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feat(library/data/fin): add more fin definition and lemmas

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Haitao Zhang 2015-07-14 19:19:26 -07:00 committed by Leonardo de Moura
parent 516333ad65
commit 9d523bae6b
1 changed files with 25 additions and 3 deletions
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library/data/fin.lean
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@ -91,6 +91,9 @@ end pigeonhole
definition zero (n : nat) : fin (succ n) :=
mk 0 !zero_lt_succ
definition mk_mod [reducible] (n i : nat) : fin (succ n) :=
mk (i mod (succ n)) (mod_lt _ !zero_lt_succ)
variable {n : nat}
theorem val_lt : ∀ i : fin n, val i < n
@ -108,8 +111,19 @@ mk n !lt_succ_self
theorem val_lift : ∀ (i : fin n) (m : nat), val i = val (lift i m)
| (mk v h) m := rfl
lemma mk_succ_ne_zero {i : nat} : ∀ {P}, mk (succ i) P ≠ zero n :=
assume P Pe, absurd (veq_of_eq Pe) !succ_ne_zero
lemma mk_mod_eq {i : fin (succ n)} : i = mk_mod n i :=
eq_of_veq begin rewrite [↑mk_mod, mod_eq_of_lt !is_lt] end
lemma mk_mod_of_lt {i : nat} (Plt : i < succ n) : mk_mod n i = mk i Plt :=
begin esimp [mk_mod], congruence, exact mod_eq_of_lt Plt end
section lift_lower
lemma lift_zero : lift_succ (zero n) = zero (succ n) := rfl
lemma ne_max_of_lt_max {i : fin (succ n)} : i < n → i ≠ maxi :=
by intro hlt he; substvars; exact absurd hlt (lt.irrefl n)
@ -196,6 +210,9 @@ section madd
definition madd (i j : fin (succ n)) : fin (succ n) :=
mk ((i + j) mod (succ n)) (mod_lt _ !zero_lt_succ)
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 val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) mod (succ n)
| (mk iv ilt) (mk jv jlt) := by esimp
@ -208,12 +225,12 @@ take j₁ j₂, fin.destruct j₁ (fin.destruct j₂ (λ jv₁ jlt₁ jv₂ jlt
apply mod_eq_mod_of_add_mod_eq_add_mod_left Peq
end))
lemma madd_mk_mod {i j : nat} : madd (mk_mod n i) (mk_mod n j) = mk_mod n (i+j) :=
eq_of_veq begin esimp [madd, mk_mod], rewrite [ mod_add_mod, add_mod_mod ] end
lemma val_mod : ∀ i : fin (succ n), (val i) mod (succ n) = val i
| (mk iv ilt) := by esimp; rewrite [(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)]
@ -255,6 +272,11 @@ definition succ : fin n → fin (succ n)
lemma val_succ : ∀ (i : fin n), val (succ i) = nat.succ (val i)
| (mk v h) := rfl
lemma succ_max : fin.succ maxi = (@maxi (nat.succ n)) := rfl
lemma lift_succ.comm : lift_succ ∘ (@succ n) = succ ∘ lift_succ :=
funext take i, eq_of_veq (begin rewrite [↑lift_succ, -val_lift, *val_succ, -val_lift] end)
definition elim0 {C : fin 0 → Type} : Π i : fin 0, C i
| (mk v h) := absurd h !not_lt_zero