(* Title: HOL/SMT_Examples/SMT_Examples_Verit.thy
Author: Sascha Boehme, TU Muenchen
Author: Mathias Fleury, JKU
Half of the examples come from the corresponding file for z3,
the others come from the Isabelle distribution or the AFP.
*)
section \<open>Examples for the (smt (verit)) binding\<close>
theory SMT_Examples_Verit
imports Complex_Main
begin
external_file \<open>SMT_Examples_Verit.certs\<close>
declare [[smt_certificates = "SMT_Examples_Verit.certs"]]
declare [[smt_read_only_certificates = true]]
section \<open>Propositional and first-order logic\<close>
lemma "True" by (smt (verit))
lemma "p \<or> \<not>p" by (smt (verit))
lemma "(p \<and> True) = p" by (smt (verit))
lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by (smt (verit))
lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" by (smt (verit))
lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by (smt (verit))
lemma "P = P = P = P = P = P = P = P = P = P" by (smt (verit))
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 (smt (verit))
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 (verit) 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 \<or> x31 \<or> x0"
and "x2 \<or> x32 \<or> x1"
and "x3 \<or> x33 \<or> x2"
and "x4 \<or> x34 \<or> x3"
and "x35 \<or> x4"
and "x5 \<or> x36 \<or> x30"
and "x6 \<or> x37 \<or> x5 \<or> x31"
and "x7 \<or> x38 \<or> x6 \<or> x32"
and "x8 \<or> x39 \<or> x7 \<or> x33"
and "x9 \<or> x40 \<or> x8 \<or> x34"
and "x41 \<or> x9 \<or> x35"
and "x10 \<or> x42 \<or> x36"
and "x11 \<or> x43 \<or> x10 \<or> x37"
and "x12 \<or> x44 \<or> x11 \<or> x38"
and "x13 \<or> x45 \<or> x12 \<or> x39"
and "x14 \<or> x46 \<or> x13 \<or> x40"
and "x47 \<or> x14 \<or> x41"
and "x15 \<or> x48 \<or> x42"
and "x16 \<or> x49 \<or> x15 \<or> x43"
and "x17 \<or> x50 \<or> x16 \<or> x44"
and "x18 \<or> x51 \<or> x17 \<or> x45"
and "x19 \<or> x52 \<or> x18 \<or> x46"
and "x53 \<or> x19 \<or> x47"
and "x20 \<or> x54 \<or> x48"
and "x21 \<or> x55 \<or> x20 \<or> x49"
and "x22 \<or> x56 \<or> x21 \<or> x50"
and "x23 \<or> x57 \<or> x22 \<or> x51"
and "x24 \<or> x58 \<or> x23 \<or> x52"
and "x59 \<or> x24 \<or> x53"
and "x25 \<or> x54"
and "x26 \<or> x25 \<or> x55"
and "x27 \<or> x26 \<or> x56"
and "x28 \<or> x27 \<or> x57"
and "x29 \<or> x28 \<or> x58"
and "~x1 \<or> ~x31"
and "~x1 \<or> ~x0"
and "~x31 \<or> ~x0"
and "~x2 \<or> ~x32"
and "~x2 \<or> ~x1"
and "~x32 \<or> ~x1"
and "~x3 \<or> ~x33"
and "~x3 \<or> ~x2"
and "~x33 \<or> ~x2"
and "~x4 \<or> ~x34"
and "~x4 \<or> ~x3"
and "~x34 \<or> ~x3"
and "~x35 \<or> ~x4"
and "~x5 \<or> ~x36"
and "~x5 \<or> ~x30"
and "~x36 \<or> ~x30"
and "~x6 \<or> ~x37"
and "~x6 \<or> ~x5"
and "~x6 \<or> ~x31"
and "~x37 \<or> ~x5"
and "~x37 \<or> ~x31"
and "~x5 \<or> ~x31"
and "~x7 \<or> ~x38"
and "~x7 \<or> ~x6"
and "~x7 \<or> ~x32"
and "~x38 \<or> ~x6"
and "~x38 \<or> ~x32"
and "~x6 \<or> ~x32"
and "~x8 \<or> ~x39"
and "~x8 \<or> ~x7"
and "~x8 \<or> ~x33"
and "~x39 \<or> ~x7"
and "~x39 \<or> ~x33"
and "~x7 \<or> ~x33"
and "~x9 \<or> ~x40"
and "~x9 \<or> ~x8"
and "~x9 \<or> ~x34"
and "~x40 \<or> ~x8"
and "~x40 \<or> ~x34"
and "~x8 \<or> ~x34"
and "~x41 \<or> ~x9"
and "~x41 \<or> ~x35"
and "~x9 \<or> ~x35"
and "~x10 \<or> ~x42"
and "~x10 \<or> ~x36"
and "~x42 \<or> ~x36"
and "~x11 \<or> ~x43"
and "~x11 \<or> ~x10"
and "~x11 \<or> ~x37"
and "~x43 \<or> ~x10"
and "~x43 \<or> ~x37"
and "~x10 \<or> ~x37"
and "~x12 \<or> ~x44"
and "~x12 \<or> ~x11"
and "~x12 \<or> ~x38"
and "~x44 \<or> ~x11"
and "~x44 \<or> ~x38"
and "~x11 \<or> ~x38"
and "~x13 \<or> ~x45"
