESON_TAC[REAL_LE_MUL] | |
]);; | |
(* }}} *) | |
let PROD_POS_GEN = prove_by_refinement | |
(`!f m n. | |
(!n. m <= n ==> &0 <= f(n)) | |
==> &0 <= prod(m,n) f`, | |
(* {{{ proof *) | |
[ | |
REPEAT STRIP_TAC THEN SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN REWRITE_TAC[prod]; | |
REAL_ARITH_TAC; | |
ASM_MESON_TAC[REAL_LE_MUL;ARITH_RULE `m <=| (m +| n)`] | |
]);; | |
(* }}} *) | |
let PROD_ABS = prove | |
(`!f m n. abs(prod(m,n) (\m. abs(f m))) = prod(m,n) (\m. abs(f m))`, | |
(* {{{ proof *) | |
REWRITE_TAC[ABS_PROD;REAL_ARITH `||. (||. x) = (||. x)`]);; | |
(* }}} *) | |
let PROD_ZERO = prove_by_refinement | |
(`!f m n. (?p. (m <= p /\ (p < (n+| m)) /\ (f p = (&.0)))) ==> | |
(prod(m,n) f = &0)`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN (REWRITE_TAC[prod]); | |
ARITH_TAC; | |
DISCH_THEN CHOOSE_TAC; | |
ASM_CASES_TAC `p <| (n+| m)`; | |
MATCH_MP_TAC (prove (`(x = (&.0)) ==> (x *. y = (&.0))`,(DISCH_THEN (fun th -> (REWRITE_TAC[th]))) THEN REAL_ARITH_TAC)); | |
FIRST_X_ASSUM MATCH_MP_TAC; | |
ASM_MESON_TAC[]; | |
POP_ASSUM (fun th -> ASSUME_TAC (MATCH_MP (ARITH_RULE `(~(p <| (n+|m)) ==> ((p <| ((SUC n) +| m)) ==> (p = ((m +| n)))))`) th)); | |
MATCH_MP_TAC (prove (`(x = (&.0)) ==> (y *. x = (&.0))`,(DISCH_THEN (fun th -> (REWRITE_TAC[th]))) THEN REAL_ARITH_TAC)); | |
ASM_MESON_TAC[] | |
]);; | |
(* }}} *) | |
let PROD_MUL = prove_by_refinement( | |
`!f g m n. prod(m,n) (\n. f(n) * g(n)) = prod(m,n) f * prod(m,n) g`, | |
(* {{{ proof *) | |
[ | |
EVERY(replicate GEN_TAC 3) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[prod]; | |
REAL_ARITH_TAC; | |
REWRITE_TAC[REAL_MUL_AC]; | |
]);; | |
(* }}} *) | |
let PROD_CMUL = prove_by_refinement( | |
`!f c m n. prod(m,n) (\n. c * f(n)) = (c **. n) * prod(m,n) f`, | |
(* {{{ proof *) | |
[ | |
EVERY(replicate GEN_TAC 3) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[prod;pow]; | |
REAL_ARITH_TAC; | |
REWRITE_TAC[REAL_MUL_AC]; | |
]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* LEMMAS ABOUT SETS *) | |
(* ------------------------------------------------------------------ *) | |
(* IN_ELIM_THM produces garbled results at times. I like this better: *) | |
(*** JRH replaced this with the "new" IN_ELIM_THM; see how it works. | |
let IN_ELIM_THM' = prove_by_refinement( | |
`(!P. !x:A. x IN (GSPEC P) <=> P x) /\ | |
(!P. !x:A. x IN (\x. P x) <=> P x) /\ | |
(!P. !x:A. (GSPEC P) x <=> P x) /\ | |
(!P (x:A) (t:A). (\t. (?y:A. P y /\ (t = y))) x <=> P x)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[IN; GSPEC]; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
****) | |
let IN_ELIM_THM' = IN_ELIM_THM;; | |
let SURJ_IMAGE = prove_by_refinement( | |
`!(f:A->B) a b. SURJ f a b ==> (b = (IMAGE f a))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
REWRITE_TAC[SURJ;IMAGE]; | |
DISCH_ALL_TAC; | |
REWRITE_TAC[EXTENSION]; | |
GEN_TAC; | |
REWRITE_TAC[IN_ELIM_THM]; | |
ASM_MESON_TAC[]] | |
(* }}} *) | |
);; | |
let SURJ_FINITE = prove_by_refinement( | |
`!a b (f:A->B). FINITE a /\ (SURJ f a b) ==> FINITE b`, | |
(* {{{ *) | |
[ | |
ASM_MESON_TAC[SURJ_IMAGE;FINITE_IMAGE] | |
]);; | |
(* }}} *) | |
let BIJ_INVERSE = prove_by_refinement( | |
`!a b (f:A->B). (SURJ f a b) ==> (?(g:B->A). (INJ g b a))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
SUBGOAL_THEN `!y. ?u. ((y IN b) ==> ((u IN a) /\ ((f:A->B) u = y)))` ASSUME_TAC; | |
ASM_MESON_TAC[SURJ]; | |
LABEL_ALL_TAC; | |
H_REWRITE_RULE[THM SKOLEM_THM] (HYP "1"); | |
LABEL_ALL_TAC; | |
H_UNDISCH_TAC (HYP"2"); | |
DISCH_THEN CHOOSE_TAC; | |
EXISTS_TAC `u:B->A`; | |
REWRITE_TAC[INJ] THEN CONJ_TAC THEN (ASM_MESON_TAC[]) | |
] | |
(* }}} *) | |
);; | |
(* complement of an intersection is a union of complements *) | |
let UNIONS_INTERS = prove_by_refinement( | |
`!(X:A->bool) V. | |
(X DIFF (INTERS V) = UNIONS (IMAGE ((DIFF) X) V))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
MATCH_MP_TAC SUBSET_ANTISYM; | |
CONJ_TAC; | |
REWRITE_TAC[SUBSET;IMAGE;IN_ELIM_THM]; | |
X_GEN_TAC `c:A`; | |
REWRITE_TAC[IN_DIFF;IN_INTERS;IN_UNIONS;NOT_FORALL_THM]; | |
DISCH_ALL_TAC; | |
UNDISCH_FIND_THEN `(?)` CHOOSE_TAC; | |
EXISTS_TAC `(X DIFF t):A->bool`; | |
REWRITE_TAC[IN_ELIM_THM]; | |
CONJ_TAC; | |
EXISTS_TAC `t:A->bool`; | |
ASM_MESON_TAC[]; | |
REWRITE_TAC[IN_DIFF]; | |
ASM_MESON_TAC[]; | |
REWRITE_TAC[SUBSET;IMAGE;IN_ELIM_THM]; | |
X_GEN_TAC `c:A`; | |
REWRITE_TAC[IN_DIFF;IN_UNIONS]; | |
DISCH_THEN CHOOSE_TAC; | |
UNDISCH_FIND_TAC `(IN)`; | |
REWRITE_TAC[IN_INTERS;IN_ELIM_THM]; | |
DISCH_ALL_TAC; | |
UNDISCH_FIND_THEN `(?)` CHOOSE_TAC; | |
CONJ_TAC; | |
ASM_MESON_TAC[SUBSET_DIFF;SUBSET]; | |
REWRITE_TAC[NOT_FORALL_THM]; | |
EXISTS_TAC `x:A->bool`; | |
ASM_MESON_TAC[IN_DIFF]; | |
]);; | |
(* }}} *) | |
let INTERS_SUBSET = prove_by_refinement ( | |
`!X (A:A->bool). (A IN X) ==> (INTERS X SUBSET A)`, | |
(* {{{ *) | |
[ | |
REPEAT GEN_TAC; | |
REWRITE_TAC[SUBSET;IN_INTERS]; | |
MESON_TAC[IN]; | |
]);; | |
(* }}} *) | |
let sub_union = prove_by_refinement( | |
`!X (U:(A->bool)->bool). (U X) ==> (X SUBSET (UNIONS U))`, | |
(* {{{ *) | |
[ | |
DISCH_ALL_TAC; | |
REWRITE_TAC[SUBSET;IN_ELIM_THM;UNIONS]; | |
REWRITE_TAC[IN]; | |
DISCH_ALL_TAC; | |
EXISTS_TAC `X:A->bool`; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let IMAGE_SURJ = prove_by_refinement( | |
`!(f:A->B) a. SURJ f a (IMAGE f a)`, | |
(* {{{ *) | |
[ | |
REWRITE_TAC[SURJ;IMAGE;IN_ELIM_THM]; | |
MESON_TAC[IN]; | |
]);; | |
(* }}} *) | |
let SUBSET_PREIMAGE = prove_by_refinement( | |
`!(f:A->B) X Y. (Y SUBSET (IMAGE f X)) ==> | |
(?Z. (Z SUBSET X) /\ (Y = IMAGE f Z))`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
EXISTS_TAC `{x | (x IN (X:A->bool))/\ (f x IN (Y:B->bool)) }`; | |
CONJ_TAC; | |
REWRITE_TAC[SUBSET;IN_ELIM_THM]; | |
MESON_TAC[]; | |
REWRITE_TAC[EXTENSION]; | |
X_GEN_TAC `y:B`; | |
UNDISCH_FIND_TAC `(SUBSET)`; | |
REWRITE_TAC[SUBSET;IN_IMAGE]; | |
REWRITE_TAC[IN_ELIM_THM]; | |
DISCH_THEN (fun t-> MP_TAC (SPEC `y:B` t)); | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let UNIONS_INTER = prove_by_refinement( | |
`!(U:(A->bool)->bool) A. (((UNIONS U) INTER A) = | |
(UNIONS (IMAGE ((INTER) A) U)))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
MATCH_MP_TAC (prove(`((C SUBSET (B:A->bool)) /\ (C SUBSET A) /\ ((A INTER B) SUBSET C)) ==> ((B INTER A) = C)`,SET_TAC[])); | |
CONJ_TAC; | |
REWRITE_TAC[SUBSET;UNIONS;IN_ELIM_THM]; | |
REWRITE_TAC[IN_IMAGE]; | |
SET_TAC[]; | |
REWRITE_TAC[SUBSET;UNIONS;IN_IMAGE]; | |
CONJ_TAC; | |
REWRITE_TAC[IN_ELIM_THM]; | |
X_GEN_TAC `y:A`; | |
DISCH_THEN CHOOSE_TAC; | |
ASM_MESON_TAC[IN_INTER]; | |
REWRITE_TAC[IN_INTER]; | |
REWRITE_TAC[IN_ELIM_THM]; | |
X_GEN_TAC `y:A`; | |
DISCH_ALL_TAC; | |
UNDISCH_FIND_THEN `(?)` CHOOSE_TAC; | |
EXISTS_TAC `A INTER (u:A->bool)`; | |
ASM SET_TAC[]; | |
]);; | |
(* }}} *) | |
let UNIONS_SUBSET = prove_by_refinement( | |
`!U (X:A->bool). (!A. (A IN U) ==> (A SUBSET X)) ==> (UNIONS U SUBSET X)`, | |
(* {{{ *) | |
[ | |
REPEAT GEN_TAC; | |
SET_TAC[]; | |
]);; | |
(* }}} *) | |
let SUBSET_INTER = prove_by_refinement( | |
`!X A (B:A->bool). (X SUBSET (A INTER B)) <=> (X SUBSET A) /\ (X SUBSET B)`, | |
(* {{{ *) | |
[ | |
REWRITE_TAC[SUBSET;INTER;IN_ELIM_THM]; | |
MESON_TAC[IN]; | |
]);; | |
(* }}} *) | |
let EMPTY_EXISTS = prove_by_refinement( | |
`!X. ~(X = {}) <=> (? (u:A). (u IN X))`, | |
(* {{{ *) | |
[ | |
REWRITE_TAC[EXTENSION]; | |
REWRITE_TAC[IN;EMPTY]; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let UNIONS_UNIONS = prove_by_refinement( | |
`!A B. (A SUBSET B) ==>(UNIONS (A:(A->bool)->bool) SUBSET (UNIONS B))`, | |
(* {{{ *) | |
[ | |
REWRITE_TAC[SUBSET;UNIONS;IN_ELIM_THM]; | |
MESON_TAC[IN]; | |
]);; | |
(* }}} *) | |
(* nested union can flatten from outside in, or inside out *) | |
