src/HOL/SMT_Examples/SMT_Examples.thy
author boehmes
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updated SMT certificates
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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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parents:
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   166
  and "~x48 | ~x42"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   167
  and "~x16 | ~x49"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   168
  and "~x16 | ~x15"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   169
  and "~x16 | ~x43"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   170
  and "~x49 | ~x15"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   171
  and "~x49 | ~x43"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   172
  and "~x15 | ~x43"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   173
  and "~x17 | ~x50"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   174
  and "~x17 | ~x16"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   175
  and "~x17 | ~x44"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   176
  and "~x50 | ~x16"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   177
  and "~x50 | ~x44"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   178
  and "~x16 | ~x44"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   179
  and "~x18 | ~x51"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   180
  and "~x18 | ~x17"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   181
  and "~x18 | ~x45"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   182
  and "~x51 | ~x17"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   183
  and "~x51 | ~x45"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   184
  and "~x17 | ~x45"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   185
  and "~x19 | ~x52"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   186
  and "~x19 | ~x18"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   187
  and "~x19 | ~x46"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
diff changeset
   188
  and "~x52 | ~x18"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   189
  and "~x52 | ~x46"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
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   190
  and "~x18 | ~x46"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   191
  and "~x53 | ~x19"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   192
  and "~x53 | ~x47"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   193
  and "~x19 | ~x47"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   194
  and "~x20 | ~x54"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   195
  and "~x20 | ~x48"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   196
  and "~x54 | ~x48"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   197
  and "~x21 | ~x55"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   198
  and "~x21 | ~x20"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   199
  and "~x21 | ~x49"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   200
  and "~x55 | ~x20"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   201
  and "~x55 | ~x49"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   202
  and "~x20 | ~x49"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   203
  and "~x22 | ~x56"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   204
  and "~x22 | ~x21"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   205
  and "~x22 | ~x50"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   206
  and "~x56 | ~x21"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   207
  and "~x56 | ~x50"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   208
  and "~x21 | ~x50"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   209
  and "~x23 | ~x57"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   210
  and "~x23 | ~x22"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   211
  and "~x23 | ~x51"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   212
  and "~x57 | ~x22"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   213
  and "~x57 | ~x51"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   214
  and "~x22 | ~x51"
8e55aa1306c5 integrated SMT into the HOL image
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   215
  and "~x24 | ~x58"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   216
  and "~x24 | ~x23"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   217
  and "~x24 | ~x52"
8e55aa1306c5 integrated SMT into the HOL image
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   218
  and "~x58 | ~x23"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   219
  and "~x58 | ~x52"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   220
  and "~x23 | ~x52"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   221
  and "~x59 | ~x24"
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   222
  and "~x59 | ~x53"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   223
  and "~x24 | ~x53"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   224
  and "~x25 | ~x54"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   225
  and "~x26 | ~x25"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   226
  and "~x26 | ~x55"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   227
  and "~x25 | ~x55"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   228
  and "~x27 | ~x26"
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   229
  and "~x27 | ~x56"
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   230
  and "~x26 | ~x56"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   231
  and "~x28 | ~x27"
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   232
  and "~x28 | ~x57"
8e55aa1306c5 integrated SMT into the HOL image
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  and "~x27 | ~x57"
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   234
  and "~x29 | ~x28"
8e55aa1306c5 integrated SMT into the HOL image
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   235
  and "~x29 | ~x58"
8e55aa1306c5 integrated SMT into the HOL image
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   236
  and "~x28 | ~x58"
8e55aa1306c5 integrated SMT into the HOL image
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  shows False
8e55aa1306c5 integrated SMT into the HOL image
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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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8e55aa1306c5 integrated SMT into the HOL image
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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"
8e55aa1306c5 integrated SMT into the HOL image
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   246
  using assms by smt
8e55aa1306c5 integrated SMT into the HOL image
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8e55aa1306c5 integrated SMT into the HOL image
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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"
8e55aa1306c5 integrated SMT into the HOL image
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  using assms by smt
8e55aa1306c5 integrated SMT into the HOL image
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   253
8e55aa1306c5 integrated SMT into the HOL image
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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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   256
  shows "P x \<longrightarrow> P y"
8e55aa1306c5 integrated SMT into the HOL image
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   257
  using assms by smt
8e55aa1306c5 integrated SMT into the HOL image
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   258
8e55aa1306c5 integrated SMT into the HOL image
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8e55aa1306c5 integrated SMT into the HOL image
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section {* Arithmetic *}
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   261
8e55aa1306c5 integrated SMT into the HOL image
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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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   267
8e55aa1306c5 integrated SMT into the HOL image
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lemma "(3 :: int) + 1 = 4" by smt
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   269
8e55aa1306c5 integrated SMT into the HOL image
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lemma "x + (y + z) = y + (z + (x::int))" by smt
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   271
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lemma "max (3::int) 8 > 5" by smt
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   273
8e55aa1306c5 integrated SMT into the HOL image
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lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt
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   275
8e55aa1306c5 integrated SMT into the HOL image
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   276