and "~x13 \<or> ~x12"
and "~x13 \<or> ~x39"
and "~x45 \<or> ~x12"
and "~x45 \<or> ~x39"
and "~x12 \<or> ~x39"
and "~x14 \<or> ~x46"
and "~x14 \<or> ~x13"
and "~x14 \<or> ~x40"
and "~x46 \<or> ~x13"
and "~x46 \<or> ~x40"
and "~x13 \<or> ~x40"
and "~x47 \<or> ~x14"
and "~x47 \<or> ~x41"
and "~x14 \<or> ~x41"
and "~x15 \<or> ~x48"
and "~x15 \<or> ~x42"
and "~x48 \<or> ~x42"
and "~x16 \<or> ~x49"
and "~x16 \<or> ~x15"
and "~x16 \<or> ~x43"
and "~x49 \<or> ~x15"
and "~x49 \<or> ~x43"
and "~x15 \<or> ~x43"
and "~x17 \<or> ~x50"
and "~x17 \<or> ~x16"
and "~x17 \<or> ~x44"
and "~x50 \<or> ~x16"
and "~x50 \<or> ~x44"
and "~x16 \<or> ~x44"
and "~x18 \<or> ~x51"
and "~x18 \<or> ~x17"
and "~x18 \<or> ~x45"
and "~x51 \<or> ~x17"
and "~x51 \<or> ~x45"
and "~x17 \<or> ~x45"
and "~x19 \<or> ~x52"
and "~x19 \<or> ~x18"
and "~x19 \<or> ~x46"
and "~x52 \<or> ~x18"
and "~x52 \<or> ~x46"
and "~x18 \<or> ~x46"
and "~x53 \<or> ~x19"
and "~x53 \<or> ~x47"
and "~x19 \<or> ~x47"
and "~x20 \<or> ~x54"
and "~x20 \<or> ~x48"
and "~x54 \<or> ~x48"
and "~x21 \<or> ~x55"
and "~x21 \<or> ~x20"
and "~x21 \<or> ~x49"
and "~x55 \<or> ~x20"
and "~x55 \<or> ~x49"
and "~x20 \<or> ~x49"
and "~x22 \<or> ~x56"
and "~x22 \<or> ~x21"
and "~x22 \<or> ~x50"
and "~x56 \<or> ~x21"
and "~x56 \<or> ~x50"
and "~x21 \<or> ~x50"
and "~x23 \<or> ~x57"
and "~x23 \<or> ~x22"
and "~x23 \<or> ~x51"
and "~x57 \<or> ~x22"
and "~x57 \<or> ~x51"
and "~x22 \<or> ~x51"
and "~x24 \<or> ~x58"
and "~x24 \<or> ~x23"
and "~x24 \<or> ~x52"
and "~x58 \<or> ~x23"
and "~x58 \<or> ~x52"
and "~x23 \<or> ~x52"
and "~x59 \<or> ~x24"
and "~x59 \<or> ~x53"
and "~x24 \<or> ~x53"
and "~x25 \<or> ~x54"
and "~x26 \<or> ~x25"
and "~x26 \<or> ~x55"
and "~x25 \<or> ~x55"
and "~x27 \<or> ~x26"
and "~x27 \<or> ~x56"
and "~x26 \<or> ~x56"
and "~x28 \<or> ~x27"
and "~x28 \<or> ~x57"
and "~x27 \<or> ~x57"
and "~x29 \<or> ~x28"
and "~x29 \<or> ~x58"
and "~x28 \<or> ~x58"
shows False
using assms by (smt (verit))
lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
by (smt (verit))
lemma
assumes "(\<forall>x y. P x y = x)"
shows "(\<exists>y. P x y) = P x c"
using assms by (smt (verit))
lemma
assumes "(\<forall>x y. P x y = x)"
and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)"
shows "(\<exists>y. P x y) = P x c"
using assms by (smt (verit))
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 (verit))
section \<open>Arithmetic\<close>
subsection \<open>Linear arithmetic over integers and reals\<close>
lemma "(3::int) = 3" by (smt (verit))
lemma "(3::real) = 3" by (smt (verit))
lemma "(3 :: int) + 1 = 4" by (smt (verit))
lemma "x + (y + z) = y + (z + (x::int))" by (smt (verit))
lemma "max (3::int) 8 > 5" by (smt (verit))
lemma "\<bar>x :: real\<bar> + \<bar>y\<bar> \<ge> \<bar>x + y\<bar>" by (smt (verit))
lemma "P ((2::int) < 3) = P True" supply[[smt_trace]] by (smt (verit))
lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by (smt (verit))
lemma
assumes "x \<ge> (3::int)" and "y = x + 4"
shows "y - x > 0"
using assms by (smt (verit))
lemma "let x = (2 :: int) in x + x \<noteq> 5" by (smt (verit))
lemma
fixes x :: int
assumes "3 * x + 7 * a < 4" and "3 < 2 * x"
shows "a < 0"
using assms by (smt (verit))
lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by (smt (verit))
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 (smt (verit))
text\<open>
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 (verit)) is able to prove this within a fraction
of one second. With proof reconstruction, it takes about 13 seconds on a Core2
processor.
