88 lines
4.4 KiB
Text
88 lines
4.4 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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-- Author: Leonardo de Moura
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import macros
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import pair
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import subtype
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import optional
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using subtype
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using optional
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-- We are encoding the (sum A B) as a subtype of (optional A) # (optional B), where
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-- (proj1 n = none) ≠ (proj2 n = none)
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definition sum_pred (A B : (Type U)) := λ p : (optional A) # (optional B), (proj1 p = none) ≠ (proj2 p = none)
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definition sum (A B : (Type U)) := subtype ((optional A) # (optional B)) (sum_pred A B)
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namespace sum
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theorem inl_pred {A : (Type U)} (a : A) (B : (Type U)) : sum_pred A B (pair (some a) none)
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:= not_intro
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(assume N : (some a = none) = (none = (@none B)),
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have eq : some a = none,
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from (symm N) ◂ (refl (@none B)),
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show false,
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from absurd eq (distinct a))
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theorem inr_pred (A : (Type U)) {B : (Type U)} (b : B) : sum_pred A B (pair none (some b))
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:= not_intro
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(assume N : (none = (@none A)) = (some b = none),
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have eq : some b = none,
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from N ◂ (refl (@none A)),
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show false,
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from absurd eq (distinct b))
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theorem inhabl {A : (Type U)} (H : inhabited A) (B : (Type U)) : inhabited (sum A B)
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:= inhabited_elim H (take w : A,
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subtype_inhabited (exists_intro (pair (some w) none) (inl_pred w B)))
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theorem inhabr (A : (Type U)) {B : (Type U)} (H : inhabited B) : inhabited (sum A B)
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:= inhabited_elim H (take w : B,
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subtype_inhabited (exists_intro (pair none (some w)) (inr_pred A w)))
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definition inl {A : (Type U)} (a : A) (B : (Type U)) : sum A B
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:= abst (pair (some a) none) (inhabl (inhabited_intro a) B)
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definition inr (A : (Type U)) {B : (Type U)} (b : B) : sum A B
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:= abst (pair none (some b)) (inhabr A (inhabited_intro b))
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theorem inl_inj {A B : (Type U)} {a1 a2 : A} : inl a1 B = inl a2 B → a1 = a2
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:= assume Heq : inl a1 B = inl a2 B,
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have eq1 : inl a1 B = abst (pair (some a1) none) (inhabl (inhabited_intro a1) B),
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from refl (inl a1 B),
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have eq2 : inl a2 B = abst (pair (some a2) none) (inhabl (inhabited_intro a1) B),
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from subst (refl (inl a2 B)) (proof_irrel (inhabl (inhabited_intro a2) B) (inhabl (inhabited_intro a1) B)),
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have rep_eq : (pair (some a1) none) = (pair (some a2) none),
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from abst_inj (inhabl (inhabited_intro a1) B) (inl_pred a1 B) (inl_pred a2 B) (trans (trans (symm eq1) Heq) eq2),
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show a1 = a2,
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from optional::injectivity
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(calc some a1 = proj1 (pair (some a1) (@none B)) : refl (some a1)
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... = proj1 (pair (some a2) (@none B)) : pair_inj1 rep_eq
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... = some a2 : refl (some a2))
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theorem inr_inj {A B : (Type U)} {b1 b2 : B} : inr A b1 = inr A b2 → b1 = b2
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:= assume Heq : inr A b1 = inr A b2,
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have eq1 : inr A b1 = abst (pair none (some b1)) (inhabr A (inhabited_intro b1)),
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from refl (inr A b1),
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have eq2 : inr A b2 = abst (pair none (some b2)) (inhabr A (inhabited_intro b1)),
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from subst (refl (inr A b2)) (proof_irrel (inhabr A (inhabited_intro b2)) (inhabr A (inhabited_intro b1))),
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have rep_eq : (pair none (some b1)) = (pair none (some b2)),
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from abst_inj (inhabr A (inhabited_intro b1)) (inr_pred A b1) (inr_pred A b2) (trans (trans (symm eq1) Heq) eq2),
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show b1 = b2,
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from optional::injectivity
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(calc some b1 = proj2 (pair (@none A) (some b1)) : refl (some b1)
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... = proj2 (pair (@none A) (some b2)) : pair_inj2 rep_eq
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... = some b2 : refl (some b2))
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theorem distinct {A B : (Type U)} (a : A) (b : B) : inl a B ≠ inr A b
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:= assume N : inl a B = inr A b,
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have eq1 : inl a B = abst (pair (some a) none) (inhabl (inhabited_intro a) B),
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from refl (inl a B),
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have eq2 : inr A b = abst (pair none (some b)) (inhabl (inhabited_intro a) B),
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from subst (refl (inr A b)) (proof_irrel (inhabr A (inhabited_intro b)) (inhabl (inhabited_intro a) B)),
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have rep_eq : (pair (some a) none) = (pair none (some b)),
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from abst_inj (inhabl (inhabited_intro a) B) (inl_pred a B) (inr_pred A b) (trans (trans (symm eq1) N) eq2),
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show false,
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from absurd (pair_inj1 rep_eq) (optional::distinct a)
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theorem dichotomy {A B : (Type U)} (n : sum A B) : (∃ a, n = inl a B) ∨ (∃ b, n = inr A b)
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:= let pred : (proj1 (rep n) = none) ≠ (proj2 (rep n) = none) := P_rep n
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in _
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