let UNIONS_IMAGE_UNIONS = prove_by_refinement( | |
`!(X:((A->bool)->bool)->bool). | |
UNIONS (UNIONS X) = (UNIONS (IMAGE UNIONS X))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
REWRITE_TAC[EXTENSION;IN_UNIONS]; | |
GEN_TAC; | |
REWRITE_TAC[EXTENSION;IN_UNIONS]; | |
EQ_TAC; | |
DISCH_THEN (CHOOSE_THEN MP_TAC); | |
DISCH_ALL_TAC; | |
FIRST_ASSUM MP_TAC; | |
DISCH_THEN CHOOSE_TAC; | |
EXISTS_TAC `UNIONS (t':(A->bool)->bool)`; | |
REWRITE_TAC[IN_UNIONS;IN_IMAGE]; | |
CONJ_TAC; | |
EXISTS_TAC `(t':(A->bool)->bool)`; | |
ASM_REWRITE_TAC[]; | |
ASM_MESON_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
FIRST_ASSUM MP_TAC; | |
REWRITE_TAC[IN_IMAGE]; | |
DISCH_ALL_TAC; | |
FIRST_ASSUM MP_TAC; | |
DISCH_THEN CHOOSE_TAC; | |
UNDISCH_TAC `(x:A) IN t`; | |
FIRST_ASSUM (fun t-> REWRITE_TAC[t]); | |
REWRITE_TAC[IN_UNIONS]; | |
DISCH_THEN (CHOOSE_TAC); | |
EXISTS_TAC `t':(A->bool)`; | |
CONJ_TAC; | |
EXISTS_TAC `x':(A->bool)->bool`; | |
ASM_REWRITE_TAC[]; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let INTERS_SUBSET2 = prove_by_refinement( | |
`!X A. (?(x:A->bool). (A x /\ (x SUBSET X))) ==> ((INTERS A) SUBSET X)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[SUBSET;INTERS;IN_ELIM_THM']; | |
REWRITE_TAC[IN]; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
(**** New proof by JRH; old one breaks because of new set comprehensions | |
let INTERS_EMPTY = prove_by_refinement( | |
`INTERS EMPTY = (UNIV:A->bool)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[INTERS;NOT_IN_EMPTY;IN_ELIM_THM';]; | |
REWRITE_TAC[UNIV;GSPEC]; | |
MATCH_MP_TAC EQ_EXT; | |
GEN_TAC; | |
REWRITE_TAC[IN_ELIM_THM']; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
****) | |
let INTERS_EMPTY = prove_by_refinement( | |
`INTERS EMPTY = (UNIV:A->bool)`, | |
[SET_TAC[]]);; | |
let preimage = new_definition `preimage dom (f:A->B) | |
Z = {x | (x IN dom) /\ (f x IN Z)}`;; | |
let in_preimage = prove_by_refinement( | |
`!f x Z dom. x IN (preimage dom (f:A->B) Z) <=> (x IN dom) /\ (f x IN Z)`, | |
(* {{{ *) | |
[ | |
REWRITE_TAC[preimage]; | |
REWRITE_TAC[IN_ELIM_THM'] | |
]);; | |
(* }}} *) | |
(* Partial functions, which we identify with functions that | |
take the canonical choice of element outside the domain. *) | |
let supp = new_definition | |
`supp (f:A->B) = \ x. ~(f x = (CHOICE (UNIV:B ->bool)) )`;; | |
let func = new_definition | |
`func a b = (\ (f:A->B). ((!x. (x IN a) ==> (f x IN b)) /\ | |
((supp f) SUBSET a))) `;; | |
(* relations *) | |
let reflexive = new_definition | |
`reflexive (f:A->A->bool) <=> (!x. f x x)`;; | |
let symmetric = new_definition | |
`symmetric (f:A->A->bool) <=> (!x y. f x y ==> f y x)`;; | |
let transitive = new_definition | |
`transitive (f:A->A->bool) <=> (!x y z. f x y /\ f y z ==> f x z)`;; | |
let equivalence_relation = new_definition | |
`equivalence_relation (f:A->A->bool) <=> | |
(reflexive f) /\ (symmetric f) /\ (transitive f)`;; | |
(* We do not introduce the equivalence class of f explicitly, because | |
it is represented directly in HOL by (f a) *) | |
let partition_DEF = new_definition | |
`partition (A:A->bool) SA <=> (UNIONS SA = A) /\ | |
(!a b. ((a IN SA) /\ (b IN SA) /\ (~(a = b)) ==> ({} = (a INTER b))))`;; | |
let DIFF_DIFF2 = prove_by_refinement( | |
`!X (A:A->bool). (A SUBSET X) ==> ((X DIFF (X DIFF A)) = A)`, | |
[ | |
SET_TAC[] | |
]);; | |
(*** Old proof replaced by JRH: no longer UNWIND_THM[12] clause in IN_ELIM_THM | |
let GSPEC_THM = prove_by_refinement( | |
`!P (x:A). (?y. P y /\ (x = y)) <=> P x`, | |
[REWRITE_TAC[IN_ELIM_THM]]);; | |
***) | |
let GSPEC_THM = prove_by_refinement( | |
`!P (x:A). (?y. P y /\ (x = y)) <=> P x`, | |
[MESON_TAC[]]);; | |
let CARD_GE_REFL = prove | |
(`!s:A->bool. s >=_c s`, | |
GEN_TAC THEN REWRITE_TAC[GE_C] THEN | |
EXISTS_TAC `\x:A. x` THEN MESON_TAC[]);; | |
let FINITE_HAS_SIZE_LEMMA = prove | |
(`!s:A->bool. FINITE s ==> ?n:num. {x | x < n} >=_c s`, | |
MATCH_MP_TAC FINITE_INDUCT THEN CONJ_TAC THENL | |
[EXISTS_TAC `0` THEN REWRITE_TAC[NOT_IN_EMPTY; GE_C; IN_ELIM_THM]; | |
REPEAT GEN_TAC THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN | |
EXISTS_TAC `SUC N` THEN POP_ASSUM MP_TAC THEN PURE_REWRITE_TAC[GE_C] THEN | |
DISCH_THEN(X_CHOOSE_TAC `f:num->A`) THEN | |
EXISTS_TAC `\n:num. if n = N then x:A else f n` THEN | |
X_GEN_TAC `y:A` THEN PURE_REWRITE_TAC[IN_INSERT] THEN | |
DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC (ANTE_RES_THEN MP_TAC)) THENL | |
[EXISTS_TAC `N:num` THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN ARITH_TAC; | |
DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN | |
REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN | |
EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN | |
UNDISCH_TAC `n:num < N` THEN COND_CASES_TAC THEN | |
ASM_REWRITE_TAC[LT_REFL] THEN ARITH_TAC]]);; | |
let NUM_COUNTABLE = prove_by_refinement( | |
`COUNTABLE (UNIV:num->bool)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[COUNTABLE;CARD_GE_REFL]; | |
]);; | |
(* }}} *) | |
let NUM2_COUNTABLE = prove_by_refinement( | |
`COUNTABLE {((x:num),(y:num)) | T}`, | |
(* {{{ proof *) | |
[ | |
CHOOSE_TAC (ISPECL[`(0,0)`;`(\ (a:num,b:num) (n:num) . if (b=0) then (0,a+b+1) else (a+1,b-1))`] num_RECURSION); | |
REWRITE_TAC[COUNTABLE;GE_C;IN_ELIM_THM']; | |
NAME_CONFLICT_TAC; | |
EXISTS_TAC `fn:num -> (num#num)`; | |
X_GEN_TAC `p:num#num`; | |
REPEAT (DISCH_THEN (CHOOSE_THEN MP_TAC)); | |
DISCH_THEN (fun t->REWRITE_TAC[t]); | |
REWRITE_TAC[IN_UNIV]; | |
SUBGOAL_TAC `?t. t = x'+|y'`; | |
MESON_TAC[]; | |
SPEC_TAC (`x':num`,`a:num`); | |
SPEC_TAC (`y':num`,`b:num`); | |
CONV_TAC (quant_left_CONV "t"); | |
CONV_TAC (quant_left_CONV "t"); | |
CONV_TAC (quant_left_CONV "t"); | |
INDUCT_TAC; | |
REDUCE_TAC; | |
REP_GEN_TAC; | |
DISCH_THEN (fun t -> REWRITE_TAC[t]); | |
EXISTS_TAC `0`; | |
ASM_REWRITE_TAC[]; | |
CONV_TAC (quant_left_CONV "a"); | |
INDUCT_TAC; | |
REDUCE_TAC; | |
GEN_TAC; | |
USE 1 (SPECL [`0`;`t:num`]); | |
UND 1 THEN REDUCE_TAC; | |
DISCH_THEN (X_CHOOSE_TAC `n:num`); | |
AND 0; | |
USE 0 (SPEC `n:num`); | |
UND 0; | |
UND 1; | |
DISCH_THEN (fun t-> REWRITE_TAC[GSYM t]); | |
CONV_TAC (ONCE_DEPTH_CONV GEN_BETA_CONV); | |
BETA_TAC; | |
REDUCE_TAC; | |
DISCH_ALL_TAC; | |
EXISTS_TAC `SUC n`; | |
EXPAND_TAC "b"; | |
KILL 0; | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC [ARITH_RULE `SUC t = t+|1`]; | |
GEN_TAC; | |
ABBREV_TAC `t' = SUC t`; | |
USE 2 (SPEC `SUC b`); | |
DISCH_TAC; | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC[ARITH_RULE `SUC a +| b = a +| SUC b`]; | |
DISCH_THEN (X_CHOOSE_TAC `n:num`); | |
EXISTS_TAC `SUC n`; | |
AND 0; | |
USE 0 (SPEC `n:num`); | |
UND 0; | |
UND 2; | |
DISCH_THEN (fun t->REWRITE_TAC[GSYM t]); | |
CONV_TAC (ONCE_DEPTH_CONV GEN_BETA_CONV); | |
BETA_TAC; | |
REDUCE_TAC; | |
DISCH_THEN (fun t->REWRITE_TAC[t]); | |
REWRITE_TAC[ARITH_RULE `SUC a = a+| 1`]; | |
]);; | |
(* }}} *) | |
let COUNTABLE_UNIONS = prove_by_refinement( | |
`!A:(A->bool)->bool. (COUNTABLE A) /\ | |
(!a. (a IN A) ==> (COUNTABLE a)) ==> (COUNTABLE (UNIONS A))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
DISCH_ALL_TAC; | |
USE 0 (REWRITE_RULE[COUNTABLE;GE_C;IN_UNIV]); | |
CHO 0; | |
USE 0 (CONV_RULE (quant_left_CONV "x")); | |
USE 0 (CONV_RULE (quant_left_CONV "x")); | |
CHO 0; | |
USE 1 (REWRITE_RULE[COUNTABLE;GE_C;IN_UNIV]); | |
USE 1 (CONV_RULE (quant_left_CONV "f")); | |
USE 1 (CONV_RULE (quant_left_CONV "f")); | |
UND 1; | |
DISCH_THEN (X_CHOOSE_TAC `g:(A->bool)->num->A`); | |
SUBGOAL_TAC `!a y. (a IN (A:(A->bool)->bool)) /\ (y IN a) ==> (? (u:num) (v:num). ( a = f u) /\ (y = g a v))`; | |
REP_GEN_TAC; | |
DISCH_ALL_TAC; | |
USE 1 (SPEC `a:A->bool`); | |
