src/HOL/SMT_Examples/SMT_Tests.thy

added function update examples and set examples

(*  Title:      HOL/SMT_Examples/SMT_Tests.thy
    Author:     Sascha Boehme, TU Muenchen
*)

header {* Tests for the SMT binding *}

theory SMT_Tests
imports Complex_Main
begin

declare [[smt_solver=z3, z3_proofs=true]]
declare [[smt_certificates="~~/src/HOL/SMT_Examples/SMT_Tests.certs"]]
declare [[smt_fixed=true]]



smt_status



text {* Most examples are taken from various Isabelle theories and from HOL4. *}



section {* Propositional logic *}

lemma
  "True"
  "\<not>False"
  "\<not>\<not>True"
  "True \<and> True"
  "True \<or> False"
  "False \<longrightarrow> True"
  "\<not>(False \<longleftrightarrow> True)"
  by smt+

lemma
  "P \<or> \<not>P"
  "\<not>(P \<and> \<not>P)"
  "(True \<and> P) \<or> \<not>P \<or> (False \<and> P) \<or> P"
  "P \<longrightarrow> P"
  "P \<and> \<not> P \<longrightarrow> False"
  "P \<and> Q \<longrightarrow> Q \<and> P"
  "P \<or> Q \<longrightarrow> Q \<or> P"
  "P \<and> Q \<longrightarrow> P \<or> Q"
  "\<not>(P \<or> Q) \<longrightarrow> \<not>P"
  "\<not>(P \<or> Q) \<longrightarrow> \<not>Q"
  "\<not>P \<longrightarrow> \<not>(P \<and> Q)"
  "\<not>Q \<longrightarrow> \<not>(P \<and> Q)"
  "(P \<and> Q) \<longleftrightarrow> (\<not>(\<not>P \<or> \<not>Q))"
  "(P \<and> Q) \<and> R \<longrightarrow> P \<and> (Q \<and> R)"
  "(P \<or> Q) \<or> R \<longrightarrow> P \<or> (Q \<or> R)"
  "(P \<and> Q) \<or> R  \<longrightarrow> (P \<or> R) \<and> (Q \<or> R)"
  "(P \<or> R) \<and> (Q \<or> R) \<longrightarrow> (P \<and> Q) \<or> R"
  "(P \<or> Q) \<and> R \<longrightarrow> (P \<and> R) \<or> (Q \<and> R)"
  "(P \<and> R) \<or> (Q \<and> R) \<longrightarrow> (P \<or> Q) \<and> R"
  "((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P"
  "(P \<longrightarrow> R) \<and> (Q \<longrightarrow> R) \<longleftrightarrow> (P \<or> Q \<longrightarrow> R)"
  "(P \<and> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> (Q \<longrightarrow> R))"
  "((P \<longrightarrow> R) \<longrightarrow> R) \<longrightarrow>  ((Q \<longrightarrow> R) \<longrightarrow> R) \<longrightarrow> (P \<and> Q \<longrightarrow> R) \<longrightarrow> R"
  "\<not>(P \<longrightarrow> R) \<longrightarrow>  \<not>(Q \<longrightarrow> R) \<longrightarrow> \<not>(P \<and> Q \<longrightarrow> R)"
  "(P \<longrightarrow> Q \<and> R) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (P \<longrightarrow> R)"
  "P \<longrightarrow> (Q \<longrightarrow> P)"
  "(P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> (P \<longrightarrow> Q)\<longrightarrow> (P \<longrightarrow> R)"
  "(P \<longrightarrow> Q) \<or> (P \<longrightarrow> R) \<longrightarrow> (P \<longrightarrow> Q \<or> R)"
  "((((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P) \<longrightarrow> Q) \<longrightarrow> Q"
  "(P \<longrightarrow> Q) \<longrightarrow> (\<not>Q \<longrightarrow> \<not>P)"
  "(P \<longrightarrow> Q \<or> R) \<longrightarrow> (P \<longrightarrow> Q) \<or> (P \<longrightarrow> R)"
  "(P \<longrightarrow> Q) \<and> (Q  \<longrightarrow> P) \<longrightarrow> (P \<longleftrightarrow> Q)"
  "(P \<longleftrightarrow> Q) \<longleftrightarrow> (Q \<longleftrightarrow> P)"
  "\<not>(P \<longleftrightarrow> \<not>P)"
  "(P \<longrightarrow> Q) \<longleftrightarrow> (\<not>Q \<longrightarrow> \<not>P)"
  "P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P \<longleftrightarrow> P"
  by smt+

