(*  Title:  HOL/ex/Arith_Examples.thy
    Author: Tjark Weber
*)

section ‹Arithmetic›

theory Arith_Examples
imports Main
begin

text ‹
 The ‹arith› method is used frequently throughout the Isabelle
 distribution. This file merely contains some additional tests and special
 corner cases. Some rather technical remarks:

 🚫‹Lin_Arith.simple_tac› is a very basic version of the tactic. It performs no
 meta-to-object-logic conversion, and only some splitting of operators.
 🚫‹Lin_Arith.tac› performs meta-to-object-logic conversion, full
 splitting of operators, and NNF normalization of the goal. The ‹arith›
 method combines them both, and tries other methods (e.g.~‹presburger›)
 as well. This is the one that you should use in your proofs!

 An ‹arith›-based simproc is available as well (see 🚫‹Lin_Arith.simproc›), which---for performance
 reasons---however does even less splitting than 🚫‹Lin_Arith.simple_tac›
 at the moment (namely inequalities only). (On the other hand, it
 does take apart conjunctions, which 🚫‹Lin_Arith.simple_tac› currently
 does not do.)
 ›


subsection ‹Splitting of Operators: term‹max›, term‹min›, term‹abs›,
 term‹minus›, term‹nat›, term‹modulo›,
 term‹divide››

lemma "(i::nat) <= max i j"
  by linarith

lemma "(i::int) <= max i j"
  by linarith

lemma "min i j <= (i::nat)"
  by linarith

lemma "min i j <= (i::int)"
  by linarith

lemma "min (i::nat) j <= max i j"
  by linarith

lemma "min (i::int) j <= max i j"
  by linarith

lemma "min (i::nat) j + max i j = i + j"
  by linarith

lemma "min (i::int) j + max i j = i + j"
  by linarith

lemma "(i::nat) < j ==> min i j < max i j"
  by linarith

lemma "(i::int) < j ==> min i j < max i j"
  by linarith

lemma "(0::int) <= ∣i∣"
  by linarith

lemma "(i::int) <= ∣i∣"
  by linarith

lemma "∣∣i::int∣∣ = ∣i∣"
  by linarith

text ‹Also testing subgoals with bound variables.›

lemma "!!x. (x::nat) <= y ==> x - y = 0"
  by linarith

lemma "!!x. (x::nat) - y = 0 ==> x <= y"
  by linarith

lemma "!!x. ((x::nat) <= y) = (x - y = 0)"
  by linarith

lemma "[| (x::nat) < y; d < 1 |] ==> x - y = d"
  by linarith

lemma "[| (x::nat) < y; d < 1 |] ==> x - y - x = d - x"
  by linarith

lemma "(x::int) < y ==> x - y < 0"
  by linarith

lemma "nat (i + j) <= nat i + nat j"
  by linarith

lemma "i < j ==> nat (i - j) = 0"
  by linarith

lemma "(i::nat) mod 0 = i"
  using split_mod [of _ _ 0, linarith_split]
    ― ‹rule 🚫‹split_mod› is only declared by default for numerals›
  by linarith

lemma "(i::nat) mod 1 = 0"
  (* rule split_mod is only declared by default for numerals *)
  using split_mod [of _ _ 1, linarith_split]
    ― ‹rule 🚫‹split_mod› is only declared by default for numerals›
  by linarith

lemma "(i::nat) mod 42 <= 41"
  by linarith

lemma "(i::int) mod 0 = i"
  using split_zmod [of _ _ 0, linarith_split]
    ― ‹rule 🚫‹split_zmod› is only declared by default for numerals›
  by linarith

lemma "(i::int) mod 1 = 0"
  using split_zmod [of _ _ "1", linarith_split]
    ― ‹rule 🚫‹split_zmod› is only declared by default for numerals›
  by linarith

lemma "(i::int) mod 42 <= 41"
  by linarith

lemma "-(i::int) * 1 = 0 ==> i = 0"
  by linarith

lemma "[| (0::int) < ∣i∣; ∣i∣ * 1 < ∣i∣ * j |] ==> 1 < ∣i∣ * j"
  by linarith


subsection ‹Meta-Logic›

lemma "x < Suc y == x <= y"
  by linarith

lemma "((x::nat) == z ==> x ~= y) ==> x ~= y | z ~= y"
  by linarith


subsection ‹Various Other Examples›

lemma "(x < Suc y) = (x <= y)"
  by linarith

lemma "[| (x::nat) < y; y < z |] ==> x < z"
  by linarith

lemma "(x::nat) < y & y < z ==> x < z"
  by linarith

text ‹This example involves no arithmetic at all, but is solved by
 preprocessing (i.e. NNF normalization) alone.›

lemma "(P::bool) = Q ==> Q = P"
  by linarith

lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> min (x::nat) y = 0"
  by linarith

lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> max (x::nat) y = x + y"
  by linarith

lemma "[| (x::nat) ~= y; a + 2 = b; a < y; y < b; a < x; x < b |] ==> False"
  by linarith

lemma "[| (x::nat) > y; y > z; z > x |] ==> False"
  by linarith

lemma "(x::nat) - 5 > y ==> y < x"
  by linarith

lemma "(x::nat) ~= 0 ==> 0 < x"
  by linarith

lemma "[| (x::nat) ~= y; x <= y |] ==> x < y"
  by linarith

lemma "[| (x::nat) < y; P (x - y) |] ==> P 0"
  by linarith

lemma "(x - y) - (x::nat) = (x - x) - y"
  by linarith

lemma "[| (a::nat) < b; c < d |] ==> (a - b) = (c - d)"
  by linarith

lemma "((a::nat) - (b - (c - (d - e)))) = (a - (b - (c - (d - e))))"
  by linarith

lemma "(n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
  (n = n' & n' < m) | (n = m & m < n') |
  (n' < m & m < n) | (n' < m & m = n) |
  (n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) |
  (m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) |
  (m = n & n < n') | (m = n' & n' < n) |
  (n' = m & m = (n::nat))"
(* FIXME: this should work in principle, but is extremely slow because     *)
(*        preprocessing negates the goal and tries to compute its negation *)
(*        normal form, which creates lots of separate cases for this       *)
(*        disjunction of conjunctions                                      *)
(* by (tactic {* Lin_Arith.tac 1 *}) *)
oops

lemma "2 * (x::nat) ~= 1"
(* FIXME: this is beyond the scope of the decision procedure at the moment, *)
(*        because its negation is satisfiable in the rationals?             *)
(* by (tactic {* Lin_Arith.simple_tac 1 *}) *)
oops

text ‹Constants.›

lemma "(0::nat) < 1"
  by linarith

lemma "(0::int) < 1"
  by linarith

lemma "(47::nat) + 11 < 8 * 15"
  by linarith

lemma "(47::int) + 11 < 8 * 15"
  by linarith

text ‹Splitting of inequalities of different type.›

lemma "[| (a::nat) ~= b; (i::int) ~= j; a < 2; b < 2 |] ==>
  a + b <= nat (max ∣i∣ ∣j∣)"
  by linarith

text ‹Again, but different order.›

lemma "[| (i::int) ~= j; (a::nat) ~= b; a < 2; b < 2 |] ==>
  a + b <= nat (max ∣i∣ ∣j∣)"
  by linarith

end