a9515ac7a4
See issue #692. The implementation still has some rough spots. It is not clear what the right semantic is. Moreover, the folds in e_closure could not be eliminated automatically.
82 lines
2.7 KiB
Text
82 lines
2.7 KiB
Text
/-
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Copyright (c) 2015 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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Formalization of Theorem 1 from the following paper:
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"The inconsistency of a Brouwerian continuity
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principle with the Curry–Howard interpretation"
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by Martín Escardó and Chuangjie Xu
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-/
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import data.nat
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open nat sigma.ops
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/- Bounded equality: α and β agree in the first n positions. -/
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definition beq (α β : nat → nat) (n : nat) : Prop :=
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∀ a, a < n → α a = β a
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notation α `=[`:50 n:50 `]` β:50 := beq α β n
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lemma pred_beq {α β : nat → nat} {n : nat} : α =[n+1] β → α =[n] β :=
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λ h a altn, h a (lt.step altn)
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definition continuous (f : (nat → nat) → nat) : Type₁ :=
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∀ α, Σ n, ∀ β, α =[n] β → f α = f β
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definition zω : nat → nat :=
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λ x, zero
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definition znkω (n : nat) (k : nat) : nat → nat :=
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λ x, if x < n then 0 else k
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lemma znkω_succ (n : nat) (k : nat) : znkω (n+1) k 0 = 0 :=
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rfl
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lemma znkω_bound (n : nat) (k : nat) : znkω n k n = k :=
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if_neg !lt.irrefl
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lemma zω_eq_znkω (n : nat) (k : nat) : zω =[n] znkω n k :=
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λ a altn, begin esimp [zω, znkω], rewrite [if_pos altn] end
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section
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hypothesis all_continuous : ∀ f, continuous f
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definition M (f : (nat → nat) → nat) : nat :=
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(all_continuous f zω).1
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lemma M_spec (f : (nat → nat) → nat) : ∀ β, zω =[M f] β → f zω = f β :=
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(all_continuous f zω).2
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definition m := M (λα, zero)
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definition f β := M (λα, β (α m))
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lemma β0_eq (β : nat → nat) : ∀ α, zω =[f β] α → β 0 = β (α m) :=
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λ α, M_spec (λα, β (α m)) α
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lemma not_all_continuous : false :=
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let β := znkω (M f + 1) 1 in
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let α := znkω m (M f + 1) in
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assert βeq₁ : zω =[M f + 1] β, from
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λ (a : nat) (h : a < M f + 1), begin unfold zω, unfold znkω, rewrite [if_pos h] end,
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assert βeq₂ : zω =[M f] β, from pred_beq βeq₁,
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assert m_eq_fβ : m = f β, from M_spec f β βeq₂,
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assert aux : ∀ α, zω =[m] α → β 0 = β (α m), by rewrite m_eq_fβ at {1}; exact (β0_eq β),
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assert zero_eq_one : 0 = 1, from calc
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0 = β 0 : by rewrite znkω_succ
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... = β (α m) : aux α (zω_eq_znkω m (M f + 1))
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... = β (M f + 1) : by rewrite znkω_bound
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... = 1 : by rewrite znkω_bound,
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by contradiction
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end
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/-
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Additional remarks:
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By using the slightly different definition of continuous
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∀ α, ∃ n, ∀ β, α =[n] β → f α = f β
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i.e., using ∃ instead of Σ, we can assume the following axiom
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all_continuous : ∀ f, continuous f
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However, the system becomes inconsistent again if we also assume Hilbert's choice,
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because with Hilbert's choice we can convert ∃ into Σ
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-/
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