(* Title: HOL/SMT_Examples/SMT_Examples.thy
Author: Sascha Boehme, TU Muenchen
*)
header {* Examples for the SMT binding *}
theory SMT_Examples
imports Complex_Main
begin
declare [[smt_solver=z3, z3_proofs=true]]
declare [[smt_certificates="~~/src/HOL/SMT_Examples/SMT_Examples.certs"]]
declare [[smt_fixed=true]]
section {* Propositional and first-order logic *}
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)"
using [[z3_proofs=false]] (* no Z3 proof *)
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 | b | c | d"
and "e | f | (a & d)"
and "~(a | (c & ~c)) | b"
and "~(b & (x | ~x)) | c"
and "~(d | False) | c"
and "~(c | (~p & (p | (q & ~q))))"
shows False
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 (smt symm_f)
(*
Taken from ~~/src/HOL/ex/SAT_Examples.thy.
Translated from TPTP problem library: PUZ015-2.006.dimacs
*)
lemma
assumes "~x0"
and "~x30"
and "~x29"
and "~x59"
and "x1 | x31 | x0"
and "x2 | x32 | x1"
and "x3 | x33 | x2"
and "x4 | x34 | x3"
and "x35 | x4"
and "x5 | x36 | x30"
and "x6 | x37 | x5 | x31"
and "x7 | x38 | x6 | x32"
and "x8 | x39 | x7 | x33"
and "x9 | x40 | x8 | x34"
and "x41 | x9 | x35"
and "x10 | x42 | x36"
and "x11 | x43 | x10 | x37"
and "x12 | x44 | x11 | x38"
and "x13 | x45 | x12 | x39"
and "x14 | x46 | x13 | x40"
and "x47 | x14 | x41"
and "x15 | x48 | x42"
and "x16 | x49 | x15 | x43"
and "x17 | x50 | x16 | x44"
and "x18 | x51 | x17 | x45"
and "x19 | x52 | x18 | x46"
and "x53 | x19 | x47"
and "x20 | x54 | x48"
and "x21 | x55 | x20 | x49"
and "x22 | x56 | x21 | x50"
and "x23 | x57 | x22 | x51"
and "x24 | x58 | x23 | x52"
and "x59 | x24 | x53"
and "x25 | x54"
and "x26 | x25 | x55"
and "x27 | x26 | x56"
and "x28 | x27 | x57"
and "x29 | x28 | x58"
and "~x1 | ~x31"
and "~x1 | ~x0"
and "~x31 | ~x0"
and "~x2 | ~x32"
and "~x2 | ~x1"
and "~x32 | ~x1"
and "~x3 | ~x33"
and "~x3 | ~x2"
and "~x33 | ~x2"
and "~x4 | ~x34"
and "~x4 | ~x3"
and "~x34 | ~x3"
and "~x35 | ~x4"
and "~x5 | ~x36"
and "~x5 | ~x30"
and "~x36 | ~x30"
and "~x6 | ~x37"
and "~x6 | ~x5"
and "~x6 | ~x31"
and "~x37 | ~x5"
and "~x37 | ~x31"
and "~x5 | ~x31"
and "~x7 | ~x38"
and "~x7 | ~x6"
and "~x7 | ~x32"
and "~x38 | ~x6"
and "~x38 | ~x32"
and "~x6 | ~x32"
and "~x8 | ~x39"
and "~x8 | ~x7"
and "~x8 | ~x33"
and "~x39 | ~x7"
and "~x39 | ~x33"
and "~x7 | ~x33"
and "~x9 | ~x40"
and "~x9 | ~x8"
and "~x9 | ~x34"
and "~x40 | ~x8"
and "~x40 | ~x34"
and "~x8 | ~x34"
and "~x41 | ~x9"
and "~x41 | ~x35"
and "~x9 | ~x35"
and "~x10 | ~x42"
and "~x10 | ~x36"
and "~x42 | ~x36"
and "~x11 | ~x43"
and "~x11 | ~x10"
and "~x11 | ~x37"
and "~x43 | ~x10"
and "~x43 | ~x37"
and "~x10 | ~x37"
and "~x12 | ~x44"
and "~x12 | ~x11"
and "~x12 | ~x38"
and "~x44 | ~x11"
and "~x44 | ~x38"
and "~x11 | ~x38"
and "~x13 | ~x45"
and "~x13 | ~x12"
and "~x13 | ~x39"
and "~x45 | ~x12"
and "~x45 | ~x39"
and "~x12 | ~x39"
and "~x14 | ~x46"
and "~x14 | ~x13"
and "~x14 | ~x40"
and "~x46 | ~x13"
and "~x46 | ~x40"
and "~x13 | ~x40"
and "~x47 | ~x14"
and "~x47 | ~x41"
and "~x14 | ~x41"
and "~x15 | ~x48"
and "~x15 | ~x42"
and "~x48 | ~x42"
and "~x16 | ~x49"
and "~x16 | ~x15"
and "~x16 | ~x43"
and "~x49 | ~x15"
and "~x49 | ~x43"
and "~x15 | ~x43"
and "~x17 | ~x50"
and "~x17 | ~x16"
and "~x17 | ~x44"
and "~x50 | ~x16"
and "~x50 | ~x44"
and "~x16 | ~x44"
and "~x18 | ~x51"
and "~x18 | ~x17"
and "~x18 | ~x45"
and "~x51 | ~x17"
and "~x51 | ~x45"
and "~x17 | ~x45"
and "~x19 | ~x52"
and "~x19 | ~x18"
and "~x19 | ~x46"
and "~x52 | ~x18"
and "~x52 | ~x46"
and "~x18 | ~x46"
and "~x53 | ~x19"
and "~x53 | ~x47"
and "~x19 | ~x47"
and "~x20 | ~x54"
and "~x20 | ~x48"
and "~x54 | ~x48"
and "~x21 | ~x55"
and "~x21 | ~x20"
and "~x21 | ~x49"
and "~x55 | ~x20"
and "~x55 | ~x49"
and "~x20 | ~x49"
and "~x22 | ~x56"
and "~x22 | ~x21"
and "~x22 | ~x50"
and "~x56 | ~x21"
and "~x56 | ~x50"
and "~x21 | ~x50"
and "~x23 | ~x57"
and "~x23 | ~x22"
and "~x23 | ~x51"
and "~x57 | ~x22"
and "~x57 | ~x51"
and "~x22 | ~x51"
and "~x24 | ~x58"
and "~x24 | ~x23"
and "~x24 | ~x52"
and "~x58 | ~x23"
and "~x58 | ~x52"
and "~x23 | ~x52"
and "~x59 | ~x24"
and "~x59 | ~x53"
and "~x24 | ~x53"
and "~x25 | ~x54"
and "~x26 | ~x25"
and "~x26 | ~x55"
and "~x25 | ~x55"
and "~x27 | ~x26"
and "~x27 | ~x56"
and "~x26 | ~x56"
and "~x28 | ~x27"
and "~x28 | ~x57"
and "~x27 | ~x57"
and "~x29 | ~x28"
and "~x29 | ~x58"
and "~x28 | ~x58"
shows False
using assms by smt
lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
by smt
lemma
assumes "(\<forall>x y. P x y = x)"
shows "(\<exists>y. P x y) = P x c"