USE 0 (SPEC `a:A->bool`); | |
EXISTS_TAC `(x:(A->bool)->num) a`; | |
ASM_SIMP_TAC[]; | |
ASSUME_TAC NUM2_COUNTABLE; | |
USE 2 (REWRITE_RULE[COUNTABLE;GE_C;IN_ELIM_THM';IN_UNIV]); | |
USE 2 (CONV_RULE NAME_CONFLICT_CONV); | |
UND 2 THEN (DISCH_THEN (X_CHOOSE_TAC `h:num->(num#num)`)); | |
DISCH_TAC; | |
REWRITE_TAC[COUNTABLE;GE_C;IN_ELIM_THM';IN_UNIV;IN_UNIONS]; | |
EXISTS_TAC `(\p. (g:(A->bool)->num->A) ((f:num->(A->bool)) (FST ((h:num->(num#num)) p))) (SND (h p)))`; | |
BETA_TAC; | |
GEN_TAC; | |
DISCH_THEN (CHOOSE_THEN MP_TAC); | |
DISCH_ALL_TAC; | |
USE 3 (SPEC `t:A->bool`); | |
USE 3 (SPEC `y:A`); | |
UND 3 THEN (ASM_REWRITE_TAC[]); | |
REPEAT (DISCH_THEN(CHOOSE_THEN (MP_TAC))); | |
DISCH_ALL_TAC; | |
USE 2 (SPEC `(u:num,v:num)`); | |
SUBGOAL_TAC `?x' y'. (u:num,v:num) = (x',y')`; | |
MESON_TAC[]; | |
DISCH_TAC; | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
DISCH_THEN (CHOOSE_THEN (ASSUME_TAC o GSYM)); | |
EXISTS_TAC `x':num`; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let COUNTABLE_IMAGE = prove_by_refinement( | |
`!(A:A->bool) (B:B->bool) . (COUNTABLE A) /\ (?f. (B SUBSET IMAGE f A)) ==> | |
(COUNTABLE B)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[COUNTABLE;GE_C;IN_UNIV;IN_ELIM_THM';SUBSET]; | |
DISCH_ALL_TAC; | |
CHO 0; | |
USE 1 (REWRITE_RULE[IMAGE;IN_ELIM_THM']); | |
CHO 1; | |
USE 1 (REWRITE_RULE[IN_ELIM_THM']); | |
USE 1 (CONV_RULE NAME_CONFLICT_CONV); | |
EXISTS_TAC `(f':A->B) o (f:num->A)`; | |
REWRITE_TAC[o_DEF]; | |
DISCH_ALL_TAC; | |
USE 1 (SPEC `y:B`); | |
UND 1; | |
ASM_REWRITE_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
USE 0 (SPEC `x':A`); | |
UND 0 THEN (ASM_REWRITE_TAC[]) THEN DISCH_TAC; | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let COUNTABLE_CARD = prove_by_refinement( | |
`!(A:A->bool) (B:B->bool). (COUNTABLE A) /\ (A >=_c B) ==> | |
(COUNTABLE B)`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
MATCH_MP_TAC COUNTABLE_IMAGE; | |
EXISTS_TAC `A:A->bool`; | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC[IMAGE;SUBSET;IN_ELIM_THM']; | |
USE 1 (REWRITE_RULE[GE_C]); | |
CHO 1; | |
EXISTS_TAC `f:A->B`; | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let COUNTABLE_NUMSEG = prove_by_refinement( | |
`!n. COUNTABLE {x | x <| n}`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
REWRITE_TAC[COUNTABLE;GE_C;IN_UNIV]; | |
EXISTS_TAC `I:num->num`; | |
REDUCE_TAC; | |
REWRITE_TAC[IN_ELIM_THM']; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let FINITE_COUNTABLE = prove_by_refinement( | |
`!(A:A->bool). (FINITE A) ==> (COUNTABLE A)`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
USE 0 (MATCH_MP FINITE_HAS_SIZE_LEMMA); | |
CHO 0; | |
ASSUME_TAC(SPEC `n:num` COUNTABLE_NUMSEG); | |
JOIN 1 0; | |
USE 0 (MATCH_MP COUNTABLE_CARD); | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let num_infinite = prove_by_refinement( | |
`~ (FINITE (UNIV:num->bool))`, | |
(* {{{ proof *) | |
[ | |
PROOF_BY_CONTR_TAC; | |
USE 0 (REWRITE_RULE[]); | |
USE 0 (MATCH_MP num_FINITE_AVOID); | |
USE 0 (REWRITE_RULE[IN_UNIV]); | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let num_SEG_UNION = prove_by_refinement( | |
`!i. ({u | i <| u} UNION {m | m <=| i}) = UNIV`, | |
(* {{{ proof *) | |
[ | |
REP_BASIC_TAC; | |
SUBGOAL_TAC `({u | i <| u} UNION {m | m <=| i}) = UNIV`; | |
MATCH_MP_TAC EQ_EXT; | |
GEN_TAC; | |
REWRITE_TAC[UNIV;UNION;IN_ELIM_THM']; | |
ARITH_TAC; | |
REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let num_above_infinite = prove_by_refinement( | |
`!i. ~ (FINITE {u | i <| u})`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
PROOF_BY_CONTR_TAC; | |
USE 0 (REWRITE_RULE[]); | |
ASSUME_TAC(SPEC `i:num` FINITE_NUMSEG_LE); | |
JOIN 0 1; | |
USE 0 (MATCH_MP FINITE_UNION_IMP); | |
SUBGOAL_TAC `({u | i <| u} UNION {m | m <=| i}) = UNIV`; | |
REWRITE_TAC[num_SEG_UNION]; | |
DISCH_TAC; | |
UND 0; | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC[num_infinite]; | |
]);; | |
(* }}} *) | |
let INTER_FINITE = prove_by_refinement( | |
`!s (t:A->bool). (FINITE s ==> FINITE(s INTER t)) /\ (FINITE t ==> FINITE (s INTER t))`, | |
(* {{{ proof *) | |
[ | |
CONV_TAC (quant_right_CONV "t"); | |
CONV_TAC (quant_right_CONV "s"); | |
SUBCONJ_TAC; | |
DISCH_ALL_TAC; | |
SUBGOAL_TAC `s INTER t SUBSET (s:A->bool)`; | |
SET_TAC[]; | |
ASM_MESON_TAC[FINITE_SUBSET]; | |
MESON_TAC[INTER_COMM]; | |
]);; | |
(* }}} *) | |
let num_above_finite = prove_by_refinement( | |
`!i J. (FINITE (J INTER {u | (i <| u)})) ==> (FINITE J)`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
SUBGOAL_TAC `J = (J INTER {u | (i <| u)}) UNION (J INTER {m | m <=| i})`; | |
REWRITE_TAC[GSYM UNION_OVER_INTER;num_SEG_UNION;INTER_UNIV]; | |
DISCH_TAC; | |
ASM (ONCE_REWRITE_TAC)[]; | |
REWRITE_TAC[FINITE_UNION]; | |
ASM_REWRITE_TAC[]; | |
MP_TAC (SPEC `i:num` FINITE_NUMSEG_LE); | |
REWRITE_TAC[INTER_FINITE]; | |
]);; | |
(* }}} *) | |
let SUBSET_SUC = prove_by_refinement( | |
`!(f:num->A->bool). (!i. f i SUBSET f (SUC i)) ==> (! i j. ( i <=| j) ==> (f i SUBSET f j))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
DISCH_TAC; | |
REP_GEN_TAC; | |
MP_TAC (prove( `?n. n = j -| i`,MESON_TAC[])); | |
CONV_TAC (quant_left_CONV "n"); | |
SPEC_TAC (`i:num`,`i:num`); | |
SPEC_TAC (`j:num`,`j:num`); | |
REP 2( CONV_TAC (quant_left_CONV "n")); | |
INDUCT_TAC; | |
REP_GEN_TAC; | |
DISCH_ALL_TAC; | |
JOIN 1 2; | |
USE 1 (CONV_RULE REDUCE_CONV); | |
ASM_REWRITE_TAC[SUBSET]; | |
REP_GEN_TAC; | |
DISCH_TAC; | |
SUBGOAL_TAC `?j'. j = SUC j'`; | |
DISJ_CASES_TAC (SPEC `j:num` num_CASES); | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
REDUCE_TAC; | |
ASM_REWRITE_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
ASM_REWRITE_TAC[]; | |
USE 0 (SPEC `j':num`); | |
USE 1(SPECL [`j':num`;`i:num`]); | |
DISCH_TAC; | |
SUBGOAL_TAC `(n = j'-|i)`; | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
ARITH_TAC; | |
DISCH_TAC; | |
SUBGOAL_TAC `(i<=| j')`; | |
USE 2 (MATCH_MP(ARITH_RULE `(SUC n = j -| i) ==> (0 < j -| i)`)); | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
ARITH_TAC; | |
UND 1; | |
ASM_REWRITE_TAC []; | |
DISCH_ALL_TAC; | |
REWR 6; | |
ASM_MESON_TAC[SUBSET_TRANS]; | |
]);; | |
(* }}} *) | |
let SUBSET_SUC2 = prove_by_refinement( | |
`!(f:num->A->bool). (!i. f (SUC i) SUBSET (f i)) ==> (! i j. ( i <=| j) ==> (f j SUBSET f i))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
DISCH_TAC; | |
REP_GEN_TAC; | |
MP_TAC (prove( `?n. n = j -| i`,MESON_TAC[])); | |
CONV_TAC (quant_left_CONV "n"); | |
SPEC_TAC (`i:num`,`i:num`); | |
SPEC_TAC (`j:num`,`j:num`); | |
REP 2( CONV_TAC (quant_left_CONV "n")); | |
INDUCT_TAC; | |
REP_GEN_TAC; | |
DISCH_ALL_TAC; | |
JOIN 1 2; | |
USE 1 (CONV_RULE REDUCE_CONV); | |
ASM_REWRITE_TAC[SUBSET]; | |
REP_GEN_TAC; | |
DISCH_TAC; | |
SUBGOAL_TAC `?j'. j = SUC j'`; | |
DISJ_CASES_TAC (SPEC `j:num` num_CASES); | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
REDUCE_TAC; | |
ASM_REWRITE_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
ASM_REWRITE_TAC[]; | |
USE 0 (SPEC `j':num`); | |
USE 1(SPECL [`j':num`;`i:num`]); | |
DISCH_TAC; | |
SUBGOAL_TAC `(n = j'-|i)`; | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
ARITH_TAC; | |
DISCH_TAC; | |
SUBGOAL_TAC `(i<=| j')`; | |
USE 2 (MATCH_MP(ARITH_RULE `(SUC n = j -| i) ==> (0 < j -| i)`)); | |
UND 2; | |
ASM_REWRITE_TAC[]; | |
ARITH_TAC; | |
UND 1; | |
ASM_REWRITE_TAC []; | |
DISCH_ALL_TAC; | |
REWR 6; | |
ASM_MESON_TAC[SUBSET_TRANS]; | |
]);; | |
(* }}} *) | |
let INFINITE_PIGEONHOLE = prove_by_refinement( | |
`!I (f:A->B) B C. (~(FINITE {i | (I i) /\ (C (f i))})) /\ (FINITE B) /\ | |
(C SUBSET (UNIONS B)) ==> | |
(?b. (B b) /\ ~(FINITE {i | (I i) /\ (C INTER b) (f i) }))`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
