author | boehmes |
Mon, 20 Dec 2010 09:31:47 +0100 | |
changeset 41303 | 939f647f0c9e |
parent 41282 | a4d1b5eef12e |
child 41601 | fda8511006f9 |
permissions | -rw-r--r-- |
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(* Title: HOL/SMT_Examples/SMT_Examples.thy |
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Author: Sascha Boehme, TU Muenchen |
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*) |
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header {* Examples for the SMT binding *} |
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theory SMT_Examples |
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imports Complex_Main |
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begin |
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declare [[smt_solver=z3, smt_oracle=false]] |
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declare [[smt_certificates="SMT_Examples.certs"]] |
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declare [[smt_fixed=true]] |
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section {* Propositional and first-order logic *} |
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lemma "True" by smt |
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lemma "p \<or> \<not>p" by smt |
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lemma "(p \<and> True) = p" by smt |
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lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt |
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lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" |
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by smt |
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lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt |
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lemma "P=P=P=P=P=P=P=P=P=P" by smt |
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lemma |
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assumes "a | b | c | d" |
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and "e | f | (a & d)" |
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and "~(a | (c & ~c)) | b" |
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and "~(b & (x | ~x)) | c" |
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and "~(d | False) | c" |
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and "~(c | (~p & (p | (q & ~q))))" |
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shows False |
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using assms by smt |
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axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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symm_f: "symm_f x y = symm_f y x" |
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lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f) |
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(* |
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Taken from ~~/src/HOL/ex/SAT_Examples.thy. |
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Translated from TPTP problem library: PUZ015-2.006.dimacs |
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*) |
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lemma |
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assumes "~x0" |
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and "~x30" |
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and "~x29" |
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and "~x59" |
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and "x1 | x31 | x0" |
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and "x2 | x32 | x1" |
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and "x3 | x33 | x2" |
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and "x4 | x34 | x3" |
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and "x35 | x4" |
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and "x5 | x36 | x30" |
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and "x6 | x37 | x5 | x31" |
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and "x7 | x38 | x6 | x32" |
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and "x8 | x39 | x7 | x33" |
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and "x9 | x40 | x8 | x34" |
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and "x41 | x9 | x35" |
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and "x10 | x42 | x36" |
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and "x11 | x43 | x10 | x37" |
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and "x12 | x44 | x11 | x38" |
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and "x13 | x45 | x12 | x39" |
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and "x14 | x46 | x13 | x40" |
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and "x47 | x14 | x41" |
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and "x15 | x48 | x42" |
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and "x16 | x49 | x15 | x43" |
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and "x17 | x50 | x16 | x44" |
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and "x18 | x51 | x17 | x45" |
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and "x19 | x52 | x18 | x46" |
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and "x53 | x19 | x47" |
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and "x20 | x54 | x48" |
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and "x21 | x55 | x20 | x49" |
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and "x22 | x56 | x21 | x50" |
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and "x23 | x57 | x22 | x51" |
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and "x24 | x58 | x23 | x52" |
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and "x59 | x24 | x53" |
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and "x25 | x54" |
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and "x26 | x25 | x55" |
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and "x27 | x26 | x56" |
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and "x28 | x27 | x57" |
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and "x29 | x28 | x58" |
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and "~x1 | ~x31" |
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and "~x1 | ~x0" |
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and "~x31 | ~x0" |
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and "~x2 | ~x32" |
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and "~x2 | ~x1" |
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and "~x32 | ~x1" |
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and "~x3 | ~x33" |
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and "~x3 | ~x2" |
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and "~x33 | ~x2" |
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and "~x4 | ~x34" |
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and "~x4 | ~x3" |
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and "~x34 | ~x3" |
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and "~x35 | ~x4" |
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and "~x5 | ~x36" |
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and "~x5 | ~x30" |
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and "~x36 | ~x30" |
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and "~x6 | ~x37" |
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and "~x6 | ~x5" |
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and "~x6 | ~x31" |
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and "~x37 | ~x5" |
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and "~x37 | ~x31" |
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and "~x5 | ~x31" |
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and "~x7 | ~x38" |
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and "~x7 | ~x6" |
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and "~x7 | ~x32" |
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and "~x38 | ~x6" |
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and "~x38 | ~x32" |