\<close>
lemma "\<lbrakk> x3 = \<bar>x2\<bar> - x1; x4 = \<bar>x3\<bar> - x2; x5 = \<bar>x4\<bar> - x3;
x6 = \<bar>x5\<bar> - x4; x7 = \<bar>x6\<bar> - x5; x8 = \<bar>x7\<bar> - x6;
x9 = \<bar>x8\<bar> - x7; x10 = \<bar>x9\<bar> - x8; x11 = \<bar>x10\<bar> - x9 \<rbrakk>
\<Longrightarrow> x1 = x10 \<and> x2 = (x11::int)"
by (smt (verit))
lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by (smt (verit))
subsection \<open>Linear arithmetic with quantifiers\<close>
lemma "~ (\<exists>x::int. False)" by (smt (verit))
lemma "~ (\<exists>x::real. False)" by (smt (verit))
lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by (smt (verit))
lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by (smt (verit))
lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by (smt (verit))
lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by (smt (verit))
lemma "(if (\<forall>x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by (smt (verit))
lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by (smt (verit))
lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by (smt (verit))
lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by (smt (verit))
subsection \<open>Linear arithmetic for natural numbers\<close>
declare [[smt_nat_as_int]]
lemma "2 * (x::nat) \<noteq> 1" by (smt (verit))
lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by (smt (verit))
lemma "let x = (1::nat) + y in x - y > 0 * x" by (smt (verit))
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 (verit))
lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by (smt (verit) int_nat_eq)
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 (verit) prime_nat_def)
lemma "2 * (x::nat) \<noteq> 1"
by (smt (verit))
lemma \<open>2*(x :: int) \<noteq> 1\<close>
by (smt (verit))
declare [[smt_nat_as_int = false]]
section \<open>Pairs\<close>
lemma "fst (x, y) = a \<Longrightarrow> x = a"
using fst_conv by (smt (verit))
lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
using fst_conv snd_conv by (smt (verit))
section \<open>Higher-order problems and recursion\<close>
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 (smt (verit))
lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
by (smt (verit))
lemma "id x = x \<and> id True = True"
by (smt (verit) 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 (smt (verit))
lemma
"f (\<exists>x. g x) \<Longrightarrow> True"
"f (\<forall>x. g x) \<Longrightarrow> True"
by (smt (verit))+
lemma True using let_rsp by (smt (verit))
lemma "le = (\<le>) \<Longrightarrow> le (3::int) 42" by (smt (verit))
lemma "map (\<lambda>i::int. i + 1) [0, 1] = [1, 2]" by (smt (verit) list.map)
lemma "(\<forall>x. P x) \<or> \<not> All P" by (smt (verit))
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 (smt (verit) dec_10.simps)
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 (smt (verit) order_trans)
end
lemma
"eq_set (List.coset xs) (set ys) = rhs"
if "\<And>ys. subset' (List.coset xs) (set ys) = (let n = card (UNIV::'a set) in 0 < n \<and> card (set (xs @ ys)) = n)"
and "\<And>uu A. (uu::'a) \<in> - A \<Longrightarrow> uu \<notin> A"
and "\<And>uu. card (set (uu::'a list)) = length (remdups uu)"
and "\<And>uu. finite (set (uu::'a list))"
and "\<And>uu. (uu::'a) \<in> UNIV"
and "(UNIV::'a set) \<noteq> {}"
and "\<And>c A B P. \<lbrakk>(c::'a) \<in> A \<union> B; c \<in> A \<Longrightarrow> P; c \<in> B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
and "\<And>a b. (a::nat) + b = b + a"
and "\<And>a b. ((a::nat) = a + b) = (b = 0)"
and "card' (set xs) = length (remdups xs)"
and "card' = (card :: 'a set \<Rightarrow> nat)"
and "\<And>A B. \<lbrakk>finite (A::'a set); finite B\<rbrakk> \<Longrightarrow> card A + card B = card (A \<union> B) + card (A \<inter> B)"
and "\<And>A. (card (A::'a set) = 0) = (A = {} \<or> infinite A)"
and "\<And>A. \<lbrakk>finite (UNIV::'a set); card (A::'a set) = card (UNIV::'a set)\<rbrakk> \<Longrightarrow> A = UNIV"
and "\<And>xs. - List.coset (xs::'a list) = set xs"
and "\<And>xs. - set (xs::'a list) = List.coset xs"
and "\<And>A B. (A \<inter> B = {}) = (\<forall>x. (x::'a) \<in> A \<longrightarrow> x \<notin> B)"
and "eq_set = (=)"
and "\<And>A. finite (A::'a set) \<Longrightarrow> finite (- A) = finite (UNIV::'a set)"
and "rhs \<equiv> let n = card (UNIV::'a set) in if n = 0 then False else let xs' = remdups xs; ys' = remdups ys in length xs' + length ys' = n \<and> (\<forall>x\<in>set xs'. x \<notin> set ys') \<and> (\<forall>y\<in>set ys'. y \<notin> set xs')"
and "\<And>xs ys. set ((xs::'a list) @ ys) = set xs \<union> set ys"
and "\<And>A B. ((A::'a set) = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
and "\<And>xs. set (remdups (xs::'a list)) = set xs"
and "subset' = (\<subseteq>)"
and "\<And>A B. (\<And>x. (x::'a) \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
and "\<And>A B. \<lbrakk>(A::'a set) \<subseteq> B; B \<subseteq> A\<rbrakk> \<Longrightarrow> A = B"
and "\<And>A ys. (A \<subseteq> List.coset ys) = (\<forall>y\<in>set ys. (y::'a) \<notin> A)"
using that by (smt (verit, default))
notepad
begin
have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
if \<open>(k, g) \<in> one_chain_typeI\<close>
\<open>\<And>A b B. ({} = (A::(real \<times> real) set) \<inter> insert (b::real \<times> real) (B::(real \<times> real) set)) = (b \<notin> A \<and> {} = A \<inter> B)\<close>
\<open>finite ({} :: (real \<times> real) set)\<close>