PROOF_BY_CONTR_TAC; | |
USE 3 ( CONV_RULE (quant_left_CONV "b")); | |
UND 0; | |
TAUT_TAC `P ==> (~P ==> F)`; | |
SUBGOAL_TAC `{i | I' i /\ (C ((f:A->B) i))} = UNIONS (IMAGE (\b. {i | I' i /\ ((C INTER b) (f i))}) B)`; | |
REWRITE_TAC[UNIONS;IN_IMAGE]; | |
MATCH_MP_TAC EQ_EXT; | |
GEN_TAC; | |
REWRITE_TAC[IN_ELIM_THM']; | |
ABBREV_TAC `j = (x:A)`; | |
EQ_TAC; | |
DISCH_ALL_TAC; | |
USE 2 (REWRITE_RULE [SUBSET;UNIONS]); | |
USE 2 (REWRITE_RULE[IN_ELIM_THM']); | |
USE 2 (SPEC `(f:A->B) j`); | |
USE 2 (REWRITE_RULE[IN]); | |
REWR 2; | |
CHO 2; | |
CONV_TAC (quant_left_CONV "x"); | |
CONV_TAC (quant_left_CONV "x"); | |
EXISTS_TAC (`u:B->bool`); | |
NAME_CONFLICT_TAC; | |
EXISTS_TAC (`{i' | I' i' /\ (C INTER u) ((f:A->B) i')}`); | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC[IN_ELIM_THM';INTER]; | |
REWRITE_TAC[IN]; | |
ASM_REWRITE_TAC[]; | |
DISCH_TAC; | |
CHO 4; | |
AND 4; | |
CHO 5; | |
REWR 4; | |
USE 4 (REWRITE_RULE[IN_ELIM_THM';INTER]); | |
USE 4 (REWRITE_RULE[IN]); | |
ASM_REWRITE_TAC[]; | |
DISCH_TAC; | |
ASM_REWRITE_TAC[]; | |
SUBGOAL_TAC `FINITE (IMAGE (\b. {i | I' i /\ (C INTER b) ((f:A->B) i)}) B)`; | |
MATCH_MP_TAC FINITE_IMAGE; | |
ASM_REWRITE_TAC[]; | |
SIMP_TAC[FINITE_UNIONS]; | |
DISCH_TAC; | |
GEN_TAC; | |
REWRITE_TAC[IN_IMAGE]; | |
DISCH_THEN (X_CHOOSE_TAC `b:B->bool`); | |
ASM_REWRITE_TAC[]; | |
USE 3 (SPEC `b:B->bool`); | |
UND 3; | |
AND 5; | |
UND 3; | |
ABBREV_TAC `r = {i | I' i /\ (C INTER b) ((f:A->B) i)}`; | |
MESON_TAC[IN]; | |
]);; | |
(* }}} *) | |
let real_FINITE = prove_by_refinement( | |
`!(s:real->bool). FINITE s ==> (?a. !x. x IN s ==> (x <=. a))`, | |
(* {{{ proof *) | |
[ | |
DISCH_ALL_TAC; | |
ASSUME_TAC REAL_ARCH_SIMPLE; | |
USE 1 (CONV_RULE (quant_left_CONV "n")); | |
CHO 1; | |
SUBGOAL_TAC `FINITE (IMAGE (n:real->num) s)`; | |
ASM_MESON_TAC[FINITE_IMAGE]; | |
(*** JRH -- num_FINITE is now an equivalence not an implication | |
ASSUME_TAC (SPEC `IMAGE (n:real->num) s` num_FINITE); | |
***) | |
ASSUME_TAC(fst(EQ_IMP_RULE(SPEC `IMAGE (n:real->num) s` num_FINITE))); | |
DISCH_TAC; | |
REWR 2; | |
CHO 2; | |
USE 2 (REWRITE_RULE[IN_IMAGE]); | |
USE 2 (CONV_RULE NAME_CONFLICT_CONV); | |
EXISTS_TAC `&.a`; | |
GEN_TAC; | |
USE 2 (CONV_RULE (quant_left_CONV "x'")); | |
USE 2 (CONV_RULE (quant_left_CONV "x'")); | |
USE 2 (SPEC `x:real`); | |
USE 2 (SPEC `(n:real->num) x`); | |
DISCH_TAC; | |
REWR 2; | |
USE 1 (SPEC `x:real`); | |
UND 1; | |
MATCH_MP_TAC (REAL_ARITH `a<=b ==> ((x <= a) ==> (x <=. b))`); | |
REDUCE_TAC; | |
ASM_REWRITE_TAC []; | |
]);; | |
(* }}} *) | |
let UNIONS_DELETE = prove_by_refinement( | |
`!s. (UNIONS (s:(A->bool)->bool)) = (UNIONS (s DELETE (EMPTY)))`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[UNIONS;DELETE;EMPTY]; | |
GEN_TAC; | |
MATCH_MP_TAC EQ_EXT; | |
REWRITE_TAC[IN_ELIM_THM']; | |
GEN_TAC; | |
REWRITE_TAC[IN]; | |
MESON_TAC[]; | |
]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* Partial functions, which we identify with functions that | |
take the canonical choice of element outside the domain. *) | |
(* ------------------------------------------------------------------ *) | |
let SUPP = new_definition | |
`SUPP (f:A->B) = \ x. ~(f x = (CHOICE (UNIV:B ->bool)) )`;; | |
let FUN = new_definition | |
`FUN a b = (\ (f:A->B). ((!x. (x IN a) ==> (f x IN b)) /\ | |
((SUPP f) SUBSET a))) `;; | |
(* ------------------------------------------------------------------ *) | |
(* compositions *) | |
(* ------------------------------------------------------------------ *) | |
let compose = new_definition | |
`compose f g = \x. (f (g x))`;; | |
let COMP_ASSOC = prove_by_refinement( | |
`!(f:num ->num) (g:num->num) (h:num->num). | |
(compose f (compose g h)) = (compose (compose f g) h)`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC THEN REWRITE_TAC[compose]; | |
]);; | |
(* }}} *) | |
let COMP_INJ = prove (`!(f:A->B) (g:B->C) s t u. | |
INJ f s t /\ (INJ g t u) ==> | |
(INJ (compose g f) s u)`, | |
(* {{{ proof *) | |
EVERY[REPEAT GEN_TAC; | |
REWRITE_TAC[INJ;compose]; | |
DISCH_ALL_TAC; | |
ASM_MESON_TAC[]]);; | |
(* }}} *) | |
let COMP_SURJ = prove (`!(f:A->B) (g:B->C) s t u. | |
SURJ f s t /\ (SURJ g t u) ==> (SURJ (compose g f) s u)`, | |
(* {{{ proof *) | |
EVERY[REWRITE_TAC[SURJ;compose]; | |
DISCH_ALL_TAC; | |
ASM_MESON_TAC[]]);; | |
(* }}} *) | |
let COMP_BIJ = prove (`!(f:A->B) s t (g:B->C) u. | |
BIJ f s t /\ (BIJ g t u) ==> (BIJ (compose g f) s u)`, | |
(* {{{ proof *) | |
EVERY[ | |
REPEAT GEN_TAC; | |
REWRITE_TAC[BIJ]; | |
DISCH_ALL_TAC; | |
ASM_MESON_TAC[COMP_INJ;COMP_SURJ]]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* general construction of an inverse function on a domain *) | |
(* ------------------------------------------------------------------ *) | |
let INVERSE_FN = prove_by_refinement( | |
`?INV. (! (f:A->B) a b. (SURJ f a b) ==> ((INJ (INV f a b) b a) /\ | |
(!(x:B). (x IN b) ==> (f ((INV f a b) x) = x))))`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[GSYM SKOLEM_THM]; | |
REPEAT GEN_TAC; | |
MATCH_MP_TAC (prove_by_refinement( `!A B. (A ==> (?x. (B x))) ==> (?(x:B->A). (A ==> (B x)))`,[MESON_TAC[]])) ; | |
REWRITE_TAC[SURJ;INJ]; | |
DISCH_ALL_TAC; | |
SUBGOAL_TAC `?u. !y. ((y IN b)==> ((u y IN a) /\ ((f:A->B) (u y) = y)))`; | |
REWRITE_TAC[GSYM SKOLEM_THM]; | |
GEN_TAC; | |
ASM_MESON_TAC[]; | |
DISCH_THEN CHOOSE_TAC; | |
EXISTS_TAC `u:B->A`; | |
REPEAT CONJ_TAC; | |
ASM_MESON_TAC[]; | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (AP_TERM `f:A->B` th)); | |
ASM_MESON_TAC[]; | |
ASM_MESON_TAC[] | |
]);; | |
(* }}} *) | |
let INVERSE_DEF = new_specification ["INV"] INVERSE_FN;; | |
let INVERSE_BIJ = prove_by_refinement( | |
`!(f:A->B) a b. (BIJ f a b) ==> ((BIJ (INV f a b) b a))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
REWRITE_TAC[BIJ]; | |
DISCH_ALL_TAC; | |
ASM_SIMP_TAC[INVERSE_DEF]; | |
REWRITE_TAC[SURJ]; | |
CONJ_TAC; | |
ASM_MESON_TAC[INVERSE_DEF;INJ]; | |
GEN_TAC THEN DISCH_TAC; | |
EXISTS_TAC `(f:A->B) x`; | |
CONJ_TAC; | |
ASM_MESON_TAC[INJ]; | |
SUBGOAL_THEN `((f:A->B) x) IN b` ASSUME_TAC; | |
ASM_MESON_TAC[INJ]; | |
SUBGOAL_THEN `(f:A->B) (INV f a b (f x)) = (f x)` ASSUME_TAC; | |
ASM_MESON_TAC[INVERSE_DEF]; | |
H_UNDISCH_TAC (HYP "0"); | |
REWRITE_TAC[INJ]; | |
DISCH_ALL_TAC; | |
FIRST_X_ASSUM (fun th -> MP_TAC (SPECL [`INV (f:A->B) a b (f x)`;`x:A`] th)); | |
ASM_REWRITE_TAC[]; | |
DISCH_ALL_TAC; | |
SUBGOAL_THEN `INV (f:A->B) a b (f x) IN a` ASSUME_TAC; | |
ASM_MESON_TAC[INVERSE_DEF;INJ]; | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let INVERSE_XY = prove_by_refinement( | |
`!(f:A->B) a b x y. (BIJ f a b) /\ (x IN a) /\ (y IN b) ==> ((INV f a b y = x) <=> (f x = y))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
EQ_TAC; | |
FIRST_X_ASSUM (fun th -> (ASSUME_TAC th THEN (ASSUME_TAC (MATCH_MP INVERSE_DEF (CONJUNCT2 (REWRITE_RULE[BIJ] th)))))); | |
ASM_MESON_TAC[]; | |
POP_ASSUM (fun th -> (ASSUME_TAC th THEN (ASSUME_TAC (CONJUNCT2 (REWRITE_RULE[INJ] (CONJUNCT1 (REWRITE_RULE[BIJ] th))))))); | |
DISCH_THEN (fun th -> ASSUME_TAC th THEN (REWRITE_TAC[GSYM th])); | |
FIRST_X_ASSUM MATCH_MP_TAC; | |
REPEAT CONJ_TAC; | |
ASM_REWRITE_TAC[]; | |
IMP_RES_THEN ASSUME_TAC INVERSE_BIJ; | |
ASM_MESON_TAC[BIJ;INJ]; | |
ASM_REWRITE_TAC[]; | |
FIRST_X_ASSUM (fun th -> (ASSUME_TAC (CONJUNCT2 (REWRITE_RULE[BIJ] th)))); | |
IMP_RES_THEN (fun th -> ASSUME_TAC (CONJUNCT2 th)) INVERSE_DEF; | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
let FINITE_BIJ = prove( | |
`!a b (f:A->B). FINITE a /\ (BIJ f a b) ==> (FINITE b)`, | |
(* {{{ proof *) | |
MESON_TAC[SURJ_IMAGE;BIJ;INJ;FINITE_IMAGE] | |
);; | |
(* }}} *) | |
let FINITE_INJ = prove_by_refinement( | |
