81 lines
2.8 KiB
Text
81 lines
2.8 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.fin
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Author: Leonardo de Moura
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Finite ordinals.
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-/
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import data.nat logic.cast
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open nat
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inductive fin : nat → Type :=
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fz : Π n, fin (succ n),
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fs : Π {n}, fin n → fin (succ n)
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namespace fin
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definition to_nat : Π {n}, fin n → nat,
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@to_nat ⌞n+1⌟ (fz n) := zero,
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@to_nat ⌞n+1⌟ (fs f) := succ (to_nat f)
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theorem to_nat_fz (n : nat) : to_nat (fz n) = zero :=
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rfl
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theorem to_nat_fs {n : nat} (f : fin n) : to_nat (fs f) = succ (to_nat f) :=
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rfl
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theorem to_nat_lt : Π {n} (f : fin n), to_nat f < n,
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to_nat_lt (fz n) := calc
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to_nat (fz n) = 0 : rfl
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... < n+1 : succ_pos n,
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to_nat_lt (@fs n f) := calc
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to_nat (fs f) = (to_nat f)+1 : rfl
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... < n+1 : succ_lt_succ (to_nat_lt f)
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definition lift : Π {n : nat}, fin n → Π (m : nat), fin (m + n),
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@lift ⌞n+1⌟ (fz n) m := fz (m + n),
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@lift ⌞n+1⌟ (@fs n f) m := fs (lift f m)
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theorem lift_fz (n m : nat) : lift (fz n) m = fz (m + n) :=
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rfl
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theorem lift_fs {n : nat} (f : fin n) (m : nat) : lift (fs f) m = fs (lift f m) :=
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rfl
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theorem to_nat_lift : ∀ {n : nat} (f : fin n) (m : nat), to_nat f = to_nat (lift f m),
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to_nat_lift (fz n) m := rfl,
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to_nat_lift (@fs n f) m := calc
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to_nat (fs f) = (to_nat f) + 1 : rfl
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... = (to_nat (lift f m)) + 1 : to_nat_lift f
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... = to_nat (lift (fs f) m) : rfl
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definition of_nat : Π (p : nat) (n : nat), p < n → fin n,
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of_nat 0 0 h := absurd h (not_lt_zero zero),
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of_nat 0 (n+1) h := fz n,
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of_nat (p+1) 0 h := absurd h (not_lt_zero (succ p)),
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of_nat (p+1) (n+1) h := fs (of_nat p n (lt_of_succ_lt_succ h))
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theorem of_nat_zero_succ (n : nat) (h : 0 < n+1) : of_nat 0 (n+1) h = fz n :=
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rfl
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theorem of_nat_succ_succ (p n : nat) (h : p+1 < n+1) :
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of_nat (p+1) (n+1) h = fs (of_nat p n (lt_of_succ_lt_succ h)) :=
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rfl
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theorem to_nat_of_nat : ∀ (p : nat) (n : nat) (h : p < n), to_nat (of_nat p n h) = p,
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to_nat_of_nat 0 0 h := absurd h (not_lt_zero 0),
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to_nat_of_nat 0 (n+1) h := rfl,
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to_nat_of_nat (p+1) 0 h := absurd h (not_lt_zero (p+1)),
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to_nat_of_nat (p+1) (n+1) h := calc
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to_nat (of_nat (p+1) (n+1) h)
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= succ (to_nat (of_nat p n _)) : rfl
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... = succ p : to_nat_of_nat p n _
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theorem of_nat_to_nat : ∀ {n : nat} (f : fin n) (h : to_nat f < n), of_nat (to_nat f) n h = f,
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of_nat_to_nat (fz n) h := rfl,
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of_nat_to_nat (@fs n f) h := calc
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of_nat (to_nat (fs f)) (succ n) h = fs (of_nat (to_nat f) n _) : rfl
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... = fs f : of_nat_to_nat f _
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end fin
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