Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Optimize soundness a bit #165

Merged
merged 1 commit into from
Sep 1, 2024
Merged

Optimize soundness a bit #165

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
76 changes: 26 additions & 50 deletions theories/Core/Soundness/NatCases.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -131,6 +131,19 @@ Qed.
#[export]
Hint Resolve glu_rel_sub_extend_nat : mcltt.

Lemma exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1 : forall {Δ σ Γ i A},
{{ Δ ⊢s σ : Γ }} ->
{{ Γ, ℕ ⊢ A : Type@i }} ->
{{ Δ, ℕ, A[q σ] ⊢ A[q σ][Wk∘Wk,,succ #1] ≈ A[Wk∘Wk,,succ #1][q (q σ)] : Type@i }}.
Proof.
intros.
autorewrite with mcltt.
rewrite -> @sub_eq_q_sigma_compose_weak_weak_extend_succ_var_1; mauto 2.
eapply exp_eq_refl.
eapply exp_sub_typ; mauto 2.
econstructor; mauto 3.
Qed.

Lemma glu_rel_exp_natrec_zero_helper : forall {i Γ SbΓ A MZ MS Δ M σ p am P El},
{{ EG Γ ∈ glu_ctx_env ↘ SbΓ }} ->
{{ Γ, ℕ ⊢ A : Type@i }} ->
Expand Down Expand Up @@ -160,6 +173,7 @@ Proof.
assert {{ Δ ⊢s σ : Γ }} by mauto 2.
assert {{ ⊢ Δ }} by mauto 2.
assert {{ ⊢ Δ, ℕ }} by mauto 3.
assert {{ Δ, ℕ ⊢s q σ : Γ, ℕ }} by mauto 3.
assert {{ ⊢ Δ, ℕ, A[q σ] }} by mauto 3.
assert {{ Δ ⊢s Id : Δ }} by mauto 2.
assert {{ Δ ⊢s σ,,M ≈ σ,,zero : Γ, ℕ }} as -> by mauto 3.
Expand All @@ -173,18 +187,8 @@ Proof.
assert {{ Δ ⊢ A[σ,,zero[σ]] ≈ A[Id,,zero][σ] : Type@i }} by mauto 3.
assert {{ Δ ⊢ A[q σ][Id,,zero] ≈ A[σ,,zero] : Type@i }} as <- by mauto 2.
assert {{ Δ ⊢ MZ[σ] : A[q σ][Id,,zero] }} by bulky_rewrite.
assert {{ Δ, ℕ, A[q σ] ⊢ succ #1 : ℕ[σ][Wk∘Wk] }}.
{
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ[Wk][Wk] }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ ℕ[σ][Wk∘Wk] ≈ ℕ : Type@0 }}; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }}.
{
autorewrite with mcltt.
rewrite -> @sub_eq_q_sigma_compose_weak_weak_extend_succ_var_1; mauto 3.
rewrite <- @exp_eq_sub_compose_typ; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢s Wk∘Wk,,succ #1 : Δ, ℕ }} by mauto 4.
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }} by (rewrite @exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1; mauto 3).
pose (R := {{{ rec zero return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end }}}).
assert
{{ Δ ⊢ R ≈ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[q σ][Id,,zero] }} as <-
Expand Down Expand Up @@ -274,20 +278,8 @@ Proof.
assert {{ Γ ⊢ zero : ℕ }} by mauto 3.
assert {{ Δ ⊢ A[σ,,zero[σ]] ≈ A[Id,,zero][σ] : Type@i }} by mauto 4.
assert {{ Δ ⊢ MZ[σ] : A[q σ][Id,,zero] }} by bulky_rewrite.
assert {{ Δ, ℕ, A[q σ] ⊢ succ #1 : ℕ[σ][Wk∘Wk] }}.
{
assert {{ Δ, ℕ ⊢s Wk : Δ }} by mauto 2.
assert {{ Δ, ℕ, A[q σ] ⊢s Wk : Δ, ℕ }} by mauto 2.
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ[Wk][Wk] }} by mauto 4.
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ ℕ[σ][Wk∘Wk] ≈ ℕ : Type@0 }}; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }}.
{
autorewrite with mcltt.
rewrite -> @sub_eq_q_sigma_compose_weak_weak_extend_succ_var_1; mauto 3.
rewrite <- @exp_eq_sub_compose_typ; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢s Wk∘Wk,,succ #1 : Δ, ℕ }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }} by (rewrite @exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1; mauto 3).
pose (R := {{{ rec M' return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end }}}).
assert (exists r, {{ rec m' ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\ {{ Δ ⊢ R : A[σ,,M'] ® r ∈ El' }}) as [r' []] by mauto 3.
assert {{ Δ ⊢ R : A[σ,,M'] }} by (erewrite <- @exp_eq_elim_sub_rhs_typ; mauto 3).
Expand Down Expand Up @@ -336,12 +328,11 @@ Proof.
unfold univ_glu_exp_pred' in *.
destruct_conjs.
assert {{ Δ ⊢s σ : Γ }} by mauto 2.
assert {{ Δ, A[σ] ⊢s q σ : Γ, A }} by mauto 2.
assert {{ Δ, A[σ] ⊢w Wk : Δ }} by mauto 3.
assert {{ ⊢ Δ, A[σ] }} by mauto 3.