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 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 smt
section {* Arithmetic *}
subsection {* Linear arithmetic over integers and reals *}
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 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 smt
lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
lemma "distinct [x < (3::int), 3 \<le> x]" by smt
lemma
assumes "a > (0::int)"
shows "distinct [a, a * 2, a - a]"
using assms by smt
lemma "
(n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
(n = n' & n' < m) | (n = m & m < n') |
(n' < m & m < n) | (n' < m & m = n) |
(n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) |
(m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) |
(m = n & n < n') | (m = n' & n' < n) |
(n' = m & m = (n::int))"
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
recurrence relation $x_{i+2} = |x_{i+1}| - x_i$ are periodic with
period 9. The example was brought to our attention by John
Harrison. It does does not require Presburger arithmetic but merely
quantifier-free linear arithmetic and holds for the rationals as well.
Warning: it takes (in 2006) over 4.2 minutes!
There, it is proved by "arith". SMT is able to prove this within a fraction
of one second. With proof reconstruction, it takes about 13 seconds on a Core2
processor.
*}
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 & x2 = (x11::int)"
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)"
sorry (* FIXME: div/mod *)
lemma "x + (let y = x mod 2 in y + y) < x + (3::int)"
sorry (* FIXME: div/mod *)
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 by smt
lemma
assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"
shows "n mod 2 = 1 & m mod 2 = (1::int)"
using assms sorry (* FIXME: div/mod *)
subsection {* Linear arithmetic with quantifiers *}
lemma "~ (\<exists>x::int. False)" by smt
lemma "~ (\<exists>x::real. False)" by smt
lemma "\<exists>x::int. 0 < x"
using [[z3_proofs=false]] (* no Z3 proof *)
by smt
lemma "\<exists>x::real. 0 < x"
using [[z3_proofs=false]] (* no Z3 proof *)
by smt
lemma "\<forall>x::int. \<exists>y. y > x"
using [[z3_proofs=false]] (* no Z3 proof *)
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. trigger [pat x] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
subsection {* Non-linear arithmetic over integers and reals *}
lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
using [[z3_proofs=false]] -- {* Isabelle's arithmetic decision procedures
are too weak to automatically prove @{thm zero_less_mult_pos}. *}
by smt (* FIXME: use z3_rule *)
lemma "(a::int) * (x + 1 + y) = a * x + a * (y + 1)" by smt
lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)" 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)"
by smt
subsection {* Linear arithmetic for natural numbers *}
lemma "2 * (x::nat) ~= 1" by smt
lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt
lemma "let x = (1::nat) + y in x - y > 0 * x" by smt
lemma
"let x = (1::nat) + y in
let P = (if x > 0 then True else False) in
False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"
by smt
lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt
lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt
definition prime_nat :: "nat \<Rightarrow> bool" where
"prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def)
section {* Pairs *}
lemma "fst (x, y) = a \<Longrightarrow> x = a" by smt
lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" 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" by smt
lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
by smt
lemma "id 3 = 3 \<and> id True = True" by (smt id_def)
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" by smt
lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps)
lemma "(ALL x. P x) | ~ All P" by smt
fun dec_10 :: "nat \<Rightarrow> nat" where
"dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
axiomatization
eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int"
where
eval_dioph_mod:
"eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"
and
eval_dioph_div_mult:
"eval_dioph ks (map (\<lambda>x. x div n) xs) * int n +
eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"
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)"
sorry (* FIXME: div/mod *)
(*
by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
*)
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 (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 (smt g1 g2 g3 list.size)
end