lemma
  "(if P then Q1 else Q2) \<longleftrightarrow> ((P \<longrightarrow> Q1) \<and> (\<not>P \<longrightarrow> Q2))"
  "if P then (Q \<longrightarrow> P) else (P \<longrightarrow> Q)"
  "(if P1 \<or> P2 then Q1 else Q2) \<longleftrightarrow> (if P1 then Q1 else if P2 then Q1 else Q2)"
  "(if P1 \<and> P2 then Q1 else Q2) \<longleftrightarrow> (if P1 then if P2 then Q1 else Q2 else Q2)"
  "(P1 \<longrightarrow> (if P2 then Q1 else Q2)) \<longleftrightarrow>
   (if P1 \<longrightarrow> P2 then P1 \<longrightarrow> Q1 else P1 \<longrightarrow> Q2)"
  by smt+

lemma
  "case P of True \<Rightarrow> P | False \<Rightarrow> \<not>P"
  "case \<not>P of True \<Rightarrow> \<not>P | False \<Rightarrow> P"
  "case P of True \<Rightarrow> (Q \<longrightarrow> P) | False \<Rightarrow> (P \<longrightarrow> Q)"
  by smt+



section {* First-order logic with equality *}

lemma
  "x = x"
  "x = y \<longrightarrow> y = x"
  "x = y \<and> y = z \<longrightarrow> x = z"
  "x = y \<longrightarrow> f x = f y"
  "x = y \<longrightarrow> g x y = g y x"
  "f (f x) = x \<and> f (f (f (f (f x)))) = x \<longrightarrow> f x = x"
  "((if a then b else c) = d) = ((a \<longrightarrow> (b = d)) \<and> (\<not> a \<longrightarrow> (c = d)))"
  by smt+

lemma
  "distinct []"
  "distinct [a]"
  "distinct [a, b, c] \<longrightarrow> a \<noteq> c"
  "distinct [a, b, c] \<longrightarrow> d = b \<longrightarrow> a \<noteq> d"
  "\<not> distinct [a, b, a, b]"
  "a = b \<longrightarrow> \<not>distinct [a, b]"
  "a = b \<and> a = c \<longrightarrow> \<not>distinct [a, b, c]"
  "distinct [a, b, c, d] \<longrightarrow> distinct [d, b, c, a]"
  "distinct [a, b, c, d] \<longrightarrow> distinct [a, b, c] \<and> distinct [b, c, d]"
  by smt+

lemma
  "\<forall>x. x = x"
  "(\<forall>x. P x) \<longleftrightarrow> (\<forall>y. P y)"
  "\<forall>x. P x \<longrightarrow> (\<forall>y. P x \<or> P y)"
  "(\<forall>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>x. P x) \<and> (\<forall>x. Q x)"
  "(\<forall>x. P x) \<or> R \<longleftrightarrow> (\<forall>x. P x \<or> R)"
  "(\<forall>x. P x) \<and> R \<longleftrightarrow> (\<forall>x. P x \<and> R)"
  "(\<forall>x y z. S x z) \<longleftrightarrow> (\<forall>x z. S x z)"
  "(\<forall>x y. S x y \<longrightarrow> S y x) \<longrightarrow> (\<forall>x. S x y) \<longrightarrow> S y x"
  "(\<forall>x. P x \<longrightarrow> P (f x)) \<and> P d \<longrightarrow> P (f(f(f(d))))"
  "(\<forall>x y. s x y = s y x) \<longrightarrow> a = a \<and> s a b = s b a"
  "(\<forall>s. q s \<longrightarrow> r s) \<and> \<not>r s \<and> (\<forall>s. \<not>r s \<and> \<not>q s \<longrightarrow> p t \<or> q t) \<longrightarrow> p t \<or> r t"
  by smt+

lemma
  "\<exists>x. x = x"
  "(\<exists>x. P x) \<longleftrightarrow> (\<exists>y. P y)"
  "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>x. P x) \<or> (\<exists>x. Q x)"
  "(\<exists>x. P x) \<and> R \<longleftrightarrow> (\<exists>x. P x \<and> R)"
  "(\<exists>x y z. S x z) \<longleftrightarrow> (\<exists>x z. S x z)"
  "\<not>((\<exists>x. \<not>P x) \<and> ((\<exists>x. P x) \<or> (\<exists>x. P x \<and> Q x)) \<and> \<not>(\<exists>x. P x))"
  by smt+

