src/HOL/SMT_Examples/SMT_Examples.thy

 
 theory SMT_Examples
 imports Complex_Main
 begin
 
-declare [[smt2_certificates = "SMT_Examples.certs2"]]
-declare [[smt2_read_only_certificates = true]]
+declare [[smt_certificates = "SMT_Examples.certs2"]]
+declare [[smt_read_only_certificates = true]]
 
 
 section {* Propositional and first-order logic *}
 
-lemma "True" by smt2
-lemma "p \<or> \<not>p" by smt2
-lemma "(p \<and> True) = p" by smt2
-lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt2
-lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" by smt2
-lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt2
-lemma "P = P = P = P = P = P = P = P = P = P" by smt2
+lemma "True" by smt
+lemma "p \<or> \<not>p" by smt
+lemma "(p \<and> True) = p" by smt
+lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt
+lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" by smt
+lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt
+lemma "P = P = P = P = P = P = P = P = P = P" by smt
 
 lemma
   assumes "a \<or> b \<or> c \<or> d"
       and "e \<or> f \<or> (a \<and> d)"
       and "\<not> (a \<or> (c \<and> ~c)) \<or> b"
       and "\<not> (b \<and> (x \<or> \<not> x)) \<or> c"
       and "\<not> (d \<or> False) \<or> c"
       and "\<not> (c \<or> (\<not> p \<and> (p \<or> (q \<and> \<not> q))))"
   shows False
-  using assms by smt2
+  using assms by smt
 
 axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   symm_f: "symm_f x y = symm_f y x"
 
-lemma "a = a \<and> symm_f a b = symm_f b a" by (smt2 symm_f)
+lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f)
 
 (*
 Taken from ~~/src/HOL/ex/SAT_Examples.thy.
 Translated from TPTP problem library: PUZ015-2.006.dimacs
 *)

   and "~x27 \<or> ~x57"
   and "~x29 \<or> ~x28"
   and "~x29 \<or> ~x58"
   and "~x28 \<or> ~x58"
   shows False
-  using assms by smt2
+  using assms by smt
 
 lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
-  by smt2
+  by smt
 
 lemma
   assumes "(\<forall>x y. P x y = x)"
   shows "(\<exists>y. P x y) = P x c"
-  using assms by smt2
+  using assms by smt
 
 lemma
   assumes "(\<forall>x y. P x y = x)"
   and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)"
   shows "(EX y. P x y) = P x c"
-  using assms by smt2
+  using assms by smt
 
 lemma
   assumes "if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)"
   shows "P x \<longrightarrow> P y"
-  using assms by smt2
+  using assms by smt
 
 
 section {* Arithmetic *}
 
 subsection {* Linear arithmetic over integers and reals *}
 
-lemma "(3::int) = 3" by smt2
-lemma "(3::real) = 3" by smt2
-lemma "(3 :: int) + 1 = 4" by smt2
-lemma "x + (y + z) = y + (z + (x::int))" by smt2
-lemma "max (3::int) 8 > 5" by smt2
-lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt2
-lemma "P ((2::int) < 3) = P True" by smt2
-lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt2
+lemma "(3::int) = 3" by smt
+lemma "(3::real) = 3" by smt
+lemma "(3 :: int) + 1 = 4" by smt
+lemma "x + (y + z) = y + (z + (x::int))" by smt
+lemma "max (3::int) 8 > 5" by smt
+lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt
+lemma "P ((2::int) < 3) = P True" by smt
+lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt
 
 lemma
   assumes "x \<ge> (3::int)" and "y = x + 4"
   shows "y - x > 0"
-  using assms by smt2
-
-lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt2
+  using assms by smt
+
+lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt
 
 lemma
   fixes x :: real
   assumes "3 * x + 7 * a < 4" and "3 < 2 * x"
   shows "a < 0"
-  using assms by smt2
-
-lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt2
+  using assms by smt
+
+lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
 