`!a b (f:A->B). FINITE b /\ (INJ f a b) ==> (FINITE a)`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
MP_TAC (SPECL [`f:A->B`;`b:B->bool`;`a:A->bool`] FINITE_IMAGE_INJ_GENERAL); | |
DISCH_ALL_TAC; | |
SUBGOAL_THEN `(a:A->bool) SUBSET ({x | (x IN a) /\ ((f:A->B) x IN b)})` ASSUME_TAC; | |
REWRITE_TAC[SUBSET]; | |
GEN_TAC ; | |
REWRITE_TAC[IN_ELIM_THM]; | |
POPL_TAC[0;1]; | |
ASM_MESON_TAC[BIJ;INJ]; | |
MATCH_MP_TAC FINITE_SUBSET; | |
EXISTS_TAC `({x | (x IN a) /\ ((f:A->B) x IN b)})` ; | |
CONJ_TAC; | |
FIRST_X_ASSUM (fun th -> MATCH_MP_TAC th); | |
CONJ_TAC; | |
ASM_MESON_TAC[BIJ;INJ]; | |
ASM_REWRITE_TAC[]; | |
ASM_REWRITE_TAC[]; | |
] | |
);; | |
(* }}} *) | |
let FINITE_BIJ2 = prove_by_refinement( | |
`!a b (f:A->B). FINITE b /\ (BIJ f a b) ==> (FINITE a)`, | |
(* {{{ proof *) | |
[ | |
MESON_TAC[BIJ;FINITE_INJ] | |
]);; | |
(* }}} *) | |
let BIJ_CARD = prove_by_refinement( | |
`!a b (f:A->B). FINITE a /\ (BIJ f a b) ==> (CARD a = (CARD b))`, | |
(* {{{ proof *) | |
[ | |
ASM_MESON_TAC[SURJ_IMAGE;BIJ;INJ;CARD_IMAGE_INJ]; | |
]);; | |
(* }}} *) | |
let PAIR_LEMMA = prove_by_refinement( | |
`!(x:num#num) i j. ((FST x = i) /\ (SND x = j)) <=> (x = (i,j))` , | |
(* {{{ proof *) | |
[ | |
MESON_TAC[FST;SND;PAIR]; | |
]);; | |
(* }}} *) | |
let CARD_SING = prove_by_refinement( | |
`!(u:A->bool). (SING u ) ==> (CARD u = 1)`, | |
(* {{{ proof *) | |
[ | |
REWRITE_TAC[SING]; | |
GEN_TAC; | |
DISCH_THEN (CHOOSE_TAC); | |
ASM_REWRITE_TAC[]; | |
ASSUME_TAC FINITE_RULES; | |
ASM_SIMP_TAC[CARD_CLAUSES;NOT_IN_EMPTY]; | |
ACCEPT_TAC (NUM_RED_CONV `SUC 0`) | |
]);; | |
(* }}} *) | |
let FINITE_SING = prove_by_refinement( | |
`!(x:A). FINITE ({x})`, | |
(* {{{ proof *) | |
[ | |
MESON_TAC[FINITE_RULES] | |
]);; | |
(* }}} *) | |
let NUM_INTRO = prove_by_refinement( | |
`!f P.((!(n:num). !(g:A). (f g = n) ==> (P g)) ==> (!g. (P g)))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
GEN_TAC; | |
H_VAL (SPECL [`(f:A->num) (g:A)`; `g:A`]) (HYP "0"); | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* Lemmas about the support of a function *) | |
(* ------------------------------------------------------------------ *) | |
(* Law of cardinal exponents B^0 = 1 *) | |
let DOMAIN_EMPTY = prove_by_refinement( | |
`!b. FUN (EMPTY:A->bool) b = { (\ (u:A). (CHOICE (UNIV:B->bool))) }`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
REWRITE_TAC[EXTENSION;FUN]; | |
X_GEN_TAC `f:A->B`; | |
REWRITE_TAC[IN_ELIM_THM;INSERT;NOT_IN_EMPTY;SUBSET_EMPTY;SUPP]; | |
REWRITE_TAC[EMPTY]; | |
ONCE_REWRITE_TAC[EXTENSION]; | |
REWRITE_TAC[IN]; | |
EQ_TAC; | |
DISCH_TAC THEN (MATCH_MP_TAC EQ_EXT); | |
BETA_TAC; | |
ASM_REWRITE_TAC[]; | |
DISCH_TAC THEN (ASM_REWRITE_TAC[]) THEN BETA_TAC; | |
]);; | |
(* }}} *) | |
(* Law of cardinal exponents B^A * B = B^(A+1) *) | |
let DOMAIN_INSERT = prove_by_refinement( | |
`!a b s. (~((s:A) IN a) ==> | |
(?F. (BIJ F (FUN (s INSERT a) b) | |
{ (u,v) | (u IN (FUN a b)) /\ ((v:B) IN b) } | |
)))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_TAC; | |
EXISTS_TAC `\ f. ((\ x. (if (x=(s:A)) then (CHOICE (UNIV:B->bool)) else (f x))),(f s))`; | |
REWRITE_TAC[BIJ;INJ;SURJ]; | |
TAUT_TAC `(A /\ (A ==> B) /\ (A ==>C)) ==> ((A/\ B) /\ (A /\ C))`; | |
REPEAT CONJ_TAC; | |
X_GEN_TAC `(f:A->B)`; | |
REWRITE_TAC[FUN;IN_ELIM_THM]; | |
REWRITE_TAC[INSERT;SUBSET]; | |
REWRITE_TAC[IN_ELIM_THM;SUPP]; | |
STRIP_TAC; | |
ABBREV_TAC `g = \ x. (if (x=(s:A)) then (CHOICE (UNIV:B->bool)) else (f x)) `; | |
EXISTS_TAC `g:A->B`; | |
EXISTS_TAC `(f:A->B) s`; | |
REWRITE_TAC[]; | |
REPEAT CONJ_TAC; | |
EXPAND_TAC "g" THEN BETA_TAC; | |
GEN_TAC; | |
REWRITE_TAC[IN;COND_ELIM_THM]; | |
ASM_MESON_TAC[IN]; | |
(* next *) ALL_TAC; | |
EXPAND_TAC "g" THEN BETA_TAC; | |
GEN_TAC; | |
ASM_CASES_TAC `(x:A) = s`; | |
ASM_REWRITE_TAC[]; | |
ASM_REWRITE_TAC[]; | |
ASM_MESON_TAC[]; | |
(* next *) ALL_TAC; | |
ASM_MESON_TAC[]; | |
(* INJ *) ALL_TAC; | |
REWRITE_TAC[FUN;SUPP]; | |
DISCH_TAC; | |
X_GEN_TAC `f1:A->B`; | |
X_GEN_TAC `f2:A->B`; | |
REWRITE_TAC[IN]; | |
DISCH_ALL_TAC; | |
MATCH_MP_TAC EQ_EXT; | |
GEN_TAC; | |
ASM_CASES_TAC `(x:A) = s`; | |
POPL_TAC[1;2;3;4;6;7]; | |
ASM_REWRITE_TAC[]; | |
ASM_MESON_TAC[PAIR;FST;SND]; | |
POPL_TAC[1;2;3;4;6;7]; | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (REWRITE_RULE[FST] (AP_TERM `FST:((A->B)#B)->(A->B)` th))) ; | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (REWRITE_RULE[COND_ELIM_THM] (BETA_RULE (AP_THM th `x:A`)))); | |
LABEL_ALL_TAC; | |
H_UNDISCH_TAC (HYP "0"); | |
COND_CASES_TAC; | |
ASM_MESON_TAC[]; | |
ASM_MESON_TAC[]; | |
(* SURJ *) ALL_TAC; | |
REWRITE_TAC[FUN;SUPP;IN_ELIM_THM]; | |
REWRITE_TAC[IN;INSERT;SUBSET]; | |
DISCH_ALL_TAC; | |
X_GEN_TAC `p:(A->B)#B`; | |
DISCH_THEN CHOOSE_TAC; | |
FIRST_X_ASSUM (fun th -> MP_TAC th); | |
DISCH_THEN CHOOSE_TAC; | |
FIRST_X_ASSUM MP_TAC; | |
DISCH_ALL_TAC; | |
ASM_REWRITE_TAC[]; | |
EXISTS_TAC `\ (x:A). if (x = s) then (v:B) else (u x)`; | |
REPEAT CONJ_TAC; | |
X_GEN_TAC `t:A`; | |
BETA_TAC; | |
REWRITE_TAC[IN_ELIM_THM;COND_ELIM_THM]; | |
POPL_TAC[1;3;4;5]; | |
ASM_MESON_TAC[]; | |
X_GEN_TAC `t:A`; | |
BETA_TAC; | |
REWRITE_TAC[IN_ELIM_THM;COND_ELIM_THM]; | |
ASM_CASES_TAC `(t:A) = s`; | |
POPL_TAC[1;3;4;5;6]; | |
ASM_REWRITE_TAC[]; | |
POPL_TAC[1;3;4;5;6]; | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (SPEC `t:A` th)); | |
ASM_SIMP_TAC[prove(`~((t:A)=s) ==> ((t=s)=F)`,MESON_TAC[])]; | |
BETA_TAC; | |
REWRITE_TAC[]; | |
POPL_TAC[0;2;3;4]; | |
AP_THM_TAC; | |
AP_TERM_TAC; | |
MATCH_MP_TAC EQ_EXT; | |
X_GEN_TAC `t:A`; | |
BETA_TAC; | |
DISJ_CASES_TAC (prove(`(((t:A)=s) <=> T) \/ ((t=s) <=> F)`,MESON_TAC[])); | |
ASM_REWRITE_TAC[]; | |
ASM_MESON_TAC[IN]; | |
ASM_REWRITE_TAC[] | |
]);; | |
(* }}} *) | |
let CARD_DELETE_CHOICE = prove_by_refinement( | |
`!(a:(A->bool)). ((FINITE a) /\ (~(a=EMPTY))) ==> | |
(SUC (CARD (a DELETE (CHOICE a))) = (CARD a))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
ASM_SIMP_TAC[CARD_DELETE]; | |
ASM_SIMP_TAC[CHOICE_DEF]; | |
MATCH_MP_TAC (ARITH_RULE `~(x=0) ==> (SUC (x -| 1) = x)`); | |
ASM_MESON_TAC[HAS_SIZE_0;HAS_SIZE]; | |
]);; | |
(* }}} *) | |
(* | |
let dets_flag = ref true;; | |
dets_flag:= !labels_flag;; | |
*) | |
labels_flag:=false;; | |
(* Law of cardinals |B^A| = |B|^|A| *) | |
let FUN_SIZE = prove_by_refinement( | |
`!b a. (FINITE (a:A->bool)) /\ (FINITE (b:B->bool)) | |
==> ((FUN a b) HAS_SIZE ((CARD b) EXP (CARD a)))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
MATCH_MP_TAC (SPEC `CARD:(A->bool)->num` ((INST_TYPE) [`:A->bool`,`:A`] NUM_INTRO)); | |
INDUCT_TAC; | |
GEN_TAC; | |
DISCH_ALL_TAC; | |
ASM_REWRITE_TAC[]; | |
REWRITE_TAC [EXP]; | |
SUBGOAL_THEN `(a:A->bool) = EMPTY` ASSUME_TAC; | |
ASM_REWRITE_TAC[GSYM HAS_SIZE_0;HAS_SIZE]; | |
ASM_REWRITE_TAC[HAS_SIZE;DOMAIN_EMPTY]; | |
CONJ_TAC; | |
REWRITE_TAC[FINITE_SING]; | |
MATCH_MP_TAC CARD_SING; | |
REWRITE_TAC[SING]; | |
MESON_TAC[]; | |
GEN_TAC; | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (SPEC `(a:A->bool) DELETE (CHOICE a)` th)) ; | |
DISCH_ALL_TAC; | |
SUBGOAL_THEN `CARD ((a:A->bool) DELETE (CHOICE a)) = n` ASSUME_TAC; | |
ASM_SIMP_TAC[CARD_DELETE]; | |
SUBGOAL_THEN `CHOICE (a:A->bool) IN a` ASSUME_TAC; | |
MATCH_MP_TAC CHOICE_DEF; | |
ASSUME_TAC( ARITH_RULE `!x. (x = (SUC n)) ==> (~(x = 0))`); | |
REWRITE_TAC[GSYM HAS_SIZE_0;HAS_SIZE]; | |
ASM_MESON_TAC[]; | |
ASM_REWRITE_TAC[]; | |
MESON_TAC[ ( ARITH_RULE `!n. (SUC n -| 1) = n`)]; | |
LABEL_ALL_TAC; | |
H_MATCH_MP (HYP "3") (HYP "4"); | |
SUBGOAL_THEN `FUN ((a:A->bool) DELETE CHOICE a) (b:B->bool) HAS_SIZE CARD b **| CARD (a DELETE CHOICE a)` ASSUME_TAC; | |
ASM_MESON_TAC[FINITE_DELETE]; | |
ASSUME_TAC (SPECL [`((a:A->bool) DELETE (CHOICE a))`;`b:B->bool`;`(CHOICE (a:A->bool))` ] DOMAIN_INSERT); | |
LABEL_ALL_TAC; | |