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and "~x6 | ~x32" |
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and "~x8 | ~x39" |
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and "~x8 | ~x7" |
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and "~x8 | ~x33" |
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and "~x39 | ~x7" |
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and "~x39 | ~x33" |
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and "~x7 | ~x33" |
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and "~x9 | ~x40" |
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and "~x9 | ~x8" |
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and "~x9 | ~x34" |
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and "~x40 | ~x8" |
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and "~x40 | ~x34" |
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and "~x8 | ~x34" |
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and "~x41 | ~x9" |
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and "~x41 | ~x35" |
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and "~x9 | ~x35" |
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and "~x10 | ~x42" |
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and "~x10 | ~x36" |
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and "~x42 | ~x36" |
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and "~x11 | ~x43" |
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and "~x11 | ~x10" |
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and "~x11 | ~x37" |
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and "~x43 | ~x10" |
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and "~x43 | ~x37" |
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and "~x10 | ~x37" |
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and "~x12 | ~x44" |
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and "~x12 | ~x11" |
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and "~x12 | ~x38" |
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and "~x44 | ~x11" |
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and "~x44 | ~x38" |
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and "~x11 | ~x38" |
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and "~x13 | ~x45" |
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and "~x13 | ~x12" |
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and "~x13 | ~x39" |
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and "~x45 | ~x12" |
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and "~x45 | ~x39" |
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and "~x12 | ~x39" |
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and "~x14 | ~x46" |
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and "~x14 | ~x13" |
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and "~x14 | ~x40" |
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and "~x46 | ~x13" |
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and "~x46 | ~x40" |
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and "~x13 | ~x40" |
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and "~x47 | ~x14" |
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and "~x47 | ~x41" |
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and "~x14 | ~x41" |
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and "~x15 | ~x48" |
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and "~x15 | ~x42" |
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and "~x48 | ~x42" |
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and "~x16 | ~x49" |
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and "~x16 | ~x15" |
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and "~x16 | ~x43" |
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and "~x49 | ~x15" |
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and "~x49 | ~x43" |
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and "~x15 | ~x43" |
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and "~x17 | ~x50" |
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and "~x17 | ~x16" |
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and "~x17 | ~x44" |
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and "~x50 | ~x16" |
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and "~x50 | ~x44" |
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and "~x16 | ~x44" |
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and "~x18 | ~x51" |
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and "~x18 | ~x17" |
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and "~x18 | ~x45" |
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and "~x51 | ~x17" |
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and "~x51 | ~x45" |
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and "~x17 | ~x45" |
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and "~x19 | ~x52" |
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and "~x19 | ~x18" |
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and "~x19 | ~x46" |
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and "~x52 | ~x18" |
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and "~x52 | ~x46" |
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and "~x18 | ~x46" |
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and "~x53 | ~x19" |
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and "~x53 | ~x47" |
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and "~x19 | ~x47" |
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and "~x20 | ~x54" |
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and "~x20 | ~x48" |
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and "~x54 | ~x48" |
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and "~x21 | ~x55" |
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and "~x21 | ~x20" |
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and "~x21 | ~x49" |
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and "~x55 | ~x20" |
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and "~x55 | ~x49" |
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and "~x20 | ~x49" |
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and "~x22 | ~x56" |
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and "~x22 | ~x21" |
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and "~x22 | ~x50" |
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and "~x56 | ~x21" |
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and "~x56 | ~x50" |
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and "~x21 | ~x50" |
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and "~x23 | ~x57" |
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and "~x23 | ~x22" |
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and "~x23 | ~x51" |
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and "~x57 | ~x22" |
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and "~x57 | ~x51" |
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and "~x22 | ~x51" |
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and "~x24 | ~x58" |
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and "~x24 | ~x23" |
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and "~x24 | ~x52" |
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and "~x58 | ~x23" |
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and "~x58 | ~x52" |
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and "~x23 | ~x52" |
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and "~x59 | ~x24" |
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and "~x59 | ~x53" |
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and "~x24 | ~x53" |
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and "~x25 | ~x54" |
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and "~x26 | ~x25" |
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and "~x26 | ~x55" |
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and "~x25 | ~x55" |
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and "~x27 | ~x26" |
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and "~x27 | ~x56" |
