Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 3,931 Bytes
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unambiguous_interface();;
let INT_NUM = prove(`!u. (integer (real_of_num u))`,
(REWRITE_TAC[is_int]) THEN GEN_TAC THEN
(EXISTS_TAC (`u:num`)) THEN (MESON_TAC[]));;
let INT_NUM_REAL = prove(`!u. (real_of_int (int_of_num u) = real_of_num u)`,
(REWRITE_TAC[int_of_num]) THEN
GEN_TAC THEN (MESON_TAC[INT_NUM;int_rep]));;
let INT_IS_INT = prove(`!(a:int). (integer (real_of_int a))`,
REWRITE_TAC[int_rep;int_abstr]);;
let INT_OF_NUM_DEST = prove(`!a n. ((real_of_int a = (real_of_num n)) =
(a = int_of_num n))`,
(REWRITE_TAC[int_eq])
THEN (REPEAT GEN_TAC)
THEN (REWRITE_TAC[int_of_num])
THEN (ASSUME_TAC (SPEC (`n:num`) INT_NUM))
THEN (UNDISCH_EL_TAC 0)
THEN (SIMP_TAC[int_rep]));;
let INT_REP = prove(`!a. ?n m. (a = (int_of_num n) - (int_of_num m))`,
GEN_TAC
THEN (let tt =(REWRITE_RULE[is_int] (SPEC (`a:int`) INT_IS_INT)) in
(CHOOSE_TAC tt))
THEN (POP_ASSUM DISJ_CASES_TAC)
THENL [
(EXISTS_TAC (`n:num`)) THEN (EXISTS_TAC (`0`)) THEN
(ASM_REWRITE_TAC[INT_SUB_RZERO;GSYM INT_OF_NUM_DEST]);
(EXISTS_TAC (`0`)) THEN (EXISTS_TAC (`n:num`)) THEN
(REWRITE_TAC[INT_SUB_LZERO]) THEN
(UNDISCH_EL_TAC 0) THEN
(REWRITE_TAC[GSYM REAL_NEG_EQ;GSYM INT_NEG_EQ;GSYM int_neg_th;GSYM
INT_OF_NUM_DEST])]);;
let INT_REP2 = prove( `!a. ?n. ((a = (&: n)) \/ (a = (--: (&: n))))`,
(GEN_TAC)
THEN ((let tt =(REWRITE_RULE[is_int] (SPEC (`a:int`) INT_IS_INT)) in
(CHOOSE_TAC tt)))
THEN ((POP_ASSUM DISJ_CASES_TAC))
THENL
[ ((EXISTS_TAC (`n:num`)))
THEN ((ASM_REWRITE_TAC[GSYM INT_OF_NUM_DEST]));
((EXISTS_TAC (`n:num`)))
(* THEN ((RULE_EL 0 (REWRITE_RULE[GSYM REAL_NEG_EQ;GSYM int_neg_th]))) *)
THEN (H_REWRITE_RULE[THM (GSYM REAL_NEG_EQ);THM (GSYM int_neg_th)] (HYP_INT 0))
THEN ((ASM_REWRITE_TAC[GSYM INT_NEG_EQ;GSYM INT_OF_NUM_DEST]))]);;
(* ------------------------------------------------------------------ *)
(* nabs : int -> num gives the natural number abs. value of an int *)
(* ------------------------------------------------------------------ *)
let nabs = new_definition(`nabs n = @u. ((n = int_of_num u) \/ (n =
int_neg (int_of_num u)))`);;
let NABS_POS = prove(`!u. (nabs (int_of_num u)) = u`,
GEN_TAC
THEN (REWRITE_TAC [nabs])
THEN (MATCH_MP_TAC SELECT_UNIQUE)
THEN (GEN_TAC THEN BETA_TAC)
THEN (EQ_TAC)
THENL [(TAUT_TAC (` ((A==>C)/\ (B==>C)) ==> (A\/B ==>C) `));
MESON_TAC[]]
THEN CONJ_TAC THENL
(let branch2 = (REWRITE_TAC[int_eq;int_neg_th;INT_NUM_REAL])
THEN (REWRITE_TAC[prove (`! u y.(((real_of_num u) = --(real_of_num y))=
((real_of_num u) +(real_of_num y) = (&0)))`,REAL_ARITH_TAC)])
THEN (REWRITE_TAC[REAL_OF_NUM_ADD;REAL_OF_NUM_EQ])
THEN (MESON_TAC[ADD_EQ_0]) in
[(REWRITE_TAC[int_eq;INT_NUM_REAL]);branch2])
THEN (REWRITE_TAC[INT_NUM_REAL])
THEN (MESON_TAC[REAL_OF_NUM_EQ]));;
let NABS_NEG = prove(`!n. (nabs (-- (int_of_num n))) = n`,
GEN_TAC
THEN (REWRITE_TAC [nabs])
THEN (MATCH_MP_TAC SELECT_UNIQUE)
THEN (GEN_TAC THEN BETA_TAC)
THEN (EQ_TAC)
THENL [(TAUT_TAC (` ((A==>C)/\ (B==>C)) ==> (A\/B ==>C) `));
MESON_TAC[]]
THEN CONJ_TAC THENL
(let branch1 = (REWRITE_TAC[int_eq;int_neg_th;INT_NUM_REAL])
THEN (REWRITE_TAC[prove (`! u y.((--(real_of_num u) = (real_of_num y))=
((real_of_num u) +(real_of_num y) = (&0)))`,REAL_ARITH_TAC)])
THEN (REWRITE_TAC[REAL_OF_NUM_ADD;REAL_OF_NUM_EQ])
THEN (MESON_TAC[ADD_EQ_0]) in
[branch1;(REWRITE_TAC[int_eq;INT_NUM_REAL])])
THEN (REWRITE_TAC[INT_NUM_REAL;int_neg_th;REAL_NEG_EQ;REAL_NEG_NEG])
THEN (MESON_TAC[REAL_OF_NUM_EQ]));;
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