H_UNDISCH_TAC (HYP "5"); | |
REWRITE_TAC[IN_DELETE]; | |
SUBGOAL_THEN `~((a:A->bool) = EMPTY)` ASSUME_TAC; | |
REWRITE_TAC[GSYM HAS_SIZE_0;HAS_SIZE]; | |
ASSUME_TAC( ARITH_RULE `!x. (x = (SUC n)) ==> (~(x = 0))`); | |
ASM_MESON_TAC[]; | |
ASM_SIMP_TAC[INSERT_DELETE;CHOICE_DEF]; | |
DISCH_THEN CHOOSE_TAC; | |
REWRITE_TAC[HAS_SIZE]; | |
SUBGOAL_THEN `FINITE (FUN (a:A->bool) (b:B->bool))` ASSUME_TAC; | |
(* CONJ_TAC; *) ALL_TAC; | |
MATCH_MP_TAC (SPEC `FUN (a:A->bool) (b:B->bool)` (PINST[(`:A->B`,`:A`);(`:(A->B)#B`,`:B`)] [] FINITE_BIJ2)); | |
EXISTS_TAC `{u,v | (u:A->B) IN FUN (a DELETE CHOICE a) b /\ (v:B) IN b}`; | |
EXISTS_TAC `F':(A->B)->((A->B)#B)`; | |
ASM_REWRITE_TAC[]; | |
MATCH_MP_TAC FINITE_PRODUCT; | |
ASM_REWRITE_TAC[]; | |
ASM_MESON_TAC[HAS_SIZE]; | |
ASM_REWRITE_TAC[]; | |
SUBGOAL_THEN `CARD (FUN (a:A->bool) (b:B->bool)) = (CARD {u,v | (u:A->B) IN FUN (a DELETE CHOICE a) b /\ (v:B) IN b})` ASSUME_TAC; | |
MATCH_MP_TAC BIJ_CARD; | |
EXISTS_TAC `F':(A->B)->((A->B)#B)`; | |
ASM_REWRITE_TAC[]; | |
(* *) ALL_TAC; | |
ASM_REWRITE_TAC[]; | |
SUBGOAL_THEN `FINITE (a DELETE CHOICE (a:A->bool))` ASSUME_TAC; | |
ASM_MESON_TAC[FINITE_DELETE]; | |
SUBGOAL_THEN `(FUN ((a:A->bool) DELETE CHOICE a) (b:B->bool)) HAS_SIZE (CARD b **| (CARD (a DELETE CHOICE a)))` ASSUME_TAC; | |
POPL_TAC[1;2;3;4;5;10;11]; | |
ASM_MESON_TAC[CARD_DELETE]; | |
POP_ASSUM (fun th -> ASSUME_TAC (REWRITE_RULE[HAS_SIZE] th) THEN (ASSUME_TAC th)); | |
ASM_SIMP_TAC[CARD_PRODUCT]; | |
REWRITE_TAC[EXP;MULT_AC] | |
]);; | |
(* }}} *) | |
labels_flag:= true;; | |
(* ------------------------------------------------------------------ *) | |
(* ------------------------------------------------------------------ *) | |
(* Definitions in math tend to be n-tuples of data. Let's make it | |
easy to pick out the individual components of a definition *) | |
(* pick out the rest of n-tuples. Indexing consistent with lib.drop *) | |
let drop0 = new_definition(`drop0 (u:A#B) = SND u`);; | |
let drop1 = new_definition(`drop1 (u:A#B#C) = SND (SND u)`);; | |
let drop2 = new_definition(`drop2 (u:A#B#C#D) = SND (SND (SND u))`);; | |
let drop3 = new_definition(`drop3 (u:A#B#C#D#E) = SND (SND (SND (SND u)))`);; | |
(* pick out parts of n-tuples *) | |
let part0 = new_definition(`part0 (u:A#B) = FST u`);; | |
let part1 = new_definition(`part1 (u:A#B#C) = FST (drop0 u)`);; | |
let part2 = new_definition(`part2 (u:A#B#C#D) = FST (drop1 u)`);; | |
let part3 = new_definition(`part3 (u:A#B#C#D#E) = FST (drop2 u)`);; | |
let part4 = new_definition(`part4 (u:A#B#C#D#E#F) = FST (drop3 u)`);; | |
let part5 = new_definition(`part5 (u:A#B#C#D#E#F#G) = | |
FST (SND (SND (SND (SND (SND u)))))`);; | |
let part6 = new_definition(`part6 (u:A#B#C#D#E#F#G#H) = | |
FST (SND (SND (SND (SND (SND (SND u))))))`);; | |
let part7 = new_definition(`part7 (u:A#B#C#D#E#F#G#H#I) = | |
FST (SND (SND (SND (SND (SND (SND (SND u)))))))`);; | |
(* ------------------------------------------------------------------ *) | |
(* Basic Definitions of Euclidean Space, Metric Spaces, and Topology *) | |
(* ------------------------------------------------------------------ *) | |
(* ------------------------------------------------------------------ *) | |
(* Interface *) | |
(* ------------------------------------------------------------------ *) | |
let euclid_def = local_definition "euclid";; | |
mk_local_interface "euclid";; | |
overload_interface | |
("+", `euclid'euclid_plus:(num->real)->(num->real)->(num->real)`);; | |
make_overloadable "*#" `:A -> B -> B`;; | |
let euclid_scale = euclid_def | |
`euclid_scale t f = \ (i:num). (t*. (f i))`;; | |
overload_interface ("*#",`euclid'euclid_scale`);; | |
parse_as_infix("*#",(20,"right"));; | |
let euclid_neg = euclid_def `euclid_neg f = \ (i:num). (--. (f i))`;; | |
(* This is highly ambiguous: -- f x can be read as | |
(-- f) x or as -- (f x). *) | |
overload_interface ("--",`euclid'euclid_neg`);; | |
overload_interface | |
("-", `euclid'euclid_minus:(num->real)->(num->real)->(num->real)`);; | |
(* ------------------------------------------------------------------ *) | |
(* Euclidean Space *) | |
(* ------------------------------------------------------------------ *) | |
let euclid_plus = euclid_def | |
`euclid_plus f g = \ (i:num). (f i) +. (g i)`;; | |
let euclid = euclid_def `euclid n v <=> !m. (n <=| m) ==> (v m = &.0)`;; | |
let euclidean = euclid_def `euclidean v <=> ?n. euclid n v`;; | |
let euclid_minus = euclid_def | |
`euclid_minus f g = \(i:num). (f i) -. (g i)`;; | |
let euclid0 = euclid_def `euclid0 = \(i:num). &.0`;; | |
let coord = euclid_def `coord i (f:num->real) = f i`;; | |
let dot = euclid_def `dot f g = | |
let (n = (min_num (\m. (euclid m f) /\ (euclid m g)))) in | |
sum (0,n) (\i. (f i)*(g i))`;; | |
let norm = euclid_def `norm f = sqrt(dot f f)`;; | |
let d_euclid = euclid_def `d_euclid f g = norm (f - g)`;; | |
(* ------------------------------------------------------------------ *) | |
(* Euclidean and Convex geometry *) | |
(* ------------------------------------------------------------------ *) | |
let sum_vector_EXISTS = prove_by_refinement( | |
`?sum_vector. (!f n. sum_vector(n,0) f = (\n. &.0)) /\ | |
(!f m n. sum_vector(n,SUC m) f = sum_vector(n,m) f + f(n + m))`, | |
(* {{{ proof *) | |
[ | |
(CHOOSE_TAC o prove_recursive_functions_exist num_RECURSION) `(!f n. sm n 0 f = (\n. &0)) /\ (!f m n. sm n (SUC m) f = sm n m f + f(n + m))`; | |
EXISTS_TAC `\(n,m) f. (sm:num->num->(num->(num->real))->(num->real)) n m f`; | |
CONV_TAC(DEPTH_CONV GEN_BETA_CONV); | |
ASM_REWRITE_TAC[]; | |
]);; | |
(* }}} *) | |
let sum_vector = new_specification ["sum_vector"] sum_vector_EXISTS;; | |
let mk_segment = euclid_def | |
`mk_segment x y = { u | ?a. (&.0 <=. a) /\ (a <=. &.1) /\ | |
(u = a *# x + (&.1 - a) *# y) }`;; | |
let mk_open_segment = euclid_def | |
`mk_open_segment x y = { u | ?a. (&.0 <. a) /\ (a <. &.1) /\ | |
(u = a *# x + (&.1 - a) *# y) }`;; | |
let convex = euclid_def | |
`convex S <=> !x y. (S x) /\ (S y) ==> (mk_segment x y SUBSET S)`;; | |
let convex_hull = euclid_def | |
`convex_hull S = { u | ?f alpha m. (!n. (n< m) ==> (S (f n))) /\ | |
(sum(0,m) alpha = &.1) /\ (!n. (n< m) ==> (&.0 <=. (alpha n))) /\ | |
(u = sum_vector(0,m) (\n. (alpha n) *# (f n)))}`;; | |
let affine_hull = euclid_def | |
`affine_hull S = { u | ?f alpha m. (!n. (n< m) ==> (S (f n))) /\ | |
(sum(0,m) alpha = &.1) /\ | |
(u = sum_vector(0,m) (\n. (alpha n) *# (f n)))}`;; | |
let mk_line = euclid_def `mk_line x y = | |
{z| ?t. (z = (t *# x) + ((&.1 - t) *# y)) }`;; | |
let affine = euclid_def | |
`affine S <=> !x y. (S x ) /\ (S y) ==> (mk_line x y SUBSET S)`;; | |
let affine_dim = euclid_def | |
`affine_dim n S <=> | |
(?T. (T HAS_SIZE (SUC n)) /\ (affine_hull T = affine_hull S)) /\ | |
(!T m. (T HAS_SIZE (SUC m)) /\ (m < n) ==> ~(affine_hull T = affine_hull S))`;; | |
let collinear = euclid_def | |
`collinear S <=> (?n. affine_dim n S /\ (n < 2))`;; | |
let coplanar = euclid_def | |
`coplanar S <=> (?n. affine_dim n S /\ (n < 3))`;; | |
let line = euclid_def | |
`line L <=> (affine L) /\ (affine_dim 1 L)`;; | |
let plane = euclid_def | |
`plane P <=> (affine P) /\ (affine_dim 2 P)`;; | |
let space = euclid_def | |
`space R <=> (affine R) /\ (affine_dim 3 R)`;; | |
(* | |
General constructor of conical objects, including | |
rays, cones, half-planes, etc. | |
L is the edge. C is the set of generators in the positive | |
direction. | |
If L is a line, and C = {c}, we get the half-plane bounded by | |
L and containing c. | |
If L is a point, and C is general, we get the cone at L generated | |
by C. | |
If L and C are both singletons, we get the ray ending at L. | |
*) | |
