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Datei: Complex_Transcendental.thy Sprache: Isabelle

Original von: Isabelle^©

(* 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 \ \p" by (smt (verit))
lemma "(p \ True) = p" by (smt (verit))
lemma "(p \ q) \ \p \ q" by (smt (verit))
lemma "(a \ b) \ (c \ d) \ (a \ b) \ (c \ d)" by (smt (verit))
lemma "(p1 \ p2) \ p3 \ (p1 \ (p3 \ p2) \ (p1 \ p3)) \ p1" by (smt (verit))
lemma "P = P = P = P = P = P = P = P = P = P" by (smt (verit))

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 (verit))

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 (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 \ 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 (verit))

lemma "\x::int. P x \ (\y::int. P x \ P y)"
 by (smt (verit))

lemma
 assumes "(\x y. P x y = x)"
 shows "(\y. P x y) = P x c"
 using assms by (smt (verit))

lemma
 assumes "(\x y. P x y = x)"
 and "(\x. \y. P x y) = (\x. P x c)"
 shows "(\y. P x y) = P x c"
 using assms by (smt (verit))

lemma
 assumes "if P x then \(\y. P y) else (\y. \P y)"
 shows "P x \ 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 "\x :: real\ + \y\ \ \x + y\" by (smt (verit))
lemma "P ((2::int) < 3) = P True" supply[[smt_trace]] by (smt (verit))
lemma "x + 3 \ 4 \ x < (1::int)" by (smt (verit))

lemma
 assumes "x \ (3::int)" and "y = x + 4"
 shows "y - x > 0"
 using assms by (smt (verit))

lemma "let x = (2 :: int) in x + x \ 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 \ y + -1 * x \ \ 0 \ x \ 0 \ (x::int)) = (\ 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' \ 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 \<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 \ 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 "\ x3 = \x2\ - x1; x4 = \x3\ - x2; x5 = \x4\ - 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)"
 supply [[smt_timeout=360]]
 by (smt (verit))

lemma "let P = 2 * x + 1 > x + (x::real) in P \ False \ P" by (smt (verit))

subsection \<open>Linear arithmetic with quantifiers\<close>

lemma "~ (\x::int. False)" by (smt (verit))
lemma "~ (\x::real. False)" by (smt (verit))

lemma "\x y::int. (x = 0 \ y = 1) \ x \ y" by (smt (verit))
lemma "\x y::int. x < y \ (2 * x + 1) < (2 * y)" by (smt (verit))
lemma "\x y::int. x + y > 2 \ x + y = 2 \ x + y < 2" by (smt (verit))
lemma "\x::int. if x > 0 then x + 1 > 0 else 1 > x" by (smt (verit))
lemma "(if (\x::int. x < 0 \ x > 0) then -1 else 3) > (0::int)" by (smt (verit))
lemma "\x::int. \x y. 0 < x \ 0 < y \ (0::int) < x + y" by (smt (verit))
lemma "\u::int. $x::int) y::real. 0 < x \ 0 < y \ -1 < x" by (smt (verit))
lemma "\(a::int) b::int. 0 Linear arithmetic for natural numbers\<close>

declare [[smt_nat_as_int]]

lemma "2 * (x::nat) \ 1" by (smt (verit))

lemma "a < 3 \ (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 \x::int$ = \x\" by (smt (verit) int_nat_eq)

definition prime_nat :: "nat \ bool" where
 "prime_nat p = (1 m = 1 \ m = p))"

lemma "prime_nat (4*m + 1) \ m \ (1::nat)" by (smt (verit) prime_nat_def)

lemma "2 * (x::nat) \ 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 \ x = a"
 using fst_conv by (smt (verit))

lemma "p1 = (x, y) \ p2 = (y, x) \ fst p1 = snd p2"
 using fst_conv snd_conv by (smt (verit))

section \<open>Higher-order problems and recursion\<close>

lemma "i \ i1 \ i \ i2 \ (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 \ True)) \ (f g x = True) \ (g x = True)"
 by (smt (verit))

lemma "id x = x \ id True = True"
 by (smt (verit) id_def)

lemma "i \ i1 \ i \ i2 \ ((f (i1 := v1)) (i2 := v2)) i = f i"
 using fun_upd_same fun_upd_apply by (smt (verit))

lemma
 "f (\x. g x) \ True"
 "f (\x. g x) \ True"
 by (smt (verit))+

lemma True using let_rsp by (smt (verit))
lemma "le = (\) \ le (3::int) 42" by (smt (verit))
lemma "map (\i::int. i + 1) [0, 1] = [1, 2]" by (smt (verit) list.map)
lemma "(\x. P x) \ \ All P" by (smt (verit))

