(* 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: @{ML 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. @{ML 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 @{ML Lin_Arith.simproc}), which---for performance reasons---however does even less splitting than @{ML Lin_Arith.simple_tac} at the moment (namely inequalities only). (On the other hand, it does take apart conjunctions, which @{ML 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" (* rule split_mod is only declared by default for numerals *) using split_mod [of _ _ "0", arith_split] by linarith lemma "(i::nat) mod 1 = 0" (* rule split_mod is only declared by default for numerals *) using split_mod [of _ _ "1", arith_split] by linarith lemma "(i::nat) mod 42 <= 41" by linarith lemma "(i::int) mod 0 = i" (* rule split_zmod is only declared by default for numerals *) using split_zmod [of _ _ "0", arith_split] by linarith lemma "(i::int) mod 1 = 0" (* rule split_zmod is only declared by default for numerals *) using split_zmod [of _ _ "1", arith_split] 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