lemma  (* only without proofs: *)
  "\<exists>x y. x = y"
  "\<exists>x. P x \<longrightarrow> (\<exists>y. P x \<and> P y)"
  "(\<exists>x. P x) \<or> R \<longleftrightarrow> (\<exists>x. P x \<or> R)"
  "\<exists>x. P x \<longrightarrow> P a \<and> P b"
  "\<exists>x. (\<exists>y. P y) \<longrightarrow> P x" 
  "(\<exists>x. Q \<longrightarrow> P x) \<longleftrightarrow> (Q \<longrightarrow> (\<exists>x. P x))"
  using [[z3_proofs=false, z3_options="AUTO_CONFIG=false SATURATE=true"]]
  by smt+

lemma
  "(\<not>(\<exists>x. P x)) \<longleftrightarrow> (\<forall>x. \<not> P x)"
  "(\<exists>x. P x \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P x) \<longrightarrow> Q"
  "(\<forall>x y. R x y = x) \<longrightarrow> (\<exists>y. R x y) = R x c"
  "\<forall>x. \<exists>y. f x y = f x (g x)"
  "(if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)) \<longrightarrow> P x \<longrightarrow> P y"
  "(\<forall>x y. R x y = x) \<and> (\<forall>x. \<exists>y. R x y) = (\<forall>x. R x c) \<longrightarrow> (\<exists>y. R x y) = R x c"
  by smt+

lemma  (* only without proofs: *)
  "(\<not>\<not>(\<exists>x. P x)) \<longleftrightarrow> (\<not>(\<forall>x. \<not> P x))"
  "\<forall>u. \<exists>v. \<forall>w. \<exists>x. f u v w x = f u (g u) w (h u w)"
  "\<exists>x. if x = y then (\<forall>y. y = x \<or> y \<noteq> x) else (\<forall>y. y = (x, x) \<or> y \<noteq> (x, x))"
  "\<exists>x. if x = y then (\<exists>y. y = x \<or> y \<noteq> x) else (\<exists>y. y = (x, x) \<or> y \<noteq> (x, x))"
  "(\<exists>x. \<forall>y. P x \<longleftrightarrow> P y) \<longrightarrow> ((\<exists>x. P x) \<longleftrightarrow> (\<forall>y. P y))"
  "\<exists>z. P z \<longrightarrow> (\<forall>x. P x)"
  "(\<exists>y. \<forall>x. R x y) \<longrightarrow> (\<forall>x. \<exists>y. R x y)"
  using [[z3_proofs=false]]
  by smt+

lemma
  "(\<exists>! x. P x) \<longrightarrow> (\<exists>x. P x)"
  "(\<exists>!x. P x) \<longleftrightarrow> (\<exists>x. P x \<and> (\<forall>y. y \<noteq> x \<longrightarrow> \<not>P y))"
  "P a \<longrightarrow> (\<forall>x. P x \<longrightarrow> x = a) \<longrightarrow> (\<exists>!x. P x)"
  "(\<exists>x. P x) \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> x = y) \<longrightarrow> (\<exists>!x. P x)"
  "(\<exists>!x. P x) \<and> (\<forall>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x) \<longrightarrow> R) \<longrightarrow> R"
  by smt+

lemma
  "let P = True in P"
  "let P = P1 \<or> P2 in P \<or> \<not>P"
  "let P1 = True; P2 = False in P1 \<and> P2 \<longrightarrow> P2 \<or> P1"
  "(let x = y in x) = y"
  "(let x = y in Q x) \<longleftrightarrow> (let z = y in Q z)"
  "(let x = y1; z = y2 in R x z) \<longleftrightarrow> (let z = y2; x = y1 in R x z)"
  "(let x = y1; z = y2 in R x z) \<longleftrightarrow> (let z = y1; x = y2 in R z x)"
  "let P = (\<forall>x. Q x) in if P then P else \<not>P"
  by smt+

lemma
  "distinct [a, b, c] \<and> (\<forall>x y. f x = f y \<longrightarrow> y = x) \<longrightarrow> f a \<noteq> f b"
  sorry  (* FIXME: injective function *)