assert {{ Δ, A[σ] ⊢w Wk : Δ }} by mauto 2.
eapply cons_glu_sub_pred_helper; mauto 2.
- eapply glu_ctx_env_sub_monotone; eassumption.
- assert {{ Δ, A[σ] ⊢ #0 : A[σ][Wk] }} by mauto 3.
assert {{ Δ, A[σ] ⊢s Wk : Δ }} by mauto 2.
- assert {{ Δ, A[σ] ⊢s Wk : Δ }} by mauto 2.
assert {{ Δ, A[σ] ⊢ A[σ∘Wk] ≈ A[σ][Wk] : Type@i }} as -> by mauto 3.
eapply var0_glu_elem; eassumption.
Qed.
Expand All @@ -362,7 +353,7 @@ Proof.
assert {{ Δ ⊢ ℕ[σ] ≈ ℕ : Type@i }} by mauto 4.
assert {{ EG Γ, ℕ ∈ glu_ctx_env ↘ cons_glu_sub_pred i Γ {{{ ℕ }}} SbΓ }}
by (invert_glu_rel_exp Hℕ; econstructor; mauto 3; reflexivity).
assert {{ ⊢ Δ }} by mauto 3.
assert {{ ⊢ Δ }} by mauto 2.
cbn.
assert {{ ⊢ Δ, ℕ[σ] ≈ Δ, ℕ }} as <- by mauto 3.
eassumption.
Expand Down Expand Up @@ -433,20 +424,8 @@ Proof.
assert {{ Γ ⊢ zero : ℕ }} by mauto 3.
assert {{ Δ ⊢ A[σ,,zero[σ]] ≈ A[Id,,zero][σ] : Type@i }} by mauto 4.
assert {{ Δ ⊢ MZ[σ] : A[q σ][Id,,zero] }} by bulky_rewrite.
assert {{ Δ, ℕ, A[q σ] ⊢ succ #1 : ℕ[σ][Wk∘Wk] }}.
{
assert {{ Δ, ℕ ⊢s Wk : Δ }} by mauto 2.
assert {{ Δ, ℕ, A[q σ] ⊢s Wk : Δ, ℕ }} by mauto 2.
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ[Wk][Wk] }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ #1 : ℕ }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ ℕ[σ][Wk∘Wk] ≈ ℕ : Type@0 }}; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }}.
{
autorewrite with mcltt.
rewrite -> @sub_eq_q_sigma_compose_weak_weak_extend_succ_var_1; mauto 3.
rewrite <- @exp_eq_sub_compose_typ; mauto 4.
}
assert {{ Δ, ℕ, A[q σ] ⊢s Wk∘Wk,,succ #1 : Δ, ℕ }} by mauto 3.
assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }} by (rewrite @exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1; mauto 3).
pose (R := {{{ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end }}}).
enough {{ Δ ⊢ R : A[σ,,M] ® rec m under p return A | zero -> mz | succ -> MS end ∈ glu_elem_bot i am }} by (eapply realize_glu_elem_bot; mauto 3).
econstructor; mauto 3.
Expand Down Expand Up @@ -580,12 +559,10 @@ Proof.
assert {{ ⊢ Δ', ℕ, A[q σ][q τ] ≈ Δ', ℕ, A[q (σ∘τ)] }} as -> by eassumption.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢s Wk∘Wk,,succ #1 : Δ', ℕ }} by mauto 2.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢ A[q σ][q τ][Wk∘Wk,,succ #1] ≈ A[q (σ∘τ)][Wk∘Wk,,succ #1] : Type@i }} as -> by mauto 3.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢ A[q (σ∘τ)][Wk∘Wk,,succ #1] ≈ A[q (σ∘τ)∘(Wk∘Wk,,succ #1)] : Type@i }} as -> by mauto 3.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢ A[q (σ∘τ)∘(Wk∘Wk,,succ #1)] ≈ A[(Wk∘Wk,,succ #1)∘q (q (σ∘τ))] : Type@i }} as -> by mauto 3.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢ A[q (σ∘τ)][Wk∘Wk,,succ #1] ≈ A[Wk∘Wk,,succ #1][q (q (σ∘τ))] : Type@i }} as -> by mauto 3 using exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1.
assert {{ Δ', ℕ ⊢s q (σ∘τ) : Γ, ℕ }} by mauto 2.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢s q (q (σ∘τ)) : Γ, ℕ, A }} by mauto 2.
assert {{ Γ, ℕ, A ⊢s Wk∘Wk,,succ #1 : Γ, ℕ }} by mauto 2.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢ A[(Wk∘Wk,,succ #1)∘q (q (σ∘τ))] ≈ A[Wk∘Wk,,succ #1][q (q (σ∘τ))] : Type@i }} as -> by mauto 3.
assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢s q (q τ) : Δ, ℕ, A[q σ] }} by mauto 2.
assert {{ Δ', ℕ, A[q σ][q τ] ⊢ MS[q (q σ)][q (q τ)] ≈ MS[q (q σ)∘q (q τ)] : A[Wk∘Wk,,succ #1][q (q σ)∘q (q τ)] }} by mauto 3.
assert {{ Δ', ℕ, A[q σ∘q τ] ⊢s q (q σ)∘q (q τ) ≈ q (q σ∘q τ) : Γ, ℕ, A }} by mauto 2.
Expand Down Expand Up @@ -633,8 +610,7 @@ Proof.
invert_glu_rel_exp HA.
assert {{ Γ, ℕ, A ⊩ A[Wk∘Wk,,succ #1] : Type@i }}.
{
assert {{ ⊩ Γ, ℕ }} by mauto 2.
assert {{ ⊩ Γ, ℕ, A }} by mauto 2.
assert {{ ⊩ Γ, ℕ, A }} by mauto 3.
assert {{ Γ, ℕ ⊩s Wk : Γ }} by mauto 3.
assert {{ Γ, ℕ, A ⊩s Wk : Γ, ℕ }} by mauto 3.
assert {{ Γ, ℕ ⊢s Wk : Γ }} by mauto 3.
Expand Down
Loading