Mcltt.Core.Soundness.SubstitutionCases

From Mcltt Require Import LibTactics.
From Mcltt.Core Require Import Base.
From Mcltt.Core.Completeness Require Import FundamentalTheorem.
From Mcltt.Core.Semantic Require Import Realizability.
From Mcltt.Core.Soundness Require Import LogicalRelation SubtypingCases TermStructureCases UniverseCases.
Import Domain_Notations.

Lemma presup_glu_rel_sub : forall {Γ σ Δ},
    {{ Γ ⊩s σ : Δ }} ->
    {{ ⊩ Γ }} /\ {{ ⊩ Δ }}.
Proof.
  intros * [].
  destruct_conjs.
  split; eexists; eassumption.
Qed.

Lemma presup_left_glu_rel_sub : forall {Γ σ Δ},
    {{ Γ ⊩s σ : Δ }} ->
    {{ ⊩ Γ }}.
Proof.
  intros * []%presup_glu_rel_sub.
  eassumption.
Qed.

#[export]
Hint Resolve presup_left_glu_rel_sub : mcltt.

Lemma presup_right_glu_rel_sub : forall {Γ σ Δ},
    {{ Γ ⊩s σ : Δ }} ->
    {{ ⊩ Δ }}.
Proof.
  intros * []%presup_glu_rel_sub.
  eassumption.
Qed.

#[export]
Hint Resolve presup_right_glu_rel_sub : mcltt.

Lemma glu_rel_sub_id : forall {Γ},
    {{ ⊩ Γ }} ->
    {{ Γ ⊩s Id : Γ }}.
Proof.
  intros * [Sb].
  do 2 eexists; repeat split; mauto.
  intros.
  econstructor; mauto.
  enough {{ Δ ⊢s σ ≈ Id∘σ : Γ }} as <-; mauto.
Qed.

#[export]
Hint Resolve glu_rel_sub_id : mcltt.

Lemma glu_rel_sub_weaken : forall {Γ A},
    {{ ⊩ Γ, A }} ->
    {{ Γ, A ⊩s Wk : Γ }}.
Proof.
  intros * [SbΓA].
  match_by_head1 glu_ctx_env invert_glu_ctx_env.
  handle_functional_glu_ctx_env.
  do 2 eexists; repeat split; [econstructor | |]; try reflexivity; mauto.
  intros.
  destruct_by_head cons_glu_sub_pred.
  econstructor; mauto.
Qed.

#[export]
Hint Resolve glu_rel_sub_weaken : mcltt.

Lemma glu_rel_sub_compose : forall {Γ1 σ2 Γ2 σ1 Γ3},
    {{ Γ1 ⊩s σ2 : Γ2 }} ->
    {{ Γ2 ⊩s σ1 : Γ3 }} ->
    {{ Γ1 ⊩s σ1 ∘ σ2 : Γ3 }}.
Proof.
  intros * Hσ2 Hσ1.
  assert {{ Γ2 ⊢s σ1 : Γ3 }} by mauto.
  assert {{ Γ1 ⊢s σ2 : Γ2 }} by mauto.
  destruct Hσ2 as [SbΓ1 [SbΓ2]].
  destruct_conjs.
  invert_glu_rel_sub Hσ1.
  do 2 eexists; repeat split; mauto.
  intros.
  destruct_glu_rel_sub_with_sub.
  destruct_glu_rel_sub_with_sub.
  handle_functional_glu_ctx_env.
  econstructor; mauto.
  enough {{ Δ ⊢s (σ1 ∘ σ2) ∘ σ ≈ σ1 ∘ (σ2 ∘ σ) : Γ3 }} as -> by eassumption.
  mauto 3.
Qed.

#[export]
Hint Resolve glu_rel_sub_compose : mcltt.