lemma
  assumes "\<forall>x. SMT.trigger [[SMT.pat (f x)]] (f x = x)"
  shows "f 1 = 1"
  using assms by smt

lemma
  assumes "\<forall>x y. SMT.trigger [[SMT.pat (f x), SMT.pat (g y)]] (f x = g y)"
  shows "f 1 = g 2"
  using assms by smt



section {* Meta logical connectives *}

lemma
  "True \<Longrightarrow> True"
  "False \<Longrightarrow> True"
  "False \<Longrightarrow> False"
  "P' x \<Longrightarrow> P' x"
  "P \<Longrightarrow> P \<or> Q"
  "Q \<Longrightarrow> P \<or> Q"
  "\<not>P \<Longrightarrow> P \<longrightarrow> Q"
  "Q \<Longrightarrow> P \<longrightarrow> Q"
  "\<lbrakk>P; \<not>Q\<rbrakk> \<Longrightarrow> \<not>(P \<longrightarrow> Q)"
  "P' x \<equiv> P' x"
  "P' x \<equiv> Q' x \<Longrightarrow> P' x = Q' x"
  "P' x = Q' x \<Longrightarrow> P' x \<equiv> Q' x"
  "x \<equiv> y \<Longrightarrow> y \<equiv> z \<Longrightarrow> x \<equiv> (z::'a::type)"
  "x \<equiv> y \<Longrightarrow> (f x :: 'b::type) \<equiv> f y"
  "(\<And>x. g x) \<Longrightarrow> g a \<or> a"
  "(\<And>x y. h x y \<and> h y x) \<Longrightarrow> \<forall>x. h x x"
  "(p \<or> q) \<and> \<not>p \<Longrightarrow> q"
  "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)"
  by smt+



section {* Natural numbers *}

lemma
  "(0::nat) = 0"
  "(1::nat) = 1"
  "(0::nat) < 1"
  "(0::nat) \<le> 1"
  "(123456789::nat) < 2345678901"
  by smt+

lemma
  "Suc 0 = 1"
  "Suc x = x + 1"
  "x < Suc x"
  "(Suc x = Suc y) = (x = y)"
  "Suc (x + y) < Suc x + Suc y"
  by smt+

lemma
  "(x::nat) + 0 = x"
  "0 + x = x"
  "x + y = y + x"
  "x + (y + z) = (x + y) + z"
  "(x + y = 0) = (x = 0 \<and> y = 0)"
  by smt+

lemma 
  "(x::nat) - 0 = x"
  "x < y \<longrightarrow> x - y = 0"
  "x - y = 0 \<or> y - x = 0"
  "(x - y) + y = (if x < y then y else x)"
  "x - y - z = x - (y + z)" 
  by smt+

lemma
  "(x::nat) * 0 = 0"
  "0 * x = 0"
  "x * 1 = x"
  "1 * x = x"
  "3 * x = x * 3"
  by smt+

lemma
  "(0::nat) div 0 = 0"
  "(x::nat) div 0 = 0"
  "(0::nat) div 1 = 0"
  "(1::nat) div 1 = 1"
  "(3::nat) div 1 = 3"
  "(x::nat) div 1 = x"
  "(0::nat) div 3 = 0"
  "(1::nat) div 3 = 0"
  "(3::nat) div 3 = 1"
  "(x::nat) div 3 \<le> x"
  "(x div 3 = x) = (x = 0)"
  by smt+

lemma
  "(0::nat) mod 0 = 0"
  "(x::nat) mod 0 = x"
  "(0::nat) mod 1 = 0"
  "(1::nat) mod 1 = 0"
  "(3::nat) mod 1 = 0"
  "(x::nat) mod 1 = 0"
  "(0::nat) mod 3 = 0"
  "(1::nat) mod 3 = 1"
  "(3::nat) mod 3 = 0"
  "x mod 3 < 3"
  "(x mod 3 = x) = (x < 3)"
  by smt+

lemma
  "(x::nat) = x div 1 * 1 + x mod 1"
  "x = x div 3 * 3 + x mod 3"
  by smt+

lemma
  "min (x::nat) y \<le> x"
  "min x y \<le> y"
  "min x y \<le> x + y"
  "z < x \<and> z < y \<longrightarrow> z < min x y"
  "min x y = min y x"
  "min x 0 = 0"
  by smt+