let mk_open_half_set = euclid_def | |
`mk_open_half_set L S = | |
{ u | ?t v c. (L v) /\ (S c) /\ (&.0 < t) /\ | |
(u = (t *# (c - v) + (&.1 - t) *# v)) }`;; | |
let mk_half_set = euclid_def | |
`mk_half_set L S = | |
{ u | ?t v c. (L v) /\ (S c) /\ (&.0 <=. t) /\ | |
(u = (t *# (c - v) + (&.1 - t) *# v)) }`;; | |
let mk_angle = euclid_def `mk_angle x y z = | |
(mk_half_set {x} {y}) UNION (mk_half_set {x} {z})`;; | |
let mk_signed_angle = euclid_def `mk_signed_angle x y z = | |
(mk_half_set {x} {y} , mk_half_set {x} {z})`;; | |
let mk_convex_cone = euclid_def | |
`mk_convex_cone v (S:(num->real)->bool) = | |
mk_half_set {v} (convex_hull S)`;; | |
(* we always normalize the radius of balls in a packing to 1 *) | |
let packing = euclid_def(`packing (S:(num->real)->bool) <=> | |
!x y. ( ((S x) /\ (S y) /\ ((d_euclid x y) < (&.2))) ==> | |
(x = y))`);; | |
let saturated_packing = euclid_def(`saturated_packing S <=> | |
(( packing S) /\ | |
(!z. (affine_hull S z) ==> | |
(?x. ((S x) /\ ((d_euclid x z) < (&.2))))))`);; | |
(* 3 dimensions specific: *) | |
let cross_product3 = euclid_def(`cross_product3 v1 v2 = | |
let (x1 = v1 0) and (x2 = v1 1) and (x3 = v1 2) in | |
let (y1 = v2 0) and (y2 = v2 1) and (y3 = v2 2) in | |
(\k. | |
(if (k=0) then (x2*y3-x3*y2) | |
else if (k=1) then (x3*y1-x1*y3) | |
else if (k=2) then (x1*y2-x2*y1) | |
else (&0)))`);; | |
let triple_product = euclid_def(`triple_product v1 v2 v3 = | |
dot v1 (cross_product3 v2 v3)`);; | |
(* the bounding edge *) | |
let mk_triangle = euclid_def `mk_triangle v1 v2 v3 = | |
(mk_segment v1 v2) UNION (mk_segment v2 v3) UNION (mk_segment v3 v1)`;; | |
(* the interior *) | |
let mk_interior_triangle = euclid_def | |
`mk_interior_triangle v1 v2 v3 = | |
mk_open_half_set (mk_line v1 v2) {v3} INTER | |
(mk_open_half_set (mk_line v2 v3) {v1}) INTER | |
(mk_open_half_set (mk_line v3 v1) {v2})`;; | |
let mk_triangular_region = euclid_def | |
`mk_triangular_region v1 v2 v3 = | |
(mk_triangle v1 v2 v3) UNION (mk_interior_triangle v1 v2 v3)`;; | |
(* ------------------------------------------------------------------ *) | |
(* Statements of Theorems in Euclidean Geometry (no proofs *) | |
(* ------------------------------------------------------------------ *) | |
let half_set_convex = `!L S. convex (mk_half_set L S)`;; | |
let open_half_set_convex = `!L S . convex (mk_open_half_set L S )`;; | |
let affine_dim0 = `!S. (affine_dim 0 S) = (SING S)`;; | |
let hull_convex = `!S. (convex (convex_hull S))`;; | |
let hull_minimal = `!S T. (convex T) /\ (S SUBSET T) ==> | |
(convex_hull S) SUBSET T`;; | |
let affine_hull_affine = `!S. (affine (affine_hull S))`;; | |
let affine_hull_minimal = `!S T. (affine T) /\ (S SUBSET T) ==> | |
(affine_hull S) SUBSET T`;; | |
let mk_line_dim = `!x y. ~(x = y) ==> affine_dim 1 (mk_line x y)`;; | |
let affine_convex_hull = `!S. (affine_hull S) = (affine_hull (convex_hull S))`;; | |
let convex_hull_hull = `!S. (convex_hull S) = (convex_hull (convex_hull S))`;; | |
let euclid_affine_dim = `!n. affine_dim n (euclid n)`;; | |
let affine_dim_subset = `!m n T S. | |
(affine_dim m T) /\ (affine_dim n S) /\ (T SUBSET S) ==> (m <= n)`;; | |
(* A few of the Birkhoff postulates of Geometry (incomplete) *) | |
let line_postulate = `!x y. ~(x = y) ==> | |
(?!L. (L x) /\ (L y) /\ (line L))`;; | |
let ruler_postulate = `!L. (line L) ==> | |
(?f. (BIJ f L UNIV) /\ | |
(!x y. (L x /\ L y ==> (d_euclid x y = abs(f x -. f y)))))`;; | |
let affine_postulate = `!n. (affine_dim n P) ==> (?S. | |
(S SUBSET P) /\ (S HAS_SIZE n) /\ (affine_dim n S))`;; | |
let line_plane = `!P x y. (plane P) /\ (P x) /\ (P y) ==> | |
(mk_line x y SUBSET P)`;; | |
let plane_of_pt = `!S. (S HAS_SIZE 3) ==> (?P. (plane P) /\ | |
(S SUBSET P))`;; | |
let plane_of_pt_unique = `!S. (S HAS_SIZE 3) ==> (collinear S) \/ | |
(?! P. (plane P) /\ (S SUBSET P))`;; | |
let plane_inter = `!P Q. (plane P) /\ (plane Q) ==> | |
(P INTER Q = EMPTY) \/ (line (P INTER Q)) \/ (P = Q)`;; | |
(* each line separates a plane into two half-planes *) | |
let plane_separation = | |
`!P L. (plane P) /\ (line L) /\ (L SUBSET P) ==> | |
(?A B. (A INTER B = EMPTY) /\ (A INTER L = EMPTY) /\ | |
(B INTER L = EMPTY) /\ (L UNION A UNION B = P) /\ | |
(!c u. (P c) /\ (u = mk_open_half_set L {c}) ==> | |
(u = A) \/ (u = B) \/ (u = L)) /\ | |
(!a b. (A a) /\ (B b) ==> ~(segment a b INTER L = EMPTY)))`;; | |
let space_separation = | |
`!R P. (space R) /\ (plane P) /\ (P SUBSET R) ==> | |
(?A B. (A INTER B = EMRTY) /\ (A INTER P = EMRTY) /\ | |
(B INTER P = EMRTY) /\ (P UNION A UNION B = R) /\ | |
(!c u. (R c) /\ (u = mk_open_half_set P {c}) ==> | |
(u = A) \/ (u = B) \/ (u = P)) /\ | |
(!a b. (A a) /\ (B b) ==> ~(segment a b INTER L = EMPTY)))`;; | |
(* ------------------------------------------------------------------ *) | |
(* Metric Space *) | |
(* ------------------------------------------------------------------ *) | |
let metric_space = euclid_def `metric_space (X:A->bool,d:A->A->real) | |
<=> | |
!x y z. | |
(X x) /\ (X y) /\ (X z) ==> | |
(((&.0) <=. (d x y)) /\ | |
((&.0 = d x y) = (x = y)) /\ | |
(d x y = d y x) /\ | |
(d x z <=. d x y + d y z))`;; | |
(* ------------------------------------------------------------------ *) | |
(* Measure *) | |
(* ------------------------------------------------------------------ *) | |
let set_translate = euclid_def | |
`set_translate v X = { z | ?x. (X x) /\ (z = v + x) }`;; | |
let set_scale = euclid_def | |
`set_scale r X = { z | ?x. (X x) /\ (z = r *# x) }`;; | |
let mk_rectangle = euclid_def | |
`mk_rectangle a b = { z | !(i:num). (a i <=. z i) /\ (z i <. b i) }`;; | |
let one_vec = euclid_def | |
`one_vec n = (\i. if (i<| n) then (&.1) else (&.0))`;; | |
let mk_cube = euclid_def | |
`mk_cube n k v = | |
let (r = twopow (--: (&: k))) in | |
let (vv = (\i. (real_of_int (v i)))) in | |
mk_rectangle (r *# vv) (r *# (vv + (one_vec n)))`;; | |
let inner_cube = euclid_def | |
`inner_cube n k A = | |
{ v | (mk_cube n k v SUBSET A) /\ | |
(!i. (n <| i) ==> (&:0 = v i)) }`;; | |
let outer_cube = euclid_def | |
`outer_cube n k A = | |
{ v | ~((mk_cube n k v) INTER A = EMPTY) /\ | |
(!i. (n <| i) ==> (&:0 = v i)) }`;; | |
let inner_vol = euclid_def | |
`inner_vol n k A = | |
(&. (CARD (inner_cube n k A)))*(twopow (--: (&: (n*k))))`;; | |
let outer_vol = euclid_def | |
`outer_vol n k A = | |
(&. (CARD (outer_cube n k A)))*(twopow (--: (&: (n*k))))`;; | |
let euclid_bounded = euclid_def | |
`euclid_bounded A = (?R. !(x:num->real) i. (A x) ==> (x i <. R))`;; | |
let vol = euclid_def | |
`vol n A = lim (\k. outer_vol n k A)`;; | |
(* ------------------------------------------------------------------ *) | |
(* COMPUTING PI *) | |
(* ------------------------------------------------------------------ *) | |
unambiguous_interface();; | |
prioritize_real();; | |
(* ------------------------------------------------------------------ *) | |
(* general series approximations *) | |
(* ------------------------------------------------------------------ *) | |
let SER_APPROX1 = prove_by_refinement( | |
`!s f g. (f sums s) /\ (summable g) ==> | |
(!k. ((!n. (||. (f (n+k)) <=. (g (n+k)))) ==> | |
( (s - (sum(0,k) f)) <=. (suminf (\n. (g (n +| k)))))))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
GEN_TAC; | |
DISCH_TAC; | |
IMP_RES_THEN ASSUME_TAC SUM_SUMMABLE; | |
IMP_RES_THEN (fun th -> (ASSUME_TAC (SPEC `k:num` th))) SER_OFFSET; | |
IMP_RES_THEN ASSUME_TAC SUM_UNIQ; | |
SUBGOAL_THEN `(\n. (f (n+ k))) sums (s - (sum(0,k) f))` ASSUME_TAC; | |
ASM_MESON_TAC[]; | |
SUBGOAL_THEN `summable (\n. (f (n+k))) /\ (suminf (\n. (f (n+k))) <=. (suminf (\n. (g (n+k)))))` ASSUME_TAC; | |
MATCH_MP_TAC SER_LE2; | |
BETA_TAC; | |