Lemma glu_rel_sub_extend : forall {Γ σ Δ M A i},
    {{ Γ ⊩s σ : Δ }} ->
    {{ Δ ⊩ A : Type@i }} ->
    {{ Γ ⊩ M : A[σ] }} ->
    {{ Γ ⊩s (σ ,, M) : Δ , A }}.
Proof.
  intros * Hσ HA HM.
  assert {{ Γ ⊢s σ : Δ }} by mauto 3.
  assert {{ Δ ⊢ A : Type@i }} by mauto 3.
  assert {{ Γ ⊢ M : A[σ] }} by mauto 3.
  assert {{ Γ ⊩ Type@i : Type@(S i) }} by mauto 3.
  assert {{ Γ ⊢ Type@i[σ] ⊆ Type@i }} by mauto 3.
  assert {{ Γ ⊩ A[σ] : Type@i }} by mauto 3.
  destruct Hσ as [SbΓ [SbΔ]].
  assert {{ Δ ⊩ Type@i : Type@(S i) }} by mauto 3.
  destruct_conjs.
  invert_glu_rel_exp HA.
  invert_glu_rel_exp HM.
  do 2 eexists; repeat split; mauto.
  - econstructor; mauto 3; try reflexivity.
  - intros.
    destruct_glu_rel_sub_with_sub.
    destruct_glu_rel_exp_with_sub.
    simplify_evals.
    match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
    apply_predicate_equivalence.
    unfold univ_glu_exp_pred' in *.
    destruct_conjs.
    handle_functional_glu_univ_elem.
    rename m0 into a.
    assert {{ Δ0 ⊢s σ0 : Γ }} by mauto 4.
    econstructor; mauto 3.
    assert {{ Δ0 ⊢s (σ,,M)∘σ0 : Δ, A }} by mauto 3.
    assert {{ Δ0 ⊢s (σ,,M)∘σ0 ≈ (σ∘σ0),,M[σ0] : Δ, A }} by mauto 3.
    assert {{ Δ0 ⊢s Wk∘((σ,,M)∘σ0) : Δ }} by mauto 4.
    assert {{ Δ0 ⊢s Wk∘((σ,,M)∘σ0) ≈ Wk∘((σ∘σ0),,M[σ0]) : Δ }} by mauto 4.
    assert {{ Δ0 ⊢ M[σ0] : A[σ][σ0] }} by mauto 3.
    assert {{ Δ0 ⊢ M[σ0] : A[σ∘σ0] }} by mauto 3.
    assert {{ Δ0 ⊢s Wk∘((σ∘σ0),,M[σ0]) ≈ σ∘σ0 : Δ }} by mauto 3.
    assert {{ Δ0 ⊢s Wk∘((σ,,M)∘σ0) ≈ σ∘σ0 : Δ }} by mauto 3.
    econstructor; mauto 4.
    + assert {{ Δ, A ⊢s Wk : Δ }} by mauto 4.
      assert {{ Δ0 ⊢ A[Wk][(σ,,M)∘σ0] ≈ A[Wk][(σ∘σ0),,M[σ0]] : Type@i }} as -> by mauto 3.
      assert {{ Δ0 ⊢ A[Wk][(σ∘σ0),,M[σ0]] ≈ A[σ∘σ0] : Type@i }} as -> by mauto 3.
      assert {{ Δ0 ⊢ A[σ∘σ0] ≈ A[σ][σ0] : Type@i }} as -> by mauto 3.
      assert {{ Δ, A ⊢ #0 : A[Wk] }} by mauto 3.
      assert {{ Δ0 ⊢ #0[(σ,,M)∘σ0] ≈ #0[(σ∘σ0),,M[σ0]] : A[Wk][(σ,,M)∘σ0] }} by mauto 3.
      assert {{ Δ0 ⊢ #0[(σ,,M)∘σ0] ≈ #0[(σ∘σ0),,M[σ0]] : A[σ][σ0] }} as -> by mauto 3.
      assert {{ Δ0 ⊢ #0[(σ∘σ0),,M[σ0]] ≈ M[σ0] : A[σ∘σ0] }} by mauto.
      assert {{ Δ0 ⊢ #0[(σ∘σ0),,M[σ0]] ≈ M[σ0] : A[σ][σ0] }} as -> by (eapply wf_exp_eq_conv; mauto 3).
      mautosolve 3.
    + enough {{ Δ0 ⊢s Wk∘((σ,,M)∘σ0) ≈ σ∘σ0 : Δ }} as ->; eassumption.
Qed.

#[export]
Hint Resolve glu_rel_sub_extend : mcltt.