lemma
  "max (x::nat) y \<ge> x"
  "max x y \<ge> y"
  "max x y \<ge> (x - y) + (y - x)"
  "z > x \<and> z > y \<longrightarrow> z > max x y"
  "max x y = max y x"
  "max x 0 = x"
  by smt+

lemma
  "0 \<le> (x::nat)"
  "0 < x \<and> x \<le> 1 \<longrightarrow> x = 1"
  "x \<le> x"
  "x \<le> y \<longrightarrow> 3 * x \<le> 3 * y"
  "x < y \<longrightarrow> 3 * x < 3 * y"
  "x < y \<longrightarrow> x \<le> y"
  "(x < y) = (x + 1 \<le> y)"
  "\<not>(x < x)"
  "x \<le> y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x \<le> y \<longrightarrow> y < z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y < z \<longrightarrow> x < z"
  "x < y \<and> y < z \<longrightarrow> \<not>(z < x)"
  by smt+



section {* Integers *}

lemma
  "(0::int) = 0"
  "(0::int) = -0"
  "(0::int) = (- 0)"
  "(1::int) = 1"
  "\<not>(-1 = (1::int))"
  "(0::int) < 1"
  "(0::int) \<le> 1"
  "-123 + 345 < (567::int)"
  "(123456789::int) < 2345678901"
  "(-123456789::int) < 2345678901"
  by smt+

lemma
  "(x::int) + 0 = x"
  "0 + x = x"
  "x + y = y + x"
  "x + (y + z) = (x + y) + z"
  "(x + y = 0) = (x = -y)"
  by smt+

lemma
  "(-1::int) = - 1"
  "(-3::int) = - 3"
  "-(x::int) < 0 \<longleftrightarrow> x > 0"
  "x > 0 \<longrightarrow> -x < 0"
  "x < 0 \<longrightarrow> -x > 0"
  by smt+

lemma 
  "(x::int) - 0 = x"
  "0 - x = -x"
  "x < y \<longrightarrow> x - y < 0"
  "x - y = -(y - x)"
  "x - y = -y + x"
  "x - y - z = x - (y + z)" 
  by smt+

lemma
  "(x::int) * 0 = 0"
  "0 * x = 0"
  "x * 1 = x"
  "1 * x = x"
  "x * -1 = -x"
  "-1 * x = -x"
  "3 * x = x * 3"
  by smt+

lemma
  "(0::int) div 0 = 0"
  "(x::int) div 0 = 0"
  "(0::int) div 1 = 0"
  "(1::int) div 1 = 1"
  "(3::int) div 1 = 3"
  "(x::int) div 1 = x"
  "(0::int) div -1 = 0"
  "(1::int) div -1 = -1"
  "(3::int) div -1 = -3"
  "(x::int) div -1 = -x"
  "(0::int) div 3 = 0"
  "(0::int) div -3 = 0"
  "(1::int) div 3 = 0"
  "(3::int) div 3 = 1"
  "(5::int) div 3 = 1"
  "(1::int) div -3 = -1"
  "(3::int) div -3 = -1"
  "(5::int) div -3 = -2"
  "(-1::int) div 3 = -1"
  "(-3::int) div 3 = -1"
  "(-5::int) div 3 = -2"
  "(-1::int) div -3 = 0"
  "(-3::int) div -3 = 1"
  "(-5::int) div -3 = 1"
  by smt+

lemma
  "(0::int) mod 0 = 0"
  "(x::int) mod 0 = x"
  "(0::int) mod 1 = 0"
  "(1::int) mod 1 = 0"
  "(3::int) mod 1 = 0"
  "(x::int) mod 1 = 0"
  "(0::int) mod -1 = 0"
  "(1::int) mod -1 = 0"
  "(3::int) mod -1 = 0"
  "(x::int) mod -1 = 0"
  "(0::int) mod 3 = 0"
  "(0::int) mod -3 = 0"
  "(1::int) mod 3 = 1"
  "(3::int) mod 3 = 0"
  "(5::int) mod 3 = 2"
  "(1::int) mod -3 = -2"
  "(3::int) mod -3 = 0"
  "(5::int) mod -3 = -1"
  "(-1::int) mod 3 = 2"
  "(-3::int) mod 3 = 0"
  "(-5::int) mod 3 = 1"
  "(-1::int) mod -3 = -1"
  "(-3::int) mod -3 = 0"
  "(-5::int) mod -3 = -2"
  "x mod 3 < 3"
  "(x mod 3 = x) \<longrightarrow> (x < 3)"
  by smt+