\<open>\<And>a A. finite (A::(real \<times> real) set) \<Longrightarrow> finite (insert (a::real \<times> real) A)\<close>
\<open>(i::real \<times> real) = (1::real, 0::real)\<close>
\<open> \<And>a A. (a::real \<times> real) \<in> (A::(real \<times> real) set) \<Longrightarrow> insert a A = A\<close> \<open>j = (0, 1)\<close>
\<open>\<And>x. (x::(real \<times> real) set) \<inter> {} = {}\<close>
\<open>\<And>y x A. insert (x::real \<times> real) (insert (y::real \<times> real) (A::(real \<times> real) set)) = insert y (insert x A)\<close>
\<open>\<And>a A. insert (a::real \<times> real) (A::(real \<times> real) set) = {a} \<union> A\<close>
\<open>\<And>F u basis2 basis1 \<gamma>. finite (u :: (real \<times> real) set) \<Longrightarrow>
line_integral_exists F basis1 \<gamma> \<Longrightarrow>
line_integral_exists F basis2 \<gamma> \<Longrightarrow>
basis1 \<union> basis2 = u \<Longrightarrow>
basis1 \<inter> basis2 = {} \<Longrightarrow>
line_integral F u \<gamma> = line_integral F basis1 \<gamma> + line_integral F basis2 \<gamma>\<close>
\<open>one_chain_line_integral F {i} one_chain_typeI =
one_chain_line_integral F {i} one_chain_typeII \<and>
(\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>) \<and>
(\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>)\<close>
\<open> one_chain_line_integral (F::real \<times> real \<Rightarrow> real \<times> real) {j::real \<times> real}
(one_chain_typeII::(int \<times> (real \<Rightarrow> real \<times> real)) set) =
one_chain_line_integral F {j} (one_chain_typeI::(int \<times> (real \<Rightarrow> real \<times> real)) set) \<and>
(\<forall>(k::int, \<gamma>::real \<Rightarrow> real \<times> real)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>) \<and>
(\<forall>(k::int, \<gamma>::real \<Rightarrow> real \<times> real)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>)\<close>
for F i j g one_chain_typeI one_chain_typeII and
line_integral :: \<open>(real \<times> real \<Rightarrow> real \<times> real) \<Rightarrow> (real \<times> real) set \<Rightarrow> (real \<Rightarrow> real \<times> real) \<Rightarrow> real\<close> and
line_integral_exists :: \<open>(real \<times> real \<Rightarrow> real \<times> real) \<Rightarrow> (real \<times> real) set \<Rightarrow> (real \<Rightarrow> real \<times> real) \<Rightarrow> bool\<close> and
one_chain_line_integral :: \<open>(real \<times> real \<Rightarrow> real \<times> real) \<Rightarrow> (real \<times> real) set \<Rightarrow> (int \<times> (real \<Rightarrow> real \<times> real)) set \<Rightarrow> real\<close> and
k
using prod.case_eq_if singleton_inject snd_conv
that
by (smt (verit))
end
lemma
fixes x y z :: real
assumes \<open>x + 2 * y > 0\<close> and
\<open>x - 2 * y > 0\<close> and
\<open>x < 0\<close>
shows False
using assms by (smt (verit))
(*test for arith reconstruction*)
lemma
fixes d :: real
assumes \<open>0 < d\<close>
\<open>diamond_y \<equiv> \<lambda>t. d / 2 - \<bar>t\<bar>\<close>
\<open>\<And>a b c :: real. (a / c < b / c) =
((0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
\<open>\<And>a b c :: real. (a / c < b / c) =
((0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
\<open>\<And>a b :: real. - a / b = - (a / b)\<close>
\<open>\<And>a b :: real. - a * b = - (a * b)\<close>
\<open>\<And>(x1 :: real) x2 y1 y2 :: real. ((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)\<close>
shows \<open>(\<lambda>y. (d / 2, (2 * y - 1) * diamond_y (d / 2))) \<noteq>
(\<lambda>x. ((x - 1 / 2) * d, - diamond_y ((x - 1 / 2) * d))) \<Longrightarrow>
(\<lambda>y. (- (d / 2), (2 * y - 1) * diamond_y (- (d / 2)))) =
(\<lambda>x. ((x - 1 / 2) * d, diamond_y ((x - 1 / 2) * d))) \<Longrightarrow>
False\<close>
using assms
by (smt (verit,del_insts))
lemma
fixes d :: real
assumes \<open>0 < d\<close>
\<open>diamond_y \<equiv> \<lambda>t. d / 2 - \<bar>t\<bar>\<close>
\<open>\<And>a b c :: real. (a / c < b / c) =
((0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
\<open>\<And>a b c :: real. (a / c < b / c) =
((0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
\<open>\<And>a b :: real. - a / b = - (a / b)\<close>
\<open>\<And>a b :: real. - a * b = - (a * b)\<close>
\<open>\<And>(x1 :: real) x2 y1 y2 :: real. ((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)\<close>
shows \<open>(\<lambda>y. (d / 2, (2 * y - 1) * diamond_y (d / 2))) \<noteq>
(\<lambda>x. ((x - 1 / 2) * d, - diamond_y ((x - 1 / 2) * d))) \<Longrightarrow>
(\<lambda>y. (- (d / 2), (2 * y - 1) * diamond_y (- (d / 2)))) =
(\<lambda>x. ((x - 1 / 2) * d, diamond_y ((x - 1 / 2) * d))) \<Longrightarrow>
False\<close>
using assms
by (smt (verit,ccfv_threshold))
(*qnt_rm_unused example*)
lemma
assumes \<open>\<forall>z y x. P z y\<close>
\<open>P z y \<Longrightarrow> False\<close>
shows False
using assms
by (smt (verit))
lemma
"max (x::int) y \<ge> y"
supply [[smt_trace]]
by (smt (verit))+
context
begin
abbreviation finite' :: "'a set \<Rightarrow> bool"
where "finite' A \<equiv> finite A \<and> A \<noteq> {}"
lemma
fixes f :: "'b \<Rightarrow> 'c :: linorder"
assumes
\<open>\<forall>(S::'b::type set) f::'b::type \<Rightarrow> 'c::linorder. finite' S \<longrightarrow> arg_min_on f S \<in> S\<close>
\<open>\<forall>(S::'a::type set) f::'a::type \<Rightarrow> 'c::linorder. finite' S \<longrightarrow> arg_min_on f S \<in> S\<close>
\<open>\<forall>(S::'b::type set) (y::'b::type) f::'b::type \<Rightarrow> 'c::linorder.
