92 lines
2.7 KiB
Text
92 lines
2.7 KiB
Text
/-
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Example from "Inductively defined types",
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from Thierry Coquand and Christine Paulin,
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COLOG-88.
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It shows it is inconsistent to allow inductive datatypes such as
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inductive A : Type :=
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| intro : ((A → Prop) → Prop) → A
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-/
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/- Phi is a positive, but not strictly positive, operator. -/
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definition Phi (A : Type) := (A → Prop) → Prop
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/- If we were allowed to form the inductive type
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inductive A: Type :=
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| introA : Phi A -> A
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we would get the following
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-/
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universe l
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-- The new type A
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axiom A : Type.{l}
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-- The constructor
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axiom introA : Phi A → A
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-- The eliminator
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axiom recA : Π {C : A → Type}, (Π (p : Phi A), C (introA p)) → (Π (a : A), C a)
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-- The "computational rule"
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axiom recA_comp : Π {C : A → Type} (H : Π (p : Phi A), C (introA p)) (p : Phi A), recA H (introA p) = H p
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-- The recursor could be used to define matchA
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noncomputable definition matchA (a : A) : Phi A :=
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recA (λ p, p) a
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-- and the computation rule would allows us to define
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lemma betaA (p : Phi A) : matchA (introA p) = p :=
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!recA_comp
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-- As in all inductive datatypes, we would be able to prove that constructors are injective.
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lemma introA_injective : ∀ {p p' : Phi A}, introA p = introA p' → p = p' :=
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λ p p' h,
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have aux : matchA (introA p) = matchA (introA p'), by rewrite h,
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by rewrite [*betaA at aux]; exact aux
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-- For any type T, there is an injection from T to (T → Prop)
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definition i {T : Type} : T → (T → Prop) :=
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λ x y, x = y
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lemma i_injective {T : Type} {a b : T} : i a = i b → a = b :=
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λ h,
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have e₁ : i a a = i b a, by rewrite [h],
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have e₂ : (a = a) = (b = a), from e₁,
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have e₃ : b = a, from eq.subst e₂ rfl,
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eq.symm e₃
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-- Hence, by composition, we get an injection f from (A → Prop) to A
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noncomputable definition f : (A → Prop) → A :=
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λ p, introA (i p)
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lemma f_injective : ∀ {p p' : A → Prop}, f p = f p' → p = p':=
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λ (p p' : A → Prop) (h : introA (i p) = introA (i p')),
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i_injective (introA_injective h)
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/-
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We are now back to the usual Cantor-Russel paradox.
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We can define
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-/
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noncomputable definition P0 (a : A) : Prop :=
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∃ (P : A → Prop), f P = a ∧ ¬ P a
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-- i.e., P0 a := codes a set P such that x∉P
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noncomputable definition x0 : A := f P0
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lemma fP0_eq : f P0 = x0 :=
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rfl
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lemma not_P0_x0 : ¬ P0 x0 :=
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λ h : P0 x0,
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obtain (P : A → Prop) (hp : f P = x0 ∧ ¬ P x0), from h,
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have fp_eq : f P = f P0, from and.elim_left hp,
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have p_eq : P = P0, from f_injective fp_eq,
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have nh : ¬ P0 x0, by rewrite [p_eq at hp]; exact (and.elim_right hp),
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absurd h nh
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lemma P0_x0 : P0 x0 :=
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exists.intro P0 (and.intro fP0_eq not_P0_x0)
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theorem inconsistent : false :=
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absurd P0_x0 not_P0_x0
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