lemma
  "(x::int) = x div 1 * 1 + x mod 1"
  "x = x div 3 * 3 + x mod 3"
  by smt+

lemma
  "abs (x::int) \<ge> 0"
  "(abs x = 0) = (x = 0)"
  "(x \<ge> 0) = (abs x = x)"
  "(x \<le> 0) = (abs x = -x)"
  "abs (abs x) = abs x"
  by smt+

lemma
  "min (x::int) y \<le> x"
  "min x y \<le> y"
  "z < x \<and> z < y \<longrightarrow> z < min x y"
  "min x y = min y x"
  "x \<ge> 0 \<longrightarrow> min x 0 = 0"
  "min x y \<le> abs (x + y)"
  by smt+

lemma
  "max (x::int) y \<ge> x"
  "max x y \<ge> y"
  "z > x \<and> z > y \<longrightarrow> z > max x y"
  "max x y = max y x"
  "x \<ge> 0 \<longrightarrow> max x 0 = x"
  "max x y \<ge> - abs x - abs y"
  by smt+

lemma
  "0 < (x::int) \<and> x \<le> 1 \<longrightarrow> x = 1"
  "x \<le> x"
  "x \<le> y \<longrightarrow> 3 * x \<le> 3 * y"
  "x < y \<longrightarrow> 3 * x < 3 * y"
  "x < y \<longrightarrow> x \<le> y"
  "(x < y) = (x + 1 \<le> y)"
  "\<not>(x < x)"
  "x \<le> y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x \<le> y \<longrightarrow> y < z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y < z \<longrightarrow> x < z"
  "x < y \<and> y < z \<longrightarrow> \<not>(z < x)"
  by smt+



section {* Reals *}

lemma
  "(0::real) = 0"
  "(0::real) = -0"
  "(0::real) = (- 0)"
  "(1::real) = 1"
  "\<not>(-1 = (1::real))"
  "(0::real) < 1"
  "(0::real) \<le> 1"
  "-123 + 345 < (567::real)"
  "(123456789::real) < 2345678901"
  "(-123456789::real) < 2345678901"
  by smt+

lemma
  "(x::real) + 0 = x"
  "0 + x = x"
  "x + y = y + x"
  "x + (y + z) = (x + y) + z"
  "(x + y = 0) = (x = -y)"
  by smt+

lemma
  "(-1::int) = - 1"
  "(-3::int) = - 3"
  "-(x::real) < 0 \<longleftrightarrow> x > 0"
  "x > 0 \<longrightarrow> -x < 0"
  "x < 0 \<longrightarrow> -x > 0"
  by smt+

lemma 
  "(x::real) - 0 = x"
  "0 - x = -x"
  "x < y \<longrightarrow> x - y < 0"
  "x - y = -(y - x)"
  "x - y = -y + x"
  "x - y - z = x - (y + z)" 
  by smt+

lemma
  "(x::int) * 0 = 0"
  "0 * x = 0"
  "x * 1 = x"
  "1 * x = x"
  "x * -1 = -x"
  "-1 * x = -x"
  "3 * x = x * 3"
  by smt+

lemma
  "(1/2 :: real) < 1"
  "(1::real) / 3 = 1 / 3"
  "(1::real) / -3 = - 1 / 3"
  "(-1::real) / 3 = - 1 / 3"
  "(-1::real) / -3 = 1 / 3"
  "(x::real) / 1 = x"
  "x > 0 \<longrightarrow> x / 3 < x"
  "x < 0 \<longrightarrow> x / 3 > x"
  by smt+

lemma
  "(3::real) * (x / 3) = x"
  "(x * 3) / 3 = x"
  "x > 0 \<longrightarrow> 2 * x / 3 < x"
  "x < 0 \<longrightarrow> 2 * x / 3 > x"
  by smt+

lemma
  "abs (x::real) \<ge> 0"
  "(abs x = 0) = (x = 0)"
  "(x \<ge> 0) = (abs x = x)"
  "(x \<le> 0) = (abs x = -x)"
  "abs (abs x) = abs x"
  by smt+

