lean2/library/logic/examples/cont.lean

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/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
Formalization of Theorem 1 from the following paper:
"The inconsistency of a Brouwerian continuity
principle with the CurryHoward interpretation"
by Martín Escardó and Chuangjie Xu
-/
import data.nat
open nat sigma.ops
/- Bounded equality: α and β agree in the first n positions. -/
definition beq (α β : nat → nat) (n : nat) : Prop :=
∀ a, a < n → α a = β a
notation α `=[`:50 n:50 `]` β:50 := beq α β n
lemma pred_beq {α β : nat → nat} {n : nat} : α =[n+1] β → α =[n] β :=
λ h a altn, h a (lt.step altn)
definition continuous (f : (nat → nat) → nat) : Type₁ :=
α, Σ n, ∀ β, α =[n] β → f α = f β
definition zω : nat → nat :=
λ x, zero
definition znkω (n : nat) (k : nat) : nat → nat :=
λ x, if x < n then 0 else k
lemma znkω_succ (n : nat) (k : nat) : znkω (n+1) k 0 = 0 :=
rfl
lemma znkω_bound (n : nat) (k : nat) : znkω n k n = k :=
if_neg !lt.irrefl
lemma zω_eq_znkω (n : nat) (k : nat) : zω =[n] znkω n k :=
λ a altn, begin esimp [zω, znkω], rewrite [if_pos altn] end
section
hypothesis all_continuous : ∀ f, continuous f
definition M (f : (nat → nat) → nat) : nat :=
(all_continuous f zω).1
lemma M_spec (f : (nat → nat) → nat) : ∀ β, zω =[M f] β → f zω = f β :=
(all_continuous f zω).2
definition m := M (λα, zero)
definition f β := M (λα, β (α m))
lemma β0_eq (β : nat → nat) : ∀ α, zω =[f β] α → β 0 = β (α m) :=
λ α, M_spec (λα, β (α m)) α
lemma not_all_continuous : false :=
let β := znkω (M f + 1) 1 in
let α := znkω m (M f + 1) in
assert βeq₁ : zω =[M f + 1] β, from
λ (a : nat) (h : a < M f + 1), begin esimp [zω, znkω], rewrite [if_pos h] end,
assert βeq₂ : zω =[M f] β, from pred_beq βeq₁,
assert m_eq_fβ : m = f β, from M_spec f β βeq₂,
assert aux : ∀ α, zω =[m] α → β 0 = β (α m), by rewrite m_eq_fβ at {1}; exact (β0_eq β),
assert zero_eq_one : 0 = 1, from calc
0 = β 0 : by rewrite znkω_succ
... = β (α m) : aux α (zω_eq_znkω m (M f + 1))
... = β (M f + 1) : by rewrite znkω_bound
... = 1 : by rewrite znkω_bound,
nat.no_confusion zero_eq_one
end
/-
Additional remarks:
By using the slightly different definition of continuous
α, ∃ n, ∀ β, α =[n] β → f α = f β
i.e., using ∃ instead of Σ, we can assume the following axiom
all_continuous : ∀ f, continuous f
However, the system becomes inconsistent again if we also assume Hilbert's choice,
because with Hilbert's choice we can convert ∃ into Σ
-/