 lemma "
   (n < m \<and> m < n') \<or> (n < m \<and> m = n') \<or> (n < n' \<and> n' < m) \<or>
   (n = n' \<and> n' < m) \<or> (n = m \<and> m < n') \<or>
   (n' < m \<and> m < n) \<or> (n' < m \<and> m = n) \<or>
   (n' < n \<and> n < m) \<or> (n' = n \<and> n < m) \<or> (n' = m \<and> m < n) \<or>
   (m < n \<and> n < n') \<or> (m < n \<and> n' = n) \<or> (m < n' \<and> n' < n) \<or>
   (m = n \<and> n < n') \<or> (m = n' \<and> n' < n) \<or>
   (n' = m \<and> m = (n::int))"
-  by smt2
+  by smt
 
 text{*
 The following example was taken from HOL/ex/PresburgerEx.thy, where it says:
 
   This following theorem proves that all solutions to the

 
 lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3;
          x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6;
          x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk>
  \<Longrightarrow> x1 = x10 \<and> x2 = (x11::int)"
-  by smt2
-
-
-lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt2
+  by smt
+
+
+lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt
 
 lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)"
-  using [[z3_new_extensions]] by smt2
+  using [[z3_extensions]] by smt
 
 lemma "x + (let y = x mod 2 in y + y) < x + (3::int)"
-  using [[z3_new_extensions]] by smt2
+  using [[z3_extensions]] by smt
 
 lemma
   assumes "x \<noteq> (0::real)"
   shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not> P then 4 else 2) * x"
-  using assms [[z3_new_extensions]] by smt2
+  using assms [[z3_extensions]] by smt
 
 lemma
   assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"
   shows "n mod 2 = 1 \<and> m mod 2 = (1::int)"
-  using assms [[z3_new_extensions]] by smt2
+  using assms [[z3_extensions]] by smt
 
 
 subsection {* Linear arithmetic with quantifiers *}
 
-lemma "~ (\<exists>x::int. False)" by smt2
-lemma "~ (\<exists>x::real. False)" by smt2
-
-lemma "\<exists>x::int. 0 < x" by smt2
+lemma "~ (\<exists>x::int. False)" by smt
+lemma "~ (\<exists>x::real. False)" by smt
+
+lemma "\<exists>x::int. 0 < x" by smt
 
 lemma "\<exists>x::real. 0 < x"
-  using [[smt2_oracle=true]] (* no Z3 proof *)
-  by smt2
-
-lemma "\<forall>x::int. \<exists>y. y > x" by smt2
-
-lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt2
-lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt2
-lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt2
-lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt2
-lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt2
-lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt2
-lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt2
-lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt2
-lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt2
-lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt2
-lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt2
-lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt2
+  using [[smt_oracle=true]] (* no Z3 proof *)
+  by smt
+
+lemma "\<forall>x::int. \<exists>y. y > x" by smt
+
+lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt
+lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt
+lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt
+lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt
+lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt
+lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt
+lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt
+lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt
+lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt
+lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt
+lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt
+lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt
 lemma "\<forall>x::int.
-  SMT2.trigger (SMT2.Symb_Cons (SMT2.Symb_Cons (SMT2.pat x) SMT2.Symb_Nil) SMT2.Symb_Nil)
-    (x < a \<longrightarrow> 2 * x < 2 * a)" by smt2
-lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by smt2
+  SMT.trigger (SMT.Symb_Cons (SMT.Symb_Cons (SMT.pat x) SMT.Symb_Nil) SMT.Symb_Nil)
+    (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
+lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by smt
 
 
 subsection {* Non-linear arithmetic over integers and reals *}
 
 lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
-  using [[smt2_oracle, z3_new_extensions]]
-  by smt2
+  using [[smt_oracle, z3_extensions]]
+  by smt
 
 lemma  "(a::int) * (x + 1 + y) = a * x + a * (y + 1)"
-  using [[z3_new_extensions]]
-  by smt2
+  using [[z3_extensions]]
+  by smt
 
 lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)"
-  using [[z3_new_extensions]]
-  by smt2
+  using [[z3_extensions]]
+  by smt
 