finite S \<and> S \<noteq> {} \<and> y \<in> S \<longrightarrow> f (arg_min_on f S) \<le> f y\<close>
\<open>\<forall>(S::'a::type set) (y::'a::type) f::'a::type \<Rightarrow> 'c::linorder.
finite S \<and> S \<noteq> {} \<and> y \<in> S \<longrightarrow> f (arg_min_on f S) \<le> f y\<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'c::linorder) (g::'a::type \<Rightarrow> 'b::type) x::'a::type. (f \<circ> g) x = f (g x)\<close>
\<open>\<forall>(F::'b::type set) h::'b::type \<Rightarrow> 'a::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'b::type set) h::'b::type \<Rightarrow> 'b::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'a::type set) h::'a::type \<Rightarrow> 'b::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'a::type set) h::'a::type \<Rightarrow> 'a::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(b::'a::type) (f::'b::type \<Rightarrow> 'a::type) A::'b::type set.
b \<in> f ` A \<and> (\<forall>x::'b::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'b::type) (f::'b::type \<Rightarrow> 'b::type) A::'b::type set.
b \<in> f ` A \<and> (\<forall>x::'b::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'b::type) (f::'a::type \<Rightarrow> 'b::type) A::'a::type set.
b \<in> f ` A \<and> (\<forall>x::'a::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'a::type) (f::'a::type \<Rightarrow> 'a::type) A::'a::type set.
b \<in> f ` A \<and> (\<forall>x::'a::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'a::type) (f::'b::type \<Rightarrow> 'a::type) (x::'b::type) A::'b::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'b::type) (f::'b::type \<Rightarrow> 'b::type) (x::'b::type) A::'b::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'b::type) (f::'a::type \<Rightarrow> 'b::type) (x::'a::type) A::'a::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'a::type) (f::'a::type \<Rightarrow> 'a::type) (x::'a::type) A::'a::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'a::type) A::'b::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'b::type) A::'b::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'a::type \<Rightarrow> 'b::type) A::'a::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'a::type \<Rightarrow> 'a::type) A::'a::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'c::linorder) (A::'b::type set) (x::'b::type) y::'b::type.
inj_on f A \<and> f x = f y \<and> x \<in> A \<and> y \<in> A \<longrightarrow> x = y\<close>
\<open>\<forall>(x::'c::linorder) y::'c::linorder. (x < y) = (x \<le> y \<and> x \<noteq> y)\<close>
\<open>inj_on (f::'b::type \<Rightarrow> 'c::linorder) ((g::'a::type \<Rightarrow> 'b::type) ` (B::'a::type set))\<close>
\<open>finite (B::'a::type set)\<close>
\<open>(B::'a::type set) \<noteq> {}\<close>
\<open>arg_min_on ((f::'b::type \<Rightarrow> 'c::linorder) \<circ> (g::'a::type \<Rightarrow> 'b::type)) (B::'a::type set) \<in> B\<close>
\<open>\<nexists>x::'a::type.
x \<in> (B::'a::type set) \<and>
((f::'b::type \<Rightarrow> 'c::linorder) \<circ> (g::'a::type \<Rightarrow> 'b::type)) x < (f \<circ> g) (arg_min_on (f \<circ> g) B)\<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'c::linorder) (P::'b::type \<Rightarrow> bool) a::'b::type.
inj_on f (Collect P) \<and> P a \<and> (\<forall>y::'b::type. P y \<longrightarrow> f a \<le> f y) \<longrightarrow> arg_min f P = a\<close>
\<open>\<forall>(S::'b::type set) f::'b::type \<Rightarrow> 'c::linorder. finite' S \<longrightarrow> arg_min_on f S \<in> S\<close>
\<open>\<forall>(S::'a::type set) f::'a::type \<Rightarrow> 'c::linorder. finite' S \<longrightarrow> arg_min_on f S \<in> S\<close>
\<open>\<forall>(S::'b::type set) (y::'b::type) f::'b::type \<Rightarrow> 'c::linorder.
finite S \<and> S \<noteq> {} \<and> y \<in> S \<longrightarrow> f (arg_min_on f S) \<le> f y\<close>
\<open>\<forall>(S::'a::type set) (y::'a::type) f::'a::type \<Rightarrow> 'c::linorder.
