/usr/share/hol88-2.02.19940316/contrib/CSP/boolarith2.ml is in hol88-contrib-source 2.02.19940316-19.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | % Another supplementary theory of Boolean Algebra and Arithmetic %
% theorems. In fact an extension to boolarith1.th. %
% %
% FILE : boolarith2.ml %
% DESCRIPTION : Extends the boolean and arithmetic built-in theories %
% with some theorems which are needed for mechanizing %
% csp. %
% %
% LOADS LIBRARY : taut %
% READS FILES : boolarith1.th %
% WRITES FILES : boolarith2.th %
% %
% AUTHOR : Albert J Camilleri %
% AFFILIATION : Hewlett-Packard Laboratories, Bristol %
% DATE : 86.04.04 %
% MODIFIED : 89.07.20 %
% REVISED : 91.10.01 %
new_theory `boolarith2`;;
new_parent `boolarith1`;;
% Load the Tautology Checker %
load_library `taut`;;
let F_IMP_EX_F =
save_thm (`F_IMP_EX_F`,
DISCH "F" (EXISTS ("?t:bool.F","F") (ASSUME "F")));;
let EX_F_IMP_F =
save_thm (`EX_F_IMP_F`,
DISCH_ALL (SELECT_RULE (ASSUME "?t:bool.F")));;
let F_FROM_EX_F =
save_thm (`F_FROM_EX_F`, IMP_ANTISYM_RULE EX_F_IMP_F F_IMP_EX_F);;
let ID_IMP =
save_thm (`ID_IMP`, TAUT_RULE "! b. b ==> b");;
let CONJ_IMP_TAUT =
save_thm (`CONJ_IMP_TAUT`,
TAUT_RULE "! a b c. (a ==> b) ==> ((a /\ c) ==> (b /\ c))");;
let CONJ2_IMP_TAUT =
save_thm (`CONJ2_IMP_TAUT`,
TAUT_RULE "! a b c d.
(a ==> b) ==>
((d /\ (a /\ c)) ==> (d /\ (b /\ c)))");;
let CONJ3_IMP_TAUT =
save_thm (`CONJ3_IMP_TAUT`,
TAUT_RULE "! a b c.
(a ==> b) ==>
((c /\ a) ==> (c /\ b))");;
let NOT_LEQ = theorem `boolarith1` `NOT_LEQ`;;
let ADD_SUC_0 =
save_thm (`ADD_SUC_0`,
(CONV_RULE (DEPTH_CONV num_CONV))
(REWRITE_RULE [SPECL ["m:num";"1"] ADD_SYM] ADD1));;
let LESS_MONO_MULT' =
save_thm (`LESS_MONO_MULT'`,
GEN_ALL
(SUBS [SPECL ["m:num";"p:num"] MULT_SYM;
SPECL ["n:num";"p:num"] MULT_SYM]
(SPEC_ALL LESS_MONO_MULT)));;
let LESS_EQ_0_N =
save_thm (`LESS_EQ_0_N`, REWRITE_RULE [NOT_LESS] NOT_LESS_0);;
let LESS_EQ_MONO_ADD_EQ' =
save_thm (`LESS_EQ_MONO_ADD_EQ'`,
GEN_ALL (SYM (SUBS [SPECL ["m:num";"p:num"] ADD_SYM;
SPECL ["n:num";"p:num"] ADD_SYM]
(SPEC_ALL LESS_EQ_MONO_ADD_EQ))));;
let LESS_EQ_MONO_ADD_EQ1 =
save_thm (`LESS_EQ_MONO_ADD_EQ1`,
GEN_ALL (REWRITE_RULE [ADD]
(SPECL ["m:num";"0:num";"p:num"]
LESS_EQ_MONO_ADD_EQ)));;
let LESS_EQ_MONO_ADD_EQ2 =
save_thm (`LESS_EQ_MONO_ADD_EQ2`,
GEN_ALL (REWRITE_RULE [ADD]
(SPECL ["0:num";"n:num";"p:num"]
LESS_EQ_MONO_ADD_EQ)));;
let LESS_EQ_MONO_ADD_EQ3 =
save_thm (`LESS_EQ_MONO_ADD_EQ3`,
GEN_ALL (REWRITE_RULE [ADD;LESS_EQ_0_N]
(SPECL ["0:num";"n:num";"p:num"]
LESS_EQ_MONO_ADD_EQ)));;
let ADD_SYM_ASSOC =
prove_thm (`ADD_SYM_ASSOC`,
"! a b c. a + (b + c) = b + (a + c)",
REPEAT GEN_TAC THEN
REWRITE_TAC [ADD_ASSOC] THEN
SUBST_TAC [SPECL ["a:num";"b:num"] ADD_SYM] THEN
REWRITE_TAC []);;
let NOT_SUC_LEQ_0 =
prove_thm (`NOT_SUC_LEQ_0`,
"! n . ~ (SUC n) <= 0",
REWRITE_TAC[NOT_LEQ;LESS_0]);;
let INV_SUC_LEQ =
prove_thm (`INV_SUC_LEQ`,
"! m n . (SUC m <= SUC n) = (m <= n)",
REWRITE_TAC [LESS_OR_EQ;LESS_MONO_EQ;INV_SUC_EQ]);;
let TWICE =
prove_thm (`TWICE`,
"! x . (x + x) = (SUC (SUC 0)) * x",
REWRITE_TAC [ADD_CLAUSES;MULT_CLAUSES]);;
let NOT_SUC_LEQ =
save_thm (`NOT_SUC_LEQ`,
NOT_INTRO
(DISCH_ALL
(REWRITE_RULE [NOT_SUC_LEQ_0]
(SPEC "0" (ASSUME "(!n m. (SUC m) <= n)")))));;
let LEQ_SPLIT =
save_thm (`LEQ_SPLIT`,
let asm_thm = ASSUME "(m + n) <= p"
in
DISCH_ALL
(CONJ
(MP (SPECL ["n:num";"m+n";"p:num"] LESS_EQ_TRANS)
(CONJ (SUBS [SPECL ["n:num";"m:num"] ADD_SYM]
(SPECL ["n:num";"m:num"] LESS_EQ_ADD))
asm_thm))
(MP (SPECL ["m:num";"m+n";"p:num"] LESS_EQ_TRANS)
(CONJ (SPEC_ALL LESS_EQ_ADD) asm_thm))));;
close_theory ();;
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