Fix the definition of semleq
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Yiyun Liu
parent
0746e9a354
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f483d63f01
1 changed files with 64 additions and 19 deletions
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@ -418,7 +418,7 @@ Proof.
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qauto l:on.
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Qed.
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Lemma InterpUniv_Sub n i (A B : PTm n) PA PB :
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Lemma InterpUniv_Sub0 n i (A B : PTm n) PA PB :
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⟦ A ⟧ i ↘ PA ->
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⟦ B ⟧ i ↘ PB ->
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Sub.R A B -> forall x, PA x -> PB x.
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@ -427,6 +427,19 @@ Proof.
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move => [+ _]. apply.
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Qed.
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Lemma InterpUniv_Sub n i j (A B : PTm n) PA PB :
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⟦ A ⟧ i ↘ PA ->
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⟦ B ⟧ j ↘ PB ->
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Sub.R A B -> forall x, PA x -> PB x.
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Proof.
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have [? ?] : i <= max i j /\ j <= max i j by lia.
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move => hPA hPB.
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have : ⟦ B ⟧ (max i j) ↘ PB by eauto using InterpUniv_cumulative.
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have : ⟦ A ⟧ (max i j) ↘ PA by eauto using InterpUniv_cumulative.
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move : InterpUniv_Sub0. repeat move/[apply].
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apply.
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Qed.
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Lemma InterpUniv_Join n i (A B : PTm n) PA PB :
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⟦ A ⟧ i ↘ PA ->
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⟦ B ⟧ i ↘ PB ->
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@ -507,12 +520,30 @@ Notation "Γ ⊨ a ∈ A" := (SemWt Γ a A) (at level 70).
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Definition SemEq {n} Γ (a b A : PTm n) := DJoin.R a b /\ forall ρ, ρ_ok Γ ρ -> exists k PA, ⟦ subst_PTm ρ A ⟧ k ↘ PA /\ PA (subst_PTm ρ a) /\ PA (subst_PTm ρ b).
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Notation "Γ ⊨ a ≡ b ∈ A" := (SemEq Γ a b A) (at level 70).
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Definition SemLEq {n} Γ (a b A : PTm n) := Sub.R a b /\ forall ρ, ρ_ok Γ ρ -> exists k PA, ⟦ subst_PTm ρ A ⟧ k ↘ PA /\ PA (subst_PTm ρ a) /\ PA (subst_PTm ρ b).
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Notation "Γ ⊨ a ≲ b ∈ A" