Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
| 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]));; | |