lemma "P ((2::int) < 3) = P True" by smt
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   277
8e55aa1306c5 integrated SMT into the HOL image
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   278
lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt
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   279
8e55aa1306c5 integrated SMT into the HOL image
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lemma
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  assumes "x \<ge> (3::int)" and "y = x + 4"
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   282
  shows "y - x > 0" 
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   283
  using assms by smt
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   284
8e55aa1306c5 integrated SMT into the HOL image
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   285
lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt
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   286
8e55aa1306c5 integrated SMT into the HOL image
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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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   290
  shows "a < 0"
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   291
  using assms by smt
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   292
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   293
lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
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   294
40681
872b08416fb4 be more precise: only treat constant 'distinct' applied to an explicit list as built-in
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parents: 40513
diff changeset
   295
lemma "distinct [x < (3::int), 3 \<le> x]" by smt
36898
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   296
8e55aa1306c5 integrated SMT into the HOL image
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lemma
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  assumes "a > (0::int)"
40681
872b08416fb4 be more precise: only treat constant 'distinct' applied to an explicit list as built-in
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parents: 40513
diff changeset
   299
  shows "distinct [a, a * 2, a - a]"
36898
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   300
  using assms by smt
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   301
8e55aa1306c5 integrated SMT into the HOL image
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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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   304
  (n = n' & n' < m) | (n = m & m < n') |
8e55aa1306c5 integrated SMT into the HOL image
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parents:
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   305
  (n' < m & m < n) | (n' < m & m = n) |
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   306
  (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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   308
  (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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   313
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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   317
  period 9.  The example was brought to our attention by John
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   318
  Harrison. It does does not require Presburger arithmetic but merely
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   319
  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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   322
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There, it is proved by "arith". SMT is able to prove this within a fraction
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   324
of one second. With proof reconstruction, it takes about 13 seconds on a Core2
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   325
processor.
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*}
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   327
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   328
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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   330
         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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parents:
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   333
8e55aa1306c5 integrated SMT into the HOL image
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   334
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   335
lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   336
37151
3e9e8dfb3c98 use Z3's builtin support for div and mod
boehmes
parents: 37124
diff changeset
   337
lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)" by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   338
37151
3e9e8dfb3c98 use Z3's builtin support for div and mod
boehmes
parents: 37124
diff changeset
   339
lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   340
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   341
lemma
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   342
  assumes "x \<noteq> (0::real)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   343
  shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   344
  using assms by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   345
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   346
lemma                                                                         
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   347
  assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"                               
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   348
  shows "n mod 2 = 1 & m mod 2 = (1::int)"      
37151
3e9e8dfb3c98 use Z3's builtin support for div and mod
boehmes
parents: 37124
diff changeset
   349
  using assms by smt
3e9e8dfb3c98 use Z3's builtin support for div and mod
boehmes
parents: 37124
diff changeset
   350
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   351
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   352
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   353
subsection {* Linear arithmetic with quantifiers *}
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   354
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   355
lemma "~ (\<exists>x::int. False)" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   356
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   357
lemma "~ (\<exists>x::real. False)" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   358
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   359
lemma "\<exists>x::int. 0 < x"
40163
a462d5207aa6 changed SMT configuration options; updated SMT certificates
boehmes
parents: 37151
diff changeset
   360
  using [[smt_oracle=true]] (* no Z3 proof *)
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   361
  by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   362
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   363
lemma "\<exists>x::real. 0 < x"
40163
a462d5207aa6 changed SMT configuration options; updated SMT certificates
boehmes
parents: 37151
diff changeset
   364
  using [[smt_oracle=true]] (* no Z3 proof *)
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   365
  by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   366
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   367
lemma "\<forall>x::int. \<exists>y. y > x"
40163
a462d5207aa6 changed SMT configuration options; updated SMT certificates
boehmes
parents: 37151
diff changeset
   368
  using [[smt_oracle=true]] (* no Z3 proof *)
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   369
  by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   370
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   371
lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   372
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   373
lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   374
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   375
lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   376
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   377
lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   378
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   379
lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   380
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   381
lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   382
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   383
lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   384
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   385
lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   386
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   387
lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   388
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   389
lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   390
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   391
lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   392