finite S \<and> S \<noteq> {} \<and> y \<in> S \<longrightarrow> f (arg_min_on f S) \<le> f y\<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'c::linorder) (g::'a::type \<Rightarrow> 'b::type) x::'a::type. (f \<circ> g) x = f (g x)\<close>
\<open>\<forall>(F::'b::type set) h::'b::type \<Rightarrow> 'a::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'b::type set) h::'b::type \<Rightarrow> 'b::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'a::type set) h::'a::type \<Rightarrow> 'b::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(F::'a::type set) h::'a::type \<Rightarrow> 'a::type. finite F \<longrightarrow> finite (h ` F)\<close>
\<open>\<forall>(b::'a::type) (f::'b::type \<Rightarrow> 'a::type) A::'b::type set.
b \<in> f ` A \<and> (\<forall>x::'b::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'b::type) (f::'b::type \<Rightarrow> 'b::type) A::'b::type set.
b \<in> f ` A \<and> (\<forall>x::'b::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'b::type) (f::'a::type \<Rightarrow> 'b::type) A::'a::type set.
b \<in> f ` A \<and> (\<forall>x::'a::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'a::type) (f::'a::type \<Rightarrow> 'a::type) A::'a::type set.
b \<in> f ` A \<and> (\<forall>x::'a::type. b = f x \<and> x \<in> A \<longrightarrow> False) \<longrightarrow> False\<close>
\<open>\<forall>(b::'a::type) (f::'b::type \<Rightarrow> 'a::type) (x::'b::type) A::'b::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'b::type) (f::'b::type \<Rightarrow> 'b::type) (x::'b::type) A::'b::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'b::type) (f::'a::type \<Rightarrow> 'b::type) (x::'a::type) A::'a::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(b::'a::type) (f::'a::type \<Rightarrow> 'a::type) (x::'a::type) A::'a::type set. b = f x \<and> x \<in> A \<longrightarrow> b \<in> f ` A \<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'a::type) A::'b::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'b::type) A::'b::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'a::type \<Rightarrow> 'b::type) A::'a::type set. (f ` A = {}) = (A = {}) \<close>
\<open>\<forall>(f::'a::type \<Rightarrow> 'a::type) A::'a::type set. (f ` A = {}) = (A = {})\<close>
\<open>\<forall>(f::'b::type \<Rightarrow> 'c::linorder) (A::'b::type set) (x::'b::type) y::'b::type.
inj_on f A \<and> f x = f y \<and> x \<in> A \<and> y \<in> A \<longrightarrow> x = y\<close>
\<open>\<forall>(x::'c::linorder) y::'c::linorder. (x < y) = (x \<le> y \<and> x \<noteq> y)\<close>
\<open>arg_min_on (f::'b::type \<Rightarrow> 'c::linorder) ((g::'a::type \<Rightarrow> 'b::type) ` (B::'a::type set)) \<noteq>
g (arg_min_on (f \<circ> g) B) \<close>
shows False
using assms
by (smt (verit))
end
experiment
begin
private datatype abort =
Rtype_error
| Rtimeout_error
private datatype ('a) error_result =
Rraise " 'a " \<comment> \<open>\<open> Should only be a value of type exn \<close>\<close>
| Rabort " abort "
private datatype( 'a, 'b) result =
Rval " 'a "
| Rerr " ('b) error_result "
lemma
fixes clock :: \<open>'astate \<Rightarrow> nat\<close> and
fun_evaluate_match :: \<open>'astate \<Rightarrow> 'vsemv_env \<Rightarrow> _ \<Rightarrow> ('pat \<times> 'exp0) list \<Rightarrow> _ \<Rightarrow>
'astate*((('v)list),('v))result\<close>
assumes
"fix_clock (st::'astate) (fun_evaluate st (env::'vsemv_env) [e::'exp0]) =
(st'::'astate, r::('v list, 'v) result)"
"clock (fst (fun_evaluate (st::'astate) (env::'vsemv_env) [e::'exp0])) \<le> clock st"
"\<forall>(b::nat) (a::nat) c::nat. b \<le> a \<and> c \<le> b \<longrightarrow> c \<le> a"
"\<forall>(a::'astate) p::'astate \<times> ('v list, 'v) result. (a = fst p) = (\<exists>b::('v list, 'v) result. p = (a, b))"
"\<forall>y::'v error_result. (\<forall>x1::'v. y = Rraise x1 \<longrightarrow> False) \<and> (\<forall>x2::abort. y = Rabort x2 \<longrightarrow> False) \<longrightarrow> False"
"\<forall>(f1::'v \<Rightarrow> 'astate \<times> ('v list, 'v) result) (f2::abort \<Rightarrow> 'astate \<times> ('v list, 'v) result) x1::'v.
(case Rraise x1 of Rraise (x::'v) \<Rightarrow> f1 x | Rabort (x::abort) \<Rightarrow> f2 x) = f1 x1"
"\<forall>(f1::'v \<Rightarrow> 'astate \<times> ('v list, 'v) result) (f2::abort \<Rightarrow> 'astate \<times> ('v list, 'v) result) x2::abort.
(case Rabort x2 of Rraise (x::'v) \<Rightarrow> f1 x | Rabort (x::abort) \<Rightarrow> f2 x) = f2 x2"
"\<forall>(s1::'astate) (s2::'astate) (x::('v list, 'v) result) s::'astate.
fix_clock s1 (s2, x) = (s, x) \<longrightarrow> clock s \<le> clock s2"
"\<forall>(s::'astate) (s'::'astate) res::('v list, 'v) result.
fix_clock s (s', res) =
(update_clock (\<lambda>_::nat. if clock s' \<le> clock s then clock s' else clock s) s', res)"
"\<forall>(x2::'v error_result) x1::'v.
