c132f555a9
The previous `decidable_bex` was using a nasty hack. First, it was relying on a bug in the local_context object that was fixed at commit6060b75e6. Note that, the type class resolution will be even more restrictive after we implement the fix described ata75b0d8ee. Second, it was built using tactics that are meant for constructing proof irrelevant code (e.g., `simp`).
68 lines
1.8 KiB
Text
68 lines
1.8 KiB
Text
/-
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Copyright (c) 2016 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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-/
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prelude
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import init.data.list.lemmas
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import init.meta.mk_dec_eq_instance
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open list
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universes u v
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local attribute [simp] join ret
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instance : monad list :=
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{pure := @list.ret, bind := @list.bind,
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id_map := begin
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intros _ xs, induction xs with x xs ih,
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{ refl },
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{ dsimp [function.comp] at ih, dsimp [function.comp], simp [*] }
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end,
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pure_bind := by simp_intros,
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bind_assoc := begin
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intros _ _ _ xs _ _, induction xs,
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{ refl },
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{ simp [*] }
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end}
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instance : alternative list :=
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{ failure := @list.nil,
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orelse := @list.append,
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..list.monad }
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namespace list
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variables {α β : Type u} (p : α → Prop) [decidable_pred p]
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instance bin_tree_to_list : has_coe (bin_tree α) (list α) :=
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⟨bin_tree.to_list⟩
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instance decidable_bex : ∀ (l : list α), decidable (∃ x ∈ l, p x)
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| [] := is_false (by simp)
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| (x::xs) :=
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if h₁ : p x
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then is_true ⟨x, mem_cons_self _ _, h₁⟩
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else match decidable_bex xs with
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| is_true h₂ := is_true
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begin
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cases h₂ with y h, cases h with hm hp,
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exact ⟨y, mem_cons_of_mem _ hm, hp⟩
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end
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| is_false h₂ := is_false
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begin
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intro h, cases h with y h, cases h with hm hp,
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cases eq_or_mem_of_mem_cons hm,
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{ rw [h] at hp, contradiction },
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{ refine absurd _ h₂,
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exact ⟨y, h, hp⟩ }
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end
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end
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instance decidable_ball (l : list α) : decidable (∀ x ∈ l, p x) :=
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if h : ∃ x ∈ l, ¬ p x then
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is_false $ let ⟨x, h, np⟩ := h in λ al, np (al x h)
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else
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is_true $ λ x hx, if h' : p x then h' else false.elim $ h ⟨x, hx, h'⟩
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end list
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