ASM_REWRITE_TAC[]; | |
IMP_RES_THEN ASSUME_TAC SER_OFFSET; | |
FIRST_X_ASSUM (fun th -> ACCEPT_TAC (MATCH_MP SUM_SUMMABLE (((SPEC `k:num`) th)))); | |
ASM_MESON_TAC[SUM_UNIQ] | |
]);; | |
(* }}} *) | |
let SER_APPROX = prove_by_refinement( | |
`!s f g. (f sums s) /\ (!n. (||. (f n) <=. (g n))) /\ | |
(summable g) ==> | |
(!k. (abs (s - (sum(0,k) f)) <=. (suminf (\n. (g (n +| k))))))`, | |
(* {{{ proof *) | |
[ | |
REPEAT GEN_TAC; | |
DISCH_ALL_TAC; | |
GEN_TAC; | |
REWRITE_TAC[REAL_ABS_BOUNDS]; | |
CONJ_TAC; | |
SUBGOAL_THEN `(!k. ((!n. (||. ((\p. (--. (f p))) (n+k))) <=. (g (n+k)))) ==> ((--.s) - (sum(0,k) (\p. (--. (f p)))) <=. (suminf (\n. (g (n +k))))))` ASSUME_TAC; | |
MATCH_MP_TAC SER_APPROX1; | |
ASM_REWRITE_TAC[]; | |
MATCH_MP_TAC SER_NEG ; | |
ASM_REWRITE_TAC[]; | |
MATCH_MP_TAC (REAL_ARITH (`(--. s -. (--. u) <=. x) ==> (--. x <=. (s -. u))`)); | |
ONCE_REWRITE_TAC[GSYM SUM_NEG]; | |
FIRST_X_ASSUM (fun th -> (MATCH_MP_TAC th)); | |
BETA_TAC; | |
ASM_REWRITE_TAC[REAL_ABS_NEG]; | |
H_VAL2 CONJ (HYP "0") (HYP "2"); | |
IMP_RES_THEN MATCH_MP_TAC SER_APPROX1 ; | |
GEN_TAC; | |
ASM_MESON_TAC[]; | |
]);; | |
(* }}} *) | |
(* ------------------------------------------------------------------ *) | |
(* now for pi calculation stuff *) | |
(* ------------------------------------------------------------------ *) | |
let local_def = local_definition "trig";; | |
let PI_EST = prove_by_refinement( | |
`!n. (1 <=| n) ==> (abs(&4 / &(8 * n + 1) - | |
&2 / &(8 * n + 4) - | |
&1 / &(8 * n + 5) - | |
&1 / &(8 * n + 6)) <= &.622/(&.819))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN DISCH_ALL_TAC; | |
REWRITE_TAC[real_div]; | |
MATCH_MP_TAC (REWRITE_RULE[real_div] (REWRITE_RULE[REAL_RAT_REDUCE_CONV `(&.4/(&.9) +(&.2/(&.12)) + (&.1/(&.13))+ (&.1/(&.14)))`] (REAL_ARITH `(abs((&.4)*.u)<=. (&.4)/(&.9)) /\ (abs((&.2)*.v)<=. (&.2)/(&.12)) /\ (abs((&.1)*w) <=. (&.1)/(&.13)) /\ (abs((&.1)*x) <=. (&.1)/(&.14)) ==> (abs((&.4)*u -(&.2)*v - (&.1)*w - (&.1)*x) <= (&.4/(&.9) +(&.2/(&.12)) + (&.1/(&.13))+ (&.1/(&.14))))`))); | |
IMP_RES_THEN ASSUME_TAC (ARITH_RULE `1 <=| n ==> (0 < n)`); | |
FIRST_X_ASSUM (fun th -> ASSUME_TAC (REWRITE_RULE[GSYM REAL_OF_NUM_LT] th)); | |
ASSUME_TAC (prove(`(a<=.b) ==> (&.n*a <=. (&.n)*b)`,MESON_TAC[REAL_PROP_LE_LMUL;REAL_POS])); | |
REWRITE_TAC[REAL_ABS_MUL;REAL_ABS_INV;prove(`||.(&.n) = (&.n)`,MESON_TAC[REAL_POS;REAL_ABS_REFL])]; | |
REPEAT CONJ_TAC THEN (POP_ASSUM (fun th -> MATCH_MP_TAC th)) THEN (MATCH_MP_TAC (prove(`((&.0 <. (&.n)) /\ (&.n <=. a)) ==> (inv(a)<=. (inv(&.n)))`,MESON_TAC[REAL_ABS_REFL;REAL_ABS_INV;REAL_LE_INV2]))) THEN | |
REWRITE_TAC[REAL_LT;REAL_LE] THEN (H_UNDISCH_TAC (HYP"0")) THEN | |
ARITH_TAC]);; | |
(* }}} *) | |
let pi_fun = local_def `pi_fun n = inv (&.16 **. n) *. | |
(&.4 / &.(8 *| n +| 1) -. | |
&.2 / &.(8 *| n +| 4) -. | |
&.1 / &.(8 *| n +| 5) -. | |
&.1 / &.(8 *| n +| 6))`;; | |
let pi_bound_fun = local_def `pi_bound_fun n = if (n=0) then (&.8) else | |
(((&.15)/(&.16))*(inv(&.16 **. n))) `;; | |
let PI_EST2 = prove_by_refinement( | |
`!k. abs(pi_fun k) <=. (pi_bound_fun k)`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
REWRITE_TAC[pi_fun;pi_bound_fun]; | |
COND_CASES_TAC; | |
ASM_REWRITE_TAC[]; | |
CONV_TAC (NUM_REDUCE_CONV); | |
(CONV_TAC (REAL_RAT_REDUCE_CONV)); | |
CONV_TAC (RAND_CONV (REWR_CONV (REAL_ARITH `a*b = b*.a`))); | |
REWRITE_TAC[REAL_ABS_MUL;REAL_ABS_INV;REAL_ABS_POW;prove(`||.(&.n) = (&.n)`,MESON_TAC[REAL_POS;REAL_ABS_REFL])]; | |
MATCH_MP_TAC (prove(`!x y z. (&.0 <. z /\ (y <=. x) ==> (z*y <=. (z*x)))`,MESON_TAC[REAL_LE_LMUL_EQ])); | |
ASSUME_TAC (REWRITE_RULE[] (REAL_RAT_REDUCE_CONV `(&.622)/(&.819) <=. (&.15)/(&.16)`)); | |
IMP_RES_THEN ASSUME_TAC (ARITH_RULE `~(k=0) ==> (1<=| k)`); | |
IMP_RES_THEN ASSUME_TAC (PI_EST); | |
CONJ_TAC; | |
SIMP_TAC[REAL_POW_LT;REAL_LT_INV;ARITH_RULE `&.0 < (&.16)`]; | |
ASM_MESON_TAC[REAL_LE_TRANS]; | |
]);; | |
(* }}} *) | |
let GP16 = prove_by_refinement( | |
`!k. (\n. inv (&16 pow k) * inv (&16 pow n)) sums | |
inv (&16 pow k) * &16 / &15`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
ASSUME_TAC (REWRITE_RULE[] (REAL_RAT_REDUCE_CONV `abs (&.1 / (&. 16)) <. (&.1)`)); | |
IMP_RES_THEN (fun th -> ASSUME_TAC (CONV_RULE REAL_RAT_REDUCE_CONV th)) GP; | |
MATCH_MP_TAC SER_CMUL; | |
ASM_REWRITE_TAC[GSYM REAL_POW_INV;REAL_INV_1OVER]; | |
]);; | |
(* }}} *) | |
let GP16a = prove_by_refinement( | |
`!k. (0<|k) ==> (\n. (pi_bound_fun (n+k))) sums (inv(&.16 **. k))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC; | |
DISCH_TAC; | |
SUBGOAL_THEN `(\n. pi_bound_fun (n+k)) = (\n. ((&.15/(&.16))* (inv(&.16)**. k) *. inv(&.16 **. n)))` (fun th-> REWRITE_TAC[th]); | |
MATCH_MP_TAC EQ_EXT; | |
X_GEN_TAC `n:num` THEN BETA_TAC; | |
REWRITE_TAC[pi_bound_fun]; | |
COND_CASES_TAC; | |
ASM_MESON_TAC[ARITH_RULE `0<| k ==> (~(n+k = 0))`]; | |
REWRITE_TAC[GSYM REAL_MUL_ASSOC]; | |
AP_TERM_TAC; | |
REWRITE_TAC[REAL_INV_MUL;REAL_POW_ADD;REAL_POW_INV;REAL_MUL_AC]; | |
SUBGOAL_THEN `(\n. (&.15/(&.16)) *. ((inv(&.16)**. k)*. inv(&.16 **. n))) sums ((&.15/(&.16)) *.(inv(&.16**. k)*. ((&.16)/(&.15))))` ASSUME_TAC; | |
MATCH_MP_TAC SER_CMUL; | |
REWRITE_TAC[REAL_POW_INV]; | |
ACCEPT_TAC (SPEC `k:num` GP16); | |
FIRST_X_ASSUM MP_TAC; | |
REWRITE_TAC[REAL_MUL_ASSOC]; | |
MATCH_MP_TAC (prove (`(x=y) ==> ((a sums x) ==> (a sums y))`,MESON_TAC[])); | |
MATCH_MP_TAC (REAL_ARITH `(b*(a*c) = (b*(&.1))) ==> ((a*b)*c = b)`); | |
AP_TERM_TAC; | |
CONV_TAC (REAL_RAT_REDUCE_CONV); | |
]);; | |
(* }}} *) | |
let PI_SER = prove_by_refinement( | |
`!k. (0<|k) ==> (abs(pi - (sum(0,k) pi_fun)) <=. (inv(&.16 **. (k))))`, | |
(* {{{ proof *) | |
[ | |
GEN_TAC THEN DISCH_TAC; | |
ASSUME_TAC (ONCE_REWRITE_RULE[ETA_AX] (REWRITE_RULE[GSYM pi_fun] POLYLOG_THM)); | |
ASSUME_TAC PI_EST2; | |
IMP_RES_THEN (ASSUME_TAC) GP16a; | |
IMP_RES_THEN (ASSUME_TAC) SUM_SUMMABLE; | |
IMP_RES_THEN (ASSUME_TAC) SER_OFFSET_REV; | |
IMP_RES_THEN (ASSUME_TAC) SUM_SUMMABLE; | |
MP_TAC (SPECL [`pi`;`pi_fun`;`pi_bound_fun` ] SER_APPROX); | |
ASM_REWRITE_TAC[]; | |
DISCH_THEN (fun th -> MP_TAC (SPEC `k:num` th)); | |
SUBGOAL_THEN `suminf (\n. pi_bound_fun (n + k)) = inv (&.16 **. k)` (fun th -> (MESON_TAC[th])); | |
ASM_MESON_TAC[SUM_UNIQ]; | |
]);; | |
(* }}} *) | |
(* replace 3 by SUC (SUC (SUC 0)) *) | |
let SUC_EXPAND_CONV tm = | |
let count = dest_numeral tm in | |
let rec add_suc i r = | |
if (i <=/ (Int 0)) then r | |
else add_suc (i -/ (Int 1)) (mk_comb (`SUC`,r)) in | |
let tm' = add_suc count `0` in | |
REWRITE_RULE[] (ARITH_REWRITE_CONV[] (mk_eq (tm,tm')));; | |
let inv_twopow = prove( | |
`!n. inv (&.16 **. n) = (twopow (--: (&:(4*n)))) `, | |
REWRITE_TAC[TWOPOW_NEG;GSYM (NUM_RED_CONV `2 EXP 4`); | |
REAL_OF_NUM_POW;EXP_MULT]);; | |
let PI_SERn n = | |
let SUM_EXPAND_CONV = | |
(ARITH_REWRITE_CONV[]) THENC | |
(TOP_DEPTH_CONV SUC_EXPAND_CONV) THENC | |
(REWRITE_CONV[sum]) THENC | |
(ARITH_REWRITE_CONV[REAL_ADD_LID;GSYM REAL_ADD_ASSOC]) in | |
let sum_thm = SUM_EXPAND_CONV (vsubst [n,`i:num`] `sum(0,i) f`) in | |
let gt_thm = ARITH_RULE (vsubst [n,`i:num`] `0 <| i`) in | |
((* CONV_RULE REAL_RAT_REDUCE_CONV *)(CONV_RULE (ARITH_REWRITE_CONV[]) (BETA_RULE (REWRITE_RULE[sum_thm;pi_fun;inv_twopow] (MATCH_MP PI_SER gt_thm)))));; | |
(* abs(pi - u ) < e *) | |
let recompute_pi bprec = | |
let n = (bprec /4) in | |
let pi_ser = PI_SERn (mk_numeral (Int n)) in | |
let _ = remove_real_constant `pi` in | |
(add_real_constant pi_ser; INTERVAL_OF_TERM bprec `pi`);; | |
(* ------------------------------------------------------------------ *) | |
(* restore defaults *) | |
(* ------------------------------------------------------------------ *) | |
reduce_local_interface("trig");; | |
pop_priority();; | |