(* Title: HOL/SMT_Examples/SMT_Examples.thy Author: Sascha Boehme, TU Muenchen *) section ‹Examples for the SMT binding› theory SMT_Examples imports Complex_Main begin declare [[smt_certificates = "SMT_Examples.certs"]] declare [[smt_read_only_certificates = true]] section ‹Propositional and first-order logic› lemma "True" by smt lemma "p ∨ ¬p" by smt lemma "(p ∧ True) = p" by smt lemma "(p ∨ q) ∧ ¬p ⟹ q" by smt lemma "(a ∧ b) ∨ (c ∧ d) ⟹ (a ∧ b) ∨ (c ∧ d)" by smt lemma "(p1 ∧ p2) ∨ p3 ⟶ (p1 ⟶ (p3 ∧ p2) ∨ (p1 ∧ p3)) ∨ 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 ⇒ 'a ⇒ 'a" where symm_f: "symm_f x y = symm_f y x" lemma "a = a ∧ 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 "∀x::int. P x ⟶ (∀y::int. P x ∨ P y)" by smt lemma assumes "(∀x y. P x y = x)" shows "(∃y. P x y) = P x c" using assms by smt lemma assumes "(∀x y. P x y = x)" and "(∀x. ∃y. P x y) = (∀x. P x c)" shows "(EX y. P x y) = P x c" using assms by smt lemma assumes "if P x then ¬(∃y. P y) else (∀y. ¬P y)" shows "P x ⟶ 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 "¦x :: real¦ + ¦y¦ ≥ ¦x + y¦" by smt lemma "P ((2::int) < 3) = P True" by smt lemma "x + 3 ≥ 4 ∨ x < (1::int)" by smt lemma assumes "x ≥ (3::int)" and "y = x + 4" shows "y - x > 0" using assms by smt lemma "let x = (2 :: int) in x + x ≠ 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 ≤ y + -1 * x ∨ ¬ 0 ≤ x ∨ 0 ≤ (x::int)) = (¬ False)" 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 "⟦ x3 = ¦x2¦ - x1; x4 = ¦x3¦ - x2; x5 = ¦x4¦ - x3; x6 = ¦x5¦ - x4; x7 = ¦x6¦ - x5; x8 = ¦x7¦ - x6; x9 = ¦x8¦ - x7; x10 = ¦x9¦ - x8; x11 = ¦x10¦ - x9 ⟧ ⟹ x1 = x10 ∧ x2 = (x11::int)" by smt lemma "let P = 2 * x + 1 > x + (x::real) in P ∨ False ∨ P" by smt lemma "x + (let y = x mod 2 in 2 * y + 1) ≥ x + (1::int)" using [[z3_extensions]] by smt lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" using [[z3_extensions]] by smt lemma assumes "x ≠ (0::real)" shows "x + x ≠ (let P = (¦x¦ > 1) in if P ∨ ¬ P then 4 else 2) * x" using assms [[z3_extensions]] by smt subsection ‹Linear arithmetic with quantifiers› lemma "~ (∃x::int. False)" by smt lemma "~ (∃x::real. False)" by smt lemma "∃x::int. 0 < x" by smt lemma "∃x::real. 0 < x" using [[smt_oracle=true]] (* no Z3 proof *) by smt lemma "∀x::int. ∃y. y > x" by smt lemma "∀x y::int. (x = 0 ∧ y = 1) ⟶ x ≠ y" by smt lemma "∃x::int. ∀y. x < y ⟶ y < 0 ∨ y >= 0" by smt lemma "∀x y::int. x < y ⟶ (2 * x + 1) < (2 * y)" by smt lemma "∀x y::int. (2 * x + 1) ≠ (2 * y)" by smt lemma "∀x y::int. x + y > 2 ∨ x + y = 2 ∨ x + y < 2" by smt lemma "∀x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt lemma "if (ALL x::int. x < 0 ∨ x > 0) then False else True" by smt lemma "(if (ALL x::int. x < 0 ∨ x > 0) then -1 else 3) > (0::int)" by smt lemma "~ (∃x y z::int. 4 * x + -6 * y = (1::int))" by smt lemma "∃x::int. ∀x y. 0 < x ∧ 0 < y ⟶ (0::int) < x + y" by smt lemma "∃u::int. ∀(x::int) y::real. 0 < x ∧ 0 < y ⟶ -1 < x" by smt lemma "∃x::int. (∀y. y ≥ x ⟶ y > 0) ⟶ x > 0" by smt lemma "∀(a::int) b::int. 0 < b ∨ b < 1" by smt subsection ‹Non-linear arithmetic over integers and reals› lemma "a > (0::int) ⟹ a*b > 0 ⟹ b > 0" using [[smt_oracle, z3_extensions]] by smt lemma "(a::int) * (x + 1 + y) = a * x + a * (y + 1)" using [[z3_extensions]] by smt lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)" 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_extensions]] by smt lemma [z3_rule]: fixes x :: "int" assumes "x * y ≤ 0" and "¬ y ≤ 0" and "¬ x ≤ 0" shows False using assms by (metis mult_le_0_iff) subsection {* Linear arithmetic for natural numbers *} declare [[smt_nat_as_int]] lemma "2 * (x::nat) ≠ 1" by smt lemma "a < 3 ⟹ (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 ∨ P = (x - 1 = y) ∨ (¬P ⟶ False)" by smt lemma "int (nat ¦x::int¦) = ¦x¦" by smt definition prime_nat :: "nat ⇒ bool" where "prime_nat p = (1 < p ∧ (∀m. m dvd p --> m = 1 ∨ m = p))" lemma "prime_nat (4*m + 1) ⟹ m ≥ (1::nat)" by (smt prime_nat_def) declare [[smt_nat_as_int = false]] section ‹Pairs› lemma "fst (x, y) = a ⟹ x = a" using fst_conv by smt lemma "p1 = (x, y) ∧ p2 = (y, x) ⟹ fst p1 = snd p2" using fst_conv snd_conv by smt section ‹Higher-order problems and recursion› lemma "i ≠ i1 ∧ i ≠ i2 ⟹ (f (i1 := v1, i2 := v2)) i = f i" using fun_upd_same fun_upd_apply by smt lemma "(f g (x::'a::type) = (g x ∧ True)) ∨ (f g x = True) ∨ (g x = True)" by smt lemma "id x = x ∧ id True = True" by (smt id_def) lemma "i ≠ i1 ∧ i ≠ i2 ⟹ ((f (i1 := v1)) (i2 := v2)) i = f i" using fun_upd_same fun_upd_apply by smt lemma "f (∃x. g x) ⟹ True" "f (∀x. g x) ⟹ True" by smt+ lemma True using let_rsp by smt lemma "le = op ≤ ⟹ le (3::int) 42" by smt lemma "map (λi::int. i + 1) [0, 1] = [1, 2]" by (smt list.map) lemma "(ALL x. P x) ∨ ~ All P" by smt fun dec_10 :: "int ⇒ int" 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 ⇒ int list ⇒ int" where eval_dioph_mod: "eval_dioph ks xs mod n = eval_dioph ks (map (λx. x mod n) xs) mod n" and eval_dioph_div_mult: "eval_dioph ks (map (λx. x div n) xs) * n + eval_dioph ks (map (λx. x mod n) xs) = eval_dioph ks xs" lemma "(eval_dioph ks xs = l) = (eval_dioph ks (map (λx. x mod 2) xs) mod 2 = l mod 2 ∧ eval_dioph ks (map (λx. x div 2) xs) = (l - eval_dioph ks (map (λx. x mod 2) xs)) div 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} ≤ Sup {b | i::bool. True}" and "Sup {b | i::bool. True} ≤ Sup {a | i::bool. True}" shows "Sup {a | i::bool. True} ≤ Sup {a | i::bool. True}" using assms by (smt order_trans) end section ‹Monomorphization examples› definition Pred :: "'a ⇒ bool" where "Pred x = True" lemma poly_Pred: "Pred x ∧ (Pred [x] ∨ ¬ Pred [x])" by (simp add: Pred_def) lemma "Pred (1::int)" by (smt poly_Pred) axiomatization g :: "'a ⇒ 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