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and "~x26 | ~x56" |
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and "~x28 | ~x27" |
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and "~x28 | ~x57" |
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and "~x27 | ~x57" |
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and "~x29 | ~x28" |
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and "~x29 | ~x58" |
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and "~x28 | ~x58" |
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shows False |
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using assms by smt |
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lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)" |
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by smt |
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lemma |
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assumes "(\<forall>x y. P x y = x)" |
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shows "(\<exists>y. P x y) = P x c" |
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using assms by smt |
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lemma |
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assumes "(\<forall>x y. P x y = x)" |
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and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)" |
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shows "(EX y. P x y) = P x c" |
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using assms by smt |
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lemma |
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assumes "if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)" |
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shows "P x \<longrightarrow> P y" |
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using assms by smt |
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section {* Arithmetic *} |
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subsection {* Linear arithmetic over integers and reals *} |
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lemma "(3::int) = 3" by smt |
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lemma "(3::real) = 3" by smt |
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lemma "(3 :: int) + 1 = 4" by smt |
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lemma "x + (y + z) = y + (z + (x::int))" by smt |
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lemma "max (3::int) 8 > 5" by smt |
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lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt |
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lemma "P ((2::int) < 3) = P True" by smt |
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lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt |
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lemma |
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assumes "x \<ge> (3::int)" and "y = x + 4" |
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shows "y - x > 0" |
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using assms by smt |
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lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt |
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lemma |
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fixes x :: real |
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assumes "3 * x + 7 * a < 4" and "3 < 2 * x" |
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shows "a < 0" |
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using assms by smt |
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lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt |
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lemma "distinct [x < (3::int), 3 \<le> x]" by smt |
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lemma |
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assumes "a > (0::int)" |
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shows "distinct [a, a * 2, a - a]" |
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using assms by smt |
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lemma " |
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(n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) | |
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(n = n' & n' < m) | (n = m & m < n') | |
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(n' < m & m < n) | (n' < m & m = n) | |
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(n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) | |
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(m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) | |
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(m = n & n < n') | (m = n' & n' < n) | |
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(n' = m & m = (n::int))" |
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by smt |
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text{* |
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The following example was taken from HOL/ex/PresburgerEx.thy, where it says: |
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This following theorem proves that all solutions to the |
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recurrence relation $x_{i+2} = |x_{i+1}| - x_i$ are periodic with |
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period 9. The example was brought to our attention by John |
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Harrison. It does does not require Presburger arithmetic but merely |
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quantifier-free linear arithmetic and holds for the rationals as well. |
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Warning: it takes (in 2006) over 4.2 minutes! |
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There, it is proved by "arith". SMT is able to prove this within a fraction |
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of one second. With proof reconstruction, it takes about 13 seconds on a Core2 |
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processor. |
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*} |
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lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3; |
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x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6; |
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x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk> |
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\<Longrightarrow> x1 = x10 & x2 = (x11::int)" |
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by smt |
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lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt |
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lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)" by smt |
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lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" by smt |
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lemma |
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assumes "x \<noteq> (0::real)" |
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shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x" |
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using assms by smt |
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lemma |
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assumes "(n + m) mod 2 = 0" and "n mod 4 = 3" |
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shows "n mod 2 = 1 & m mod 2 = (1::int)" |
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using assms by smt |
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subsection {* Linear arithmetic with quantifiers *} |
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lemma "~ (\<exists>x::int. False)" by smt |
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lemma "~ (\<exists>x::real. False)" by smt |
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lemma "\<exists>x::int. 0 < x" |