 lemma
   "(U::int) + (1 + p) * (b + e) + p * d =
    U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)"
-  using [[z3_new_extensions]] by smt2
-
-lemma [z3_new_rule]:
+  using [[z3_extensions]] by smt
+
+lemma [z3_rule]:
   fixes x :: "int"
   assumes "x * y \<le> 0" and "\<not> y \<le> 0" and "\<not> x \<le> 0"
   shows False
   using assms by (metis mult_le_0_iff)
 
 
 section {* Pairs *}
 
 lemma "fst (x, y) = a \<Longrightarrow> x = a"
-  using fst_conv by smt2
+  using fst_conv by smt
 
 lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
-  using fst_conv snd_conv by smt2
+  using fst_conv snd_conv by smt
 
 
 section {* Higher-order problems and recursion *}
 
 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i"
-  using fun_upd_same fun_upd_apply by smt2
+  using fun_upd_same fun_upd_apply by smt
 
 lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
-  by smt2
+  by smt
 
 lemma "id x = x \<and> id True = True"
-  by (smt2 id_def)
+  by (smt id_def)
 
 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i"
-  using fun_upd_same fun_upd_apply by smt2
+  using fun_upd_same fun_upd_apply by smt
 
 lemma
   "f (\<exists>x. g x) \<Longrightarrow> True"
   "f (\<forall>x. g x) \<Longrightarrow> True"
-  by smt2+
-
-lemma True using let_rsp by smt2
-lemma "le = op \<le> \<Longrightarrow> le (3::int) 42" by smt2
-lemma "map (\<lambda>i::int. i + 1) [0, 1] = [1, 2]" by (smt2 list.map)
-lemma "(ALL x. P x) \<or> ~ All P" by smt2
+  by smt+
+
+lemma True using let_rsp by smt
+lemma "le = op \<le> \<Longrightarrow> le (3::int) 42" by smt
+lemma "map (\<lambda>i::int. i + 1) [0, 1] = [1, 2]" by (smt list.map)
+lemma "(ALL x. P x) \<or> ~ All P" by smt
 
 fun dec_10 :: "int \<Rightarrow> int" where
   "dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
 
-lemma "dec_10 (4 * dec_10 4) = 6" by (smt2 dec_10.simps)
+lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
 
 axiomatization
   eval_dioph :: "int list \<Rightarrow> int list \<Rightarrow> int"
 where
   eval_dioph_mod: "eval_dioph ks xs mod n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod n"

 
 lemma
   "(eval_dioph ks xs = l) =
    (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and>
     eval_dioph ks (map (\<lambda>x. x div 2) xs) = (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"
-  using [[smt2_oracle = true]] (*FIXME*)
-  using [[z3_new_extensions]]
-  by (smt2 eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
+  using [[smt_oracle = true]] (*FIXME*)
+  using [[z3_extensions]]
+  by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
 
 
 context complete_lattice
 begin
 
 lemma
   assumes "Sup {a | i::bool. True} \<le> Sup {b | i::bool. True}"
   and "Sup {b | i::bool. True} \<le> Sup {a | i::bool. True}"
   shows "Sup {a | i::bool. True} \<le> Sup {a | i::bool. True}"
-  using assms by (smt2 order_trans)
+  using assms by (smt order_trans)
 
 end
 
 
 section {* Monomorphization examples *}
 
 definition Pred :: "'a \<Rightarrow> bool" where "Pred x = True"
 
 lemma poly_Pred: "Pred x \<and> (Pred [x] \<or> \<not> Pred [x])" by (simp add: Pred_def)
 
-lemma "Pred (1::int)" by (smt2 poly_Pred)
+lemma "Pred (1::int)" by (smt poly_Pred)
 
 axiomatization g :: "'a \<Rightarrow> nat"
 axiomatization where
   g1: "g (Some x) = g [x]" and
   g2: "g None = g []" and
   g3: "g xs = length xs"
 
-lemma "g (Some (3::int)) = g (Some True)" by (smt2 g1 g2 g3 list.size)
+lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size)
 
 end