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   393
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   394
lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   395
37124
fe22fc54b876 hide constants and types introduced by SMT,
boehmes
parents: 36900
diff changeset
   396
lemma "\<forall>x::int. SMT.trigger [[SMT.pat x]] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   397
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   398
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   399
subsection {* Non-linear arithmetic over integers and reals *}
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   400
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   401
lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
41303
939f647f0c9e updated SMT certificates
boehmes
parents: 41282
diff changeset
   402
  using [[smt_oracle=true]]
41282
a4d1b5eef12e updated SMT certificates
boehmes
parents: 41132
diff changeset
   403
  by smt
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   404
41282
a4d1b5eef12e updated SMT certificates
boehmes
parents: 41132
diff changeset
   405
lemma  "(a::int) * (x + 1 + y) = a * x + a * (y + 1)"
a4d1b5eef12e updated SMT certificates
boehmes
parents: 41132
diff changeset
   406
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   407
41282
a4d1b5eef12e updated SMT certificates
boehmes
parents: 41132
diff changeset
   408
lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)"
41303
939f647f0c9e updated SMT certificates
boehmes
parents: 41282
diff changeset
   409
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   410
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   411
lemma
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   412
  "(U::int) + (1 + p) * (b + e) + p * d =
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   413
   U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)"
41303
939f647f0c9e updated SMT certificates
boehmes
parents: 41282
diff changeset
   414
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   415
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   416
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   417
subsection {* Linear arithmetic for natural numbers *}
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   418
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   419
lemma "2 * (x::nat) ~= 1" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   420
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   421
lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   422
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   423
lemma "let x = (1::nat) + y in x - y > 0 * x" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   424
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   425
lemma
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   426
  "let x = (1::nat) + y in
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   427
   let P = (if x > 0 then True else False) in
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   428
   False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   429
  by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   430
40681
872b08416fb4 be more precise: only treat constant 'distinct' applied to an explicit list as built-in
boehmes
parents: 40513
diff changeset
   431
lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   432
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   433
lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   434
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   435
definition prime_nat :: "nat \<Rightarrow> bool" where
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   436
  "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   437
lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def)
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   438
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   439
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   440
section {* Pairs *}
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   441
41132
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   442
lemma "fst (x, y) = a \<Longrightarrow> x = a"
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   443
  using fst_conv
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   444
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   445
41132
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   446
lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   447
  using fst_conv snd_conv
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   448
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   449
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   450
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   451
section {* Higher-order problems and recursion *}
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   452
41132
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   453
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i"
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   454
  using fun_upd_same fun_upd_apply
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   455
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   456
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   457
lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   458
  by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   459
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   460
lemma "id 3 = 3 \<and> id True = True" by (smt id_def)
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   461
41132
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   462
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i"
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   463
  using fun_upd_same fun_upd_apply
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   464
  by smt
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   465
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   466
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   467
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   468
lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps)
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   469
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   470
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   471
lemma "(ALL x. P x) | ~ All P" by smt
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   472
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   473
fun dec_10 :: "nat \<Rightarrow> nat" where
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   474
  "dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   475
lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   476
36899
bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents: 36898
diff changeset
   477
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   478
axiomatization
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   479
  eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   480
  where
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   481
  eval_dioph_mod:
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   482
  "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   483
  and
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   484
  eval_dioph_div_mult:
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   485
  "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n +
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   486
   eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   487
lemma
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   488
  "(eval_dioph ks xs = l) =
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   489
   (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and>
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   490
    eval_dioph ks (map (\<lambda>x. x div 2) xs) =
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   491
      (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"
41132
42384824b732 updated SMT certificates
boehmes
parents: 40681
diff changeset
   492
  using [[smt_oracle=true]] (*FIXME*)
36898
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   493
  by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   494
8e55aa1306c5 integrated SMT into the HOL image
boehmes
parents:
diff changeset
   495
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section {* Monomorphization examples *}
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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)
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axiomatization g :: "'a \<Rightarrow> nat"
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axiomatization where
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  g1: "g (Some x) = g [x]" and
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  g2: "g None = g []" and
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  g3: "g xs = length xs"
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lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size)
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end