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using [[smt_oracle=true]] (* no Z3 proof *) |
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by smt |
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lemma "\<exists>x::real. 0 < x" |
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using [[smt_oracle=true]] (* no Z3 proof *) |
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by smt |
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lemma "\<forall>x::int. \<exists>y. y > x" |
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using [[smt_oracle=true]] (* no Z3 proof *) |
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by smt |
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lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt |
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lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt |
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36899
bcd6fce5bf06
layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents:
36898
diff
changeset
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375 |
lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt |
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lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt |
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lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt |
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lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt |
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lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt |
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lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt |
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lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt |
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lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt |
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lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt |
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|
36898 | 394 |
lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt |
395 |
||
37124 | 396 |
lemma "\<forall>x::int. SMT.trigger [[SMT.pat x]] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt |
36898 | 397 |
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398 |
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subsection {* Non-linear arithmetic over integers and reals *} |
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lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0" |
|
41303 | 402 |
using [[smt_oracle=true]] |
41282 | 403 |
by smt |
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404 |
|
41282 | 405 |
lemma "(a::int) * (x + 1 + y) = a * x + a * (y + 1)" |
406 |
by smt |
|
36898 | 407 |
|
41282 | 408 |
lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)" |
41303 | 409 |
by smt |
36898 | 410 |
|
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lemma |
|
412 |
"(U::int) + (1 + p) * (b + e) + p * d = |
|
413 |
U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)" |
|
41303 | 414 |
by smt |
36898 | 415 |
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416 |
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subsection {* Linear arithmetic for natural numbers *} |
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418 |
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419 |
lemma "2 * (x::nat) ~= 1" by smt |
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420 |
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lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt |
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422 |
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lemma "let x = (1::nat) + y in x - y > 0 * x" by smt |
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424 |
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425 |
lemma |
|
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"let x = (1::nat) + y in |
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let P = (if x > 0 then True else False) in |
|
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False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)" |
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429 |
by smt |
|
430 |
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lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt |
36898 | 432 |
|
433 |
lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt |
|
434 |
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435 |
definition prime_nat :: "nat \<Rightarrow> bool" where |
|
436 |
"prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))" |
|
437 |
lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def) |
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438 |
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439 |
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440 |
section {* Pairs *} |
|
441 |
||
41132 | 442 |
lemma "fst (x, y) = a \<Longrightarrow> x = a" |
443 |
using fst_conv |
|
444 |
by smt |
|
36898 | 445 |
|
41132 | 446 |
lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" |
447 |
using fst_conv snd_conv |
|
448 |
by smt |
|
36898 | 449 |
|
450 |
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451 |
section {* Higher-order problems and recursion *} |
|
452 |
||
41132 | 453 |
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i" |
454 |
using fun_upd_same fun_upd_apply |
|
455 |
by smt |
|
36898 | 456 |
|
457 |
lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)" |
|
458 |
by smt |
|
459 |
||
460 |
lemma "id 3 = 3 \<and> id True = True" by (smt id_def) |
|
461 |
||
41132 | 462 |
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" |
463 |
using fun_upd_same fun_upd_apply |
|
464 |
by smt |
|
36898 | 465 |
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466 |
|
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467 |
|
36898 | 468 |
lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps) |
469 |
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470 |
|
36898 | 471 |
lemma "(ALL x. P x) | ~ All P" by smt |
472 |
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473 |
fun dec_10 :: "nat \<Rightarrow> nat" where |
|
474 |
"dec_10 n = (if n < 10 then n else dec_10 (n - 10))" |
|
475 |
lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps) |
|
476 |
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477 |
|
36898 | 478 |
axiomatization |
479 |
eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int" |
|
480 |
where |
|
481 |
eval_dioph_mod: |
|
482 |
"eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n" |
|
483 |
and |
|
484 |
eval_dioph_div_mult: |
|
485 |
"eval_dioph ks (map (\<lambda>x. x div n) xs) * int n + |
|
486 |
eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs" |
|
487 |
lemma |
|
488 |
"(eval_dioph ks xs = l) = |
|
489 |
(eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and> |
|
490 |
eval_dioph ks (map (\<lambda>x. x div 2) xs) = |
|
491 |
(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)" |
|
41132 | 492 |
using [[smt_oracle=true]] (*FIXME*) |
36898 | 493 |
by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2]) |
494 |
||
495 |
||
496 |
section {* Monomorphization examples *} |
|
497 |
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498 |
definition Pred :: "'a \<Rightarrow> bool" where "Pred x = True" |
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lemma poly_Pred: "Pred x \<and> (Pred [x] \<or> \<not>Pred[x])" by (simp add: Pred_def) |
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lemma "Pred (1::int)" by (smt poly_Pred) |
36898 | 501 |
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502 |
axiomatization g :: "'a \<Rightarrow> nat" |
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503 |
axiomatization where |
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g1: "g (Some x) = g [x]" and |
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505 |
g2: "g None = g []" and |
36898 | 506 |
g3: "g xs = length xs" |
507 |
lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size) |
|
508 |
||
509 |
end |