(r::('v list, 'v) result) = Rerr x2 \<and> x2 = Rraise x1 \<longrightarrow>
clock (fst (fun_evaluate_match (st'::'astate) (env::'vsemv_env) x1 (pes::('pat \<times> 'exp0) list) x1))
\<le> clock st'"
shows "((r::('v list, 'v) result) = Rerr (x2::'v error_result) \<longrightarrow>
clock
(fst (case x2 of
Rraise (v2::'v) \<Rightarrow>
fun_evaluate_match (st'::'astate) (env::'vsemv_env) v2 (pes::('pat \<times> 'exp0) list) v2
| Rabort (abort::abort) \<Rightarrow> (st', Rerr (Rabort abort))))
\<le> clock (st::'astate))"
using assms by (smt (verit))
end
context
fixes piecewise_C1 :: "('real :: {one,zero,ord} \<Rightarrow> 'a :: {one,zero,ord}) \<Rightarrow> 'real set \<Rightarrow> bool" and
joinpaths :: "('real \<Rightarrow> 'a) \<Rightarrow> ('real \<Rightarrow> 'a) \<Rightarrow> 'real \<Rightarrow> 'a"
begin
notation piecewise_C1 (infixr \<open>piecewise'_C1'_differentiable'_on\<close> 50)
notation joinpaths (infixr \<open>+++\<close> 75)
lemma
\<open>(\<And>v1. \<forall>v0. (rec_join v0 = v1 \<and>
(v0 = [] \<and> (\<lambda>uu. 0) = v1 \<longrightarrow> False) \<and>
(\<forall>v2. v0 = [v2] \<and> v1 = coeff_cube_to_path v2 \<longrightarrow> False) \<and>
(\<forall>v2 v3 v4.
v0 = v2 # v3 # v4 \<and> v1 = coeff_cube_to_path v2 +++ rec_join (v3 # v4) \<longrightarrow> False) \<longrightarrow>
False) =
(rec_join v0 = rec_join v0 \<and>
(v0 = [] \<and> (\<lambda>uu. 0) = rec_join v0 \<longrightarrow> False) \<and>
(\<forall>v2. v0 = [v2] \<and> rec_join v0 = coeff_cube_to_path v2 \<longrightarrow> False) \<and>
(\<forall>v2 v3 v4.
v0 = v2 # v3 # v4 \<and> rec_join v0 = coeff_cube_to_path v2 +++ rec_join (v3 # v4) \<longrightarrow>
False) \<longrightarrow>
False)) \<Longrightarrow>
(\<forall>v0 v1.
rec_join v0 = v1 \<and>
(v0 = [] \<and> (\<lambda>uu. 0) = v1 \<longrightarrow> False) \<and>
(\<forall>v2. v0 = [v2] \<and> v1 = coeff_cube_to_path v2 \<longrightarrow> False) \<and>
(\<forall>v2 v3 v4. v0 = v2 # v3 # v4 \<and> v1 = coeff_cube_to_path v2 +++ rec_join (v3 # v4) \<longrightarrow> False) \<longrightarrow>
False) =
(\<forall>v0. rec_join v0 = rec_join v0 \<and>
(v0 = [] \<and> (\<lambda>uu. 0) = rec_join v0 \<longrightarrow> False) \<and>
(\<forall>v2. v0 = [v2] \<and> rec_join v0 = coeff_cube_to_path v2 \<longrightarrow> False) \<and>
(\<forall>v2 v3 v4.
v0 = v2 # v3 # v4 \<and> rec_join v0 = coeff_cube_to_path v2 +++ rec_join (v3 # v4) \<longrightarrow>
False) \<longrightarrow>
False)\<close>
by (smt (verit))
end
section \<open>Monomorphization examples\<close>
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 (verit) 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 (verit) g1 g2 g3 list.size)
experiment
begin
lemma duplicate_goal: \<open>A \<Longrightarrow> A \<Longrightarrow> A\<close>
by auto
datatype 'a M_nres = is_fail: FAIL | SPEC "'a \<Rightarrow> bool"
definition "is_res m x \<equiv> case m of FAIL \<Rightarrow> True | SPEC P \<Rightarrow> P x"
datatype ('a,'s) M_state = M_STATE (run: "'s \<Rightarrow> ('a\<times>'s) M_nres")
(*Courtesy of Peter Lammich
https://isabelle.zulipchat.com/#narrow/stream/247541-Mirror.3A-Isabelle-Users-Mailing-List/topic/.5Bisabelle.5D.20smt.20.28verit.29.3A.20exception.20THM.200.20raised.20.28line.20312.20.2E.2E.2E/near/290088165
*)
lemma "\<lbrakk>\<forall>x y. (\<forall>xa s. is_fail (run (x xa) s) \<or>
is_fail (run (y xa) s) = is_fail (run (x xa) s) \<and>
(\<forall>a b. is_res (run (y xa) s) (a, b) = is_res (run (x xa) s) (a, b)))
\<longrightarrow>
(\<forall>s. is_fail (run (B x) s) \<or>
is_fail (run (B y) s) = is_fail (run (B x) s) \<and>
(\<forall>a b. is_res (run (B y) s) (a, b) = is_res (run (B x) s) (a, b)));
\<And>y. \<forall>x ya. (\<forall>xa s. is_fail (run (x xa) s) \<or>
is_fail (run (ya xa) s) = is_fail (run (x xa) s) \<and>
(\<forall>a b. is_res (run (ya xa) s) (a, b) = is_res (run (x xa) s) (a, b)))
\<longrightarrow>
(\<forall>s. is_fail (run (C y x) s) \<or>
is_fail (run (C y ya) s) = is_fail (run (C y x) s) \<and>
(\<forall>a b. is_res (run (C y ya) s) (a, b) = is_res (run (C y x) s) (a,
b)))\<rbrakk>
\<Longrightarrow> \<forall>x y. (\<forall>xa s.