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 (verit) dec_10.simps)

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 (verit) order_trans)

end

lemma
"eq_set (List.coset xs) (set ys) = rhs"
 if "\ys. subset' (List.coset xs) (set ys) = (let n = card (UNIV::'a set) in 0 < n \ card (set (xs @ ys)) = n)"
 and "\uu A. (uu::'a) \ - A \ uu \ A"
 and "\uu. card (set (uu::'a list)) = length (remdups uu)"
 and "\uu. finite (set (uu::'a list))"
 and "\uu. (uu::'a) \ UNIV"
 and "(UNIV::'a set) \ {}"
 and "\c A B P. \(c::'a) \ A \ B; c \ A \ P; c \ B \ P\ \ P"
 and "\a b. (a::nat) + b = b + a"
 and "\a b. ((a::nat) = a + b) = (b = 0)"
 and "card' (set xs) = length (remdups xs)"
 and "card' = (card :: 'a set \ nat)"
 and "\A B. \finite (A::'a set); finite B\ \ card A + card B = card (A \ B) + card (A \ B)"
 and "\A. (card (A::'a set) = 0) = (A = {} \ infinite A)"
 and "\A. \finite (UNIV::'a set); card (A::'a set) = card (UNIV::'a set)\ \ A = UNIV"
 and "\xs. - List.coset (xs::'a list) = set xs"
 and "\xs. - set (xs::'a list) = List.coset xs"
 and "\A B. (A \ B = {}) = (\x. (x::'a) \ A \ x \ B)"
 and "eq_set = (=)"
 and "\A. finite (A::'a set) \ finite (- A) = finite (UNIV::'a set)"
 and "rhs \ let n = card (UNIV::'a set) in if n = 0 then False else let xs' = remdups xs; ys' = remdups ys in length xs' + length ys' = n \ (\x\set xs'. x \ set ys') \ (\y\set ys'. y \ set xs')"
 and "\xs ys. set ((xs::'a list) @ ys) = set xs \ set ys"
 and "\A B. ((A::'a set) = B) = (A \ B \ B \ A)"
 and "\xs. set (remdups (xs::'a list)) = set xs"
 and "subset' = (\)"
 and "\A B. (\x. (x::'a) \ A \ x \ B) \ A \ B"
 and "\A B. \(A::'a set) \ B; B \ A\ \ A = B"
 and "\A ys. (A \ List.coset ys) = (\y\set ys. (y::'a) \ 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 a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
 \<open>\<And>a b c :: real. (a / c 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 a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0)\<close>
 \<open>\<And>a b c :: real. (a / c 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 \ y"
 supply [[smt_trace]]
 by (smt (verit))+

context
begin
abbreviation finite' :: "'a set \<Rightarrow> bool"
 where "finite' A \ finite A \ A \ {}"

lemma
 fixes f :: "'b \ '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 \ 'c::linorder) \ (g::'a::type \ 'b::type)) x < (f \ g) (arg_min_on (f \ g) B)\
 \<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\
 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])) \ clock st"
 "\(b::nat) (a::nat) c::nat. b \ a \ c \ b \ c \ a"
 "\(a::'astate) p::'astate \ ('v list, 'v) result. (a = fst p) = (\b::('v list, 'v) result. p = (a, b))"
 "\y::'v error_result. (\x1::'v. y = Rraise x1 \ False) \ (\x2::abort. y = Rabort x2 \ False) \ False"
 "\(f1::'v \ 'astate \ ('v list, 'v) result) (f2::abort \ 'astate \ ('v list, 'v) result) x1::'v.
 (case Rraise x1 of Rraise (x::'v) \ f1 x | Rabort (x::abort) \ f2 x) = f1 x1"
 " \(f1::'v \ 'astate \ ('v list, 'v) result) (f2::abort \ 'astate \ ('v list, 'v) result) x2::abort.
 (case Rabort x2 of Rraise (x::'v) \ f1 x | Rabort (x::abort) \ f2 x) = f2 x2"
 "\(s1::'astate) (s2::'astate) (x::('v list, 'v) result) s::'astate.
 fix_clock s1 (s2, x) = (s, x) \<longrightarrow> clock s \<le> clock s2"
 "\(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)"
 "\(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) \
 clock
 (fst (case x2 of
 Rraise (v2::'v) \
 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} \ 'a :: {one,zero,ord}) \ 'real set \ bool" and
 joinpaths :: "('real \ 'a) \ ('real \ 'a) \ 'real \ 'a"
begin
notation piecewise_C1 (infixr "piecewise'_C1'_differentiable'_on" 50)
notation joinpaths (infixr "+++" 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 \ 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 (verit) 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 (verit) g1 g2 g3 list.size)

end

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