(*  Title:       SMT_Examples.thy

    Author:      Sascha Böhme, TU Muenchen

*)

header {* Examples for the 'smt' tactic. *}

theory SMT_Examples

imports "../SMT"

begin

declare [[smt_solver=z3, z3_proofs=false]]

declare [[smt_trace=true]] (*FIXME*)

section {* Propositional and first-order logic *}

lemma "True" by smt

lemma "p \<or> \<not>p" by smt

lemma "(p \<and> True) = p" by smt

lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt

lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" by smt

lemma "P=P=P=P=P=P=P=P=P=P" by smt

axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where

  symm_f: "symm_f x y = symm_f y x"

lemma "a = a \<and> symm_f a b = symm_f b a" by (smt add: symm_f)

section {* Linear arithmetic *}

lemma "(3::int) = 3" by smt

lemma "(3::real) = 3" by smt

lemma "(3 :: int) + 1 = 4" by smt

lemma "max (3::int) 8 > 5" by smt

lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt

lemma "let x = (2 :: int) in x + x \<noteq> 5" 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.

*}

lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3;

         x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6;

         x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk>

 \<Longrightarrow> x1 = x10 & x2 = (x11::int)"

  by smt

lemma "\<exists>x::int. 0 < x" by smt

lemma "\<exists>x::real. 0 < x" by smt

lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt

lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt

lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt

lemma "~ (\<exists>x::int. False)" by smt

section {* Non-linear arithmetic *}

lemma "((x::int) * (1 + y) - x * (1 - y)) = (2 * x * y)" by smt

lemma

  "(U::int) + (1 + p) * (b + e) + p * d =

   U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)"

  by smt

section {* Linear arithmetic for natural numbers *}

lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt

lemma "let x = (1::nat) + y in x - y > 0 * x" by smt

lemma

  "let x = (1::nat) + y in

   let P = (if x > 0 then True else False) in

   False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"

  by smt

section {* Bitvectors *}

locale bv

begin

declare [[smt_solver=z3]]

lemma "(27 :: 4 word) = -5" by smt

lemma "(27 :: 4 word) = 11" by smt

lemma "23 < (27::8 word)" by smt

lemma "27 + 11 = (6::5 word)" by smt

lemma "7 * 3 = (21::8 word)" by smt

lemma "11 - 27 = (-16::8 word)" by smt

lemma "- -11 = (11::5 word)" by smt

lemma "-40 + 1 = (-39::7 word)" by smt

lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by smt

lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by smt

lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by smt

lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by smt

lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by smt

lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" by smt

lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" 

  by smt

lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" by smt

lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by smt

lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by smt

lemma "bv_lshr 0b10011 2 = (0b100::8 word)" by smt

lemma "bv_ashr 0b10011 2 = (0b100::8 word)" by smt

lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by smt

lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt

lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" by smt

lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" by smt

end

section {* Pairs *}

lemma "fst (x, y) = a \<Longrightarrow> x = a" by smt

lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" by smt

section {* Higher-order problems and recursion *}

lemma "(f g x = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)" by smt

lemma "P ((2::int) < 3) = P True" by smt

lemma "P ((2::int) < 3) = (P True :: bool)" by smt

lemma "P (0 \<le> (a :: 4 word)) = P True" using [[smt_solver=z3]] by smt

lemma "id 3 = 3 \<and> id True = True" by (smt add: id_def)

lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" by smt

lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt add: map.simps)

lemma "(ALL x. P x) | ~ All P" by smt

fun dec_10 :: "nat \<Rightarrow> nat" where

  "dec_10 n = (if n < 10 then n else dec_10 (n - 10))"

lemma "dec_10 (4 * dec_10 4) = 6" by (smt add: dec_10.simps)

axiomatization

  eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int"

  where

  eval_dioph_mod:

  "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"

  and

  eval_dioph_div_mult:

  "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n +

   eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"

lemma

  "(eval_dioph ks xs = l) =

   (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and>

    eval_dioph ks (map (\<lambda>x. x div 2) xs) =

      (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"

  by (smt add: eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])

section {* Monomorphization examples *}

definition P :: "'a \<Rightarrow> bool" where "P x = True"

lemma poly_P: "P x \<and> (P [x] \<or> \<not>P[x])" by (simp add: P_def)

lemma "P (1::int)" by (smt add: poly_P)

consts g :: "'a \<Rightarrow> nat"

axioms

  g1: "g (Some x) = g [x]"

  g2: "g None = g []"

  g3: "g xs = length xs"

lemma "g (Some (3::int)) = g (Some True)" by (smt add: g1 g2 g3 list.size)

end