Library SSProve.Mon.SPropBase

(*From Coq Require Export Logic.StrictProp.*)
(*This file was originally referring to SProp. Not anymore*)
From SSProve.Mon Require Import Base.
From mathcomp Require Import ssreflect.

From Coq Require ClassicalFacts.

Axiom ax_proof_irrel : ClassicalFacts.proof_irrelevance.

Set Primitive Projections.

Module Redefined_sprop_constructs.

Record Box (A:Prop) : Prop := box { unbox : A }.

End Redefined_sprop_constructs.
Export Redefined_sprop_constructs.

Definition sand := and.

Module SPropNotations.

  Notation "⦑ t ⦒" := (exist _ t _).

  Notation " x ∙1" := (proj1_sig x) (at level 2).
  Notation " x ∙2" := (proj2_sig x) (at level 2).

End SPropNotations.

Section sigLemmas.

  Lemma sig_eq {A} (P : A → Prop) :
    ∀ (mx my : sig P), proj1_sig mx = proj1_sig my → mx = my.
  Proof.
    intros [cx ex] [cy ey]. simpl.
    induction 1.
    have hintUnif : ex = ey.
      by apply ax_proof_irrel.
    rewrite hintUnif. reflexivity.
  Defined.

  Lemma transport_sig :
    ∀ {A B} (F : B → A → Prop) {x y} h z,
      eq_rect x (fun x ⇒ sig (fun b ⇒ F b x)) z y h
      = exist _ (proj1_sig z) (@eq_ind A x (F (proj1_sig z)) (proj2_sig z) y h).
  Proof.
    intros.
    dependent inversion h. compute. destruct z. reflexivity.
  Qed.

  Lemma eq_above_sig {A B} (F : B → A → Prop)
        (G := fun x ⇒ sig (fun b ⇒ F b x)) {x1 x2 : A} {h : x1 = x2}
        {z1 : G x1} {z2 : G x2} :
    proj1_sig z1 = proj1_sig z2 → z1 =⟨ h ⟩ z2.
  Proof.
    intro Hz.
    unfold eq_above.
    unfold G.
    rewrite (transport_sig F h z1).
    apply sig_eq.
    assumption.
  Qed.

End sigLemmas.

Module SPropAxioms.

  Import SPropNotations.

  Axiom sprop_ext : ∀ {p q : Prop}, p = q ↔ Box (sand (p → q) (q → p)).

End SPropAxioms.