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feat(library/data/fin) : establish add_comm_group on fin using madd

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Haitao Zhang 2015-06-16 12:38:13 -07:00 committed by Leonardo de Moura
parent f60cdee14b
commit 8817042318
1 changed files with 41 additions and 4 deletions
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library/data/fin.lean
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@ -5,7 +5,7 @@ Authors: Haitao Zhang, Leonardo de Moura
Finite ordinal types. Finite ordinal types.
-/ -/
import data.list.basic data.finset.basic data.fintype.card import data.list.basic data.finset.basic data.fintype.card algebra.group
open eq.ops nat function list finset fintype open eq.ops nat function list finset fintype
structure fin (n : nat) := (val : nat) (is_lt : val < n) structure fin (n : nat) := (val : nat) (is_lt : val < n)
@ -104,9 +104,6 @@ lift i 1
definition maxi [reducible] : fin (succ n) := definition maxi [reducible] : fin (succ n) :=
mk n !lt_succ_self mk n !lt_succ_self
definition sub_max : fin (succ n) → fin (succ n)
| (mk iv ilt) := mk (succ iv mod (succ n)) (mod_lt _ !zero_lt_succ)
theorem val_lift : ∀ (i : fin n) (m : nat), val i = val (lift i m) theorem val_lift : ∀ (i : fin n) (m : nat), val i = val (lift i m)
| (mk v h) m := rfl | (mk v h) m := rfl
@ -187,6 +184,10 @@ take f₁ f₂ Peq, funext (λ i,
assert Peqi : lift_fun f₁ (lift_succ i) = lift_fun f₂ (lift_succ i), from congr_fun Peq _, assert Peqi : lift_fun f₁ (lift_succ i) = lift_fun f₂ (lift_succ i), from congr_fun Peq _,
begin revert Peqi, rewrite [*lift_fun_eq], apply lift_succ_inj end) begin revert Peqi, rewrite [*lift_fun_eq], apply lift_succ_inj end)
lemma lower_inj_apply {f Pinj Pmax} (i : fin n) :
val (lower_inj f Pinj Pmax i) = val (f (lift_succ i)) :=
by rewrite [↑lower_inj]
end lift_lower end lift_lower
section madd section madd
@ -197,6 +198,42 @@ mk ((i + j) mod (succ n)) (mod_lt _ !zero_lt_succ)
lemma val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) mod (succ n) lemma val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) mod (succ n)
| (mk iv ilt) (mk jv jlt) := by rewrite [val_mk] | (mk iv ilt) (mk jv jlt) := by rewrite [val_mk]
lemma madd_inj : ∀ {i : fin (succ n)}, injective (madd i)
| (mk iv ilt) :=
take j₁ j₂, fin.destruct j₁ (fin.destruct j₂ (λ jv₁ jlt₁ jv₂ jlt₂, begin
rewrite [↑madd, -eq_iff_veq],
intro Peq, congruence,
rewrite [-(mod_eq_of_lt jlt₁), -(mod_eq_of_lt jlt₂)],
apply mod_eq_mod_of_add_mod_eq_add_mod_left Peq
end))
lemma val_mod : ∀ i : fin (succ n), (val i) mod (succ n) = val i
| (mk iv ilt) := by rewrite [*val_mk, mod_eq_of_lt ilt]
definition minv : ∀ i : fin (succ n), fin (succ n)
| (mk iv ilt) := mk ((succ n - iv) mod succ n) (mod_lt _ !zero_lt_succ)
lemma madd_comm (i j : fin (succ n)) : madd i j = madd j i :=
by apply eq_of_veq; rewrite [*val_madd, add.comm (val i)]
lemma zero_madd (i : fin (succ n)) : madd (zero n) i = i :=
by apply eq_of_veq; rewrite [val_madd, ↑zero, nat.zero_add, mod_eq_of_lt (is_lt i)]
lemma madd_zero (i : fin (succ n)) : madd i (zero n) = i :=
!madd_comm ▸ zero_madd i
lemma madd_assoc (i j k : fin (succ n)) : madd (madd i j) k = madd i (madd j k) :=
by apply eq_of_veq; rewrite [*val_madd, mod_add_mod, add_mod_mod, add.assoc (val i)]
lemma madd_left_inv : ∀ i : fin (succ n), madd (minv i) i = zero n
| (mk iv ilt) := eq_of_veq (by
rewrite [val_madd, ↑minv, ↑zero, mod_add_mod, sub_add_cancel (le_of_lt ilt), mod_self])
open algebra
definition madd_is_comm_group [instance] : add_comm_group (fin (succ n)) :=
add_comm_group.mk madd madd_assoc (zero n) zero_madd madd_zero minv madd_left_inv madd_comm
end madd end madd
definition pred : fin n → fin n definition pred : fin n → fin n