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refactor(library/data/int/basic): cleanup proof

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Leonardo de Moura 2015-10-22 13:45:20 -07:00
parent bceac9ece5
commit ac01e9e352
1 changed files with 17 additions and 2 deletions
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19
library/data/int/basic.lean
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@ -95,6 +95,18 @@ definition int_has_add [reducible] [instance] [priority int.prio] : has_add int
definition int_has_neg [reducible] [instance] [priority int.prio] : has_neg int := has_neg.mk int.neg
definition int_has_mul [reducible] [instance] [priority int.prio] : has_mul int := has_mul.mk int.mul
lemma mul_of_nat_of_nat (m n : nat) : of_nat m * of_nat n = of_nat (m * n) :=
rfl
lemma mul_of_nat_neg_succ_of_nat (m n : nat) : of_nat m * -[1+ n] = neg_of_nat (m * succ n) :=
rfl
lemma mul_neg_succ_of_nat_of_nat (m n : nat) : -[1+ m] * of_nat n = neg_of_nat (succ m * n) :=
rfl
lemma mul_neg_succ_of_nat_neg_succ_of_nat (m n : nat) : -[1+ m] * -[1+ n] = succ m * succ n :=
rfl
/- some basic functions and properties -/
theorem of_nat.inj {m n : ℕ} (H : of_nat m = of_nat n) : m = n :=
@ -412,12 +424,15 @@ calc
... = pabs (repr a) + nat_abs b : nat_abs_eq_pabs_repr
... = nat_abs a + nat_abs b : nat_abs_eq_pabs_repr
theorem nat_abs_neg_of_nat (n : nat) : nat_abs (neg_of_nat n) = n :=
begin cases n, reflexivity, reflexivity end
section
local attribute nat_abs [reducible]
theorem nat_abs_mul : Π (a b : ℤ), nat_abs (a * b) = (nat_abs a) * (nat_abs b)
| (of_nat m) (of_nat n) := rfl
| (of_nat m) -[1+ n] := !nat_abs_neg ▸ rfl
| -[1+ m] (of_nat n) := !nat_abs_neg ▸ rfl
| (of_nat m) -[1+ n] := by rewrite [mul_of_nat_neg_succ_of_nat, nat_abs_neg_of_nat]
| -[1+ m] (of_nat n) := by rewrite [mul_neg_succ_of_nat_of_nat, nat_abs_neg_of_nat]
| -[1+ m] -[1+ n] := rfl
end