is_fail (run (x xa) s) \<or>
is_fail (run (y xa) s) = is_fail (run (x xa) s) \<and>
(\<forall>a b. is_res (run (y xa) s) (a, b) = is_res (run (x xa) s) (a, b)))
\<longrightarrow>
(\<forall>s. is_fail (run (B x) s) \<or>
(\<exists>a b. is_res (run (B x) s) (a, b) \<and> is_fail (run (C a x) b)) \<or>
(is_fail (run (B y) s) \<or> (\<exists>a b. is_res (run (B y) s) (a, b) \<and>
is_fail (run (C a y) b))) =
(is_fail (run (B x) s) \<or> (\<exists>a b. is_res (run (B x) s) (a, b) \<and>
is_fail (run (C a x) b))) \<and>
(\<forall>a b. (is_fail (run (B y) s) \<or>
(\<exists>aa ba. is_res (run (B y) s) (aa, ba) \<and> is_res (run (C aa y)
ba) (a, b))) =
(is_fail (run (B x) s) \<or>
(\<exists>aa ba. is_res (run (B x) s) (aa, ba) \<and> is_res (run (C aa x)
ba) (a, b)))))"
apply (rule duplicate_goal)
subgoal
supply [[verit_compress_proofs=true]]
by (smt (verit))
subgoal
supply [[verit_compress_proofs=false]]
by (smt (verit))
done
(*Example of Reordering in skolemization*)
lemma
fixes Abs_ExpList :: "'freeExp_list \<Rightarrow> 'exp_list" and
Abs_Exp:: "'freeExp_set \<Rightarrow> 'exp" and
exprel:: "('freeExp \<times> 'freeExp) set" and
map2 :: "('freeExp \<Rightarrow> 'exp) \<Rightarrow> 'freeExp_list \<Rightarrow> 'exp_list"
assumes "\<And>Xs. Abs_ExpList Xs \<equiv> map2 (\<lambda>U. Abs_Exp (myImage exprel {U})) Xs"
"\<And>P z. (\<And>U. z = Abs_Exp (myImage exprel {U}) \<Longrightarrow> P) \<Longrightarrow> P"
"\<And>(ys::'exp_list) (f::'freeExp \<Rightarrow> _). (\<exists>xs. ys = map2 f xs) = (\<forall>y\<in>myset ys. \<exists>x. y = f x)"
shows "\<exists>Us. z = Abs_ExpList Us"
apply (rule duplicate_goal)
subgoal
supply [[verit_compress_proofs=true]]
using assms
by (smt (verit,del_insts))
subgoal
using assms
supply [[verit_compress_proofs=false]]
by (smt (verit,del_insts))
done
end
context
fixes
L2_final :: "'afset \<Rightarrow> 'afset \<times> 'afset \<Rightarrow> bool" and
L3_final :: "'afset \<Rightarrow> 'afset \<times> 'afset \<Rightarrow> bool" and
ground_resolution :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" and
is_least_false_clause :: "'afset \<Rightarrow> 'a \<Rightarrow> bool" and
fset :: "'afset \<Rightarrow> 'a set" and
union_fset :: "'afset \<Rightarrow> 'afset \<Rightarrow> 'afset" (infixr \<open>|\<union>|\<close> 50)
begin
term "a |\<union>| b"
fun L2_matches_L3 where
"L2_matches_L3 N2 (Ur2, Uff2) N3 (Urr3, Uff3) \<longleftrightarrow>
N2 = N3 \<and> Uff2 = Uff3 \<and>
(\<forall>Cr \<in> fset Ur2. \<exists>C \<in> fset (N3 |\<union>| Urr3 |\<union>| Uff3). \<exists>D \<in> fset (N3 |\<union>| Urr3 |\<union>| Uff3).
(ground_resolution D)\<^sup>+\<^sup>+ C Cr \<and>
(\<exists>Crr \<in> fset Urr3. (ground_resolution D)\<^sup>*\<^sup>* Cr Crr) \<or> (is_least_false_clause (N2 |\<union>| Ur2 |\<union>| Uff2) Cr))"
lemma
assumes match: "L2_matches_L3 Const2 S2 Const3 S3"
shows "L2_final Const2 S2 \<longleftrightarrow> L2_final Const3 S3"
proof -
from match obtain N Ur Uff Urr where
state_simps:
"Const2 = N"
"Const3 = N"
"S2 = (Ur, Uff)"
"S3 = (Urr, Uff)" and
Ur_spec: "
\<forall>Cr \<in> fset Ur.
\<exists>C \<in> fset (N |\<union>| Urr |\<union>| Uff).
\<exists>D \<in> fset (N |\<union>| Urr |\<union>| Uff).
(ground_resolution D)\<^sup>+\<^sup>+ C Cr \<and>
(\<exists>Crr \<in> fset Urr. (ground_resolution D)\<^sup>*\<^sup>* Cr Crr) \<or>
(is_least_false_clause (N |\<union>| Ur |\<union>| Uff) Cr)"
by (smt (verit) L2_matches_L3.elims(2))
oops
end
end