lemma
  "min (x::real) y \<le> x"
  "min x y \<le> y"
  "z < x \<and> z < y \<longrightarrow> z < min x y"
  "min x y = min y x"
  "x \<ge> 0 \<longrightarrow> min x 0 = 0"
  "min x y \<le> abs (x + y)"
  by smt+

lemma
  "max (x::real) y \<ge> x"
  "max x y \<ge> y"
  "z > x \<and> z > y \<longrightarrow> z > max x y"
  "max x y = max y x"
  "x \<ge> 0 \<longrightarrow> max x 0 = x"
  "max x y \<ge> - abs x - abs y"
  by smt+

lemma
  "x \<le> (x::real)"
  "x \<le> y \<longrightarrow> 3 * x \<le> 3 * y"
  "x < y \<longrightarrow> 3 * x < 3 * y"
  "x < y \<longrightarrow> x \<le> y"
  "\<not>(x < x)"
  "x \<le> y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y \<le> z \<longrightarrow> x \<le> z"
  "x \<le> y \<longrightarrow> y < z \<longrightarrow> x \<le> z"
  "x < y \<longrightarrow> y < z \<longrightarrow> x < z"
  "x < y \<and> y < z \<longrightarrow> \<not>(z < x)"
  by smt+



section {* Pairs *}

lemma
  "x = fst (x, y)"
  "y = snd (x, y)"
  "((x, y) = (y, x)) = (x = y)"
  "((x, y) = (u, v)) = (x = u \<and> y = v)"
  "(fst (x, y, z) = fst (u, v, w)) = (x = u)"
  "(snd (x, y, z) = snd (u, v, w)) = (y = v \<and> z = w)"
  "(fst (snd (x, y, z)) = fst (snd (u, v, w))) = (y = v)"
  "(snd (snd (x, y, z)) = snd (snd (u, v, w))) = (z = w)"
  "(fst (x, y) = snd (x, y)) = (x = y)"
  "p1 = (x, y) \<and> p2 = (y, x) \<longrightarrow> fst p1 = snd p2"
  "(fst (x, y) = snd (x, y)) = (x = y)"
  "(fst p = snd p) = (p = (snd p, fst p))"
  by smt+



section {* Function updates *}

lemma
  "(f (i := v)) i = v"
  "i1 \<noteq> i2 \<longrightarrow> (f (i1 := v)) i2 = f i2"
  "i1 \<noteq> i2 \<longrightarrow> (f (i1 := v1, i2 := v2)) i1 = v1"
  "i1 \<noteq> i2 \<longrightarrow> (f (i1 := v1, i2 := v2)) i2 = v2"
  "i1 = i2 \<longrightarrow> (f (i1 := v1, i2 := v2)) i1 = v2"
  "i1 = i2 \<longrightarrow> (f (i1 := v1, i2 := v2)) i1 = v2"
  "distinct [i1, i2, i3] \<longrightarrow> (f (i1 := v1, i2 := v2)) i3 = f i3"
  by smt+



section {* Sets *}

lemma smt_sets:
  "\<not>({} x)"
  "UNIV x"
  "(A \<union> B) x = (A x \<or> B x)"
  "(A \<inter> B) x = (A x \<and> B x)"
  by auto

lemma
  "x \<in> P = P x"
  "x \<in> {x. P x} = P x"
  "x \<notin> {}"
  "x \<in> UNIV"
  "x \<in> P \<union> Q = (P x \<or> Q x)"
  "x \<in> P \<union> {} = P x"
  "x \<in> P \<union> UNIV"
  "(x \<in> P \<union> Q) = (x \<in> Q \<union> P)"
  "(x \<in> P \<union> P) = (x \<in> P)"
  "(x \<in> P \<union> (Q \<union> R)) = (x \<in> (P \<union> Q) \<union> R)"
  "(x \<in> P \<inter> Q) = (P x \<and> Q x)"
  "x \<notin> P \<inter> {}"
  "(x \<in> P \<inter> UNIV) = (x \<in> P)"
  "(x \<in> P \<inter> Q) = (x \<in> Q \<inter> P)"
  "(x \<in> P \<inter> P) = (x \<in> P)"
  "(x \<in> P \<inter> (Q \<inter> R)) = (x \<in> (P \<inter> Q) \<inter> R)"
  "{x. P x} = {y. P y}"
  unfolding mem_def Collect_def
  by (smt smt_sets)+

end