Hacky version of the Collection axiom
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czf.v
59
czf.v
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@ -1,5 +1,4 @@
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From stdpp Require Import base.
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From Coq Require Import Logic.PropExtensionality.
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(** * Cumulative hierarchy *)
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Inductive V : Type :=
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@ -205,6 +204,10 @@ Qed.
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(** * Axioms *)
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(** First, we define a class of set predicates: predicates that respect the equality on V *)
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Class SetPredicate (ϕ : V → Prop) :=
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pred_proper :> Proper ((≡) ==> impl) ϕ.
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(** Extensionality *)
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Theorem CZF_extensionality (a b : V) :
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a ⊆ b ∧ b ⊆ a ↔ a ≡ b.
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@ -215,7 +218,7 @@ Proof.
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Qed.
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(** Paring *)
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Lemma CZF_pairing (x y : V) :
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Theorem CZF_pairing (x y : V) :
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forall (w : V), w ∈ ({[x; y]} : V)
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↔ w ≡ x ∨ w ≡ y.
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Proof.
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@ -230,13 +233,23 @@ Qed.
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(* TODO: need big union operation *)
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(** Empty set *)
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Lemma CZF_empty :
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Theorem CZF_empty :
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forall y, ¬ (y ∈ (∅ : V)).
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Proof. intros ? [[] ?]. Qed.
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(** Set Induction *)
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Theorem CZF_induction ϕ `{SetPredicate ϕ} :
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(forall x, (forall y, y ∈ x → ϕ y) → ϕ x) → (forall x, ϕ x).
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Proof.
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intros Hϕ x.
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induction x as [A f IH].
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apply Hϕ.
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intros y [a ->]. apply IH.
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Qed.
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(** Infinity *)
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Definition suc (x : V) := x ∪ {[x]}.
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Lemma CZF_infinity :
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Theorem CZF_infinity :
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∅ ∈ NN ∧ (forall y, y ∈ NN → suc y ∈ NN).
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Proof.
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split.
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@ -254,9 +267,8 @@ Proof.
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intros [x ?]. exact (f x).
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Defined.
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Class SetPredicate (ϕ : V → Prop) :=
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pred_proper :> Proper ((≡) ==> impl) ϕ.
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Lemma CZF_separation a ϕ `{SetPredicate ϕ} :
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(* Note that we have non-restricted separation *)
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Theorem CZF_separation a ϕ `{SetPredicate ϕ} :
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forall y, (y ∈ sep a ϕ ↔ y ∈ a ∧ ϕ y).
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Proof.
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intros y. destruct a as [A f]. split.
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@ -266,3 +278,36 @@ Proof.
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rewrite Hx in Hϕ.
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exists (existT x Hϕ). assumption.
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Qed.
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(** Collection *)
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(** Collection holds only if we use types-as-proposition
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interpretation, by using Σ-types (landing in Type) instead of ∃
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(landing in Prop). Curiously, the predicate ϕ doesn't have to be
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proper. *)
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Definition coll (a : V) (ϕ : V → V → Prop) :
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(forall (x : V), x ∈ a → { y | ϕ x y }) →
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V.
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Proof.
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intros Hϕ.
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destruct a as [A f].
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refine (sup A _).
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intros x.
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assert (f x ∈ sup A f) as Hfx. { exists x; reflexivity. }
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exact (proj1_sig (Hϕ (f x) Hfx)).
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Defined.
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(* ∀a[∀x∈a ∃y φ(x,y) → ∃b∀x∈a ∃y∈b φ(x,y)] *)
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Theorem CZF_collection (a : V) (ϕ : V → V → Prop)
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`{Proper _ ((≡) ==> (≡) ==> (impl)) ϕ} :
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(forall (x : V), x ∈ a → { y | ϕ x y }) →
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exists (b : V), (forall (x : V), x ∈ a → exists y, y ∈ b ∧ ϕ x y ).
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Proof.
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intros Hϕ.
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exists (coll a ϕ Hϕ). intros xx Hxx.
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destruct a as [A f].
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destruct Hxx as [x Hx]. simpl in *.
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set (Hϕ':=Hϕ (f x) (ex_intro (λ y, f x ≡ f y) x (reflexivity _))).
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exists (proj1_sig Hϕ'). split.
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2: { rewrite Hx. apply (proj2_sig Hϕ'). }
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exists x. cbn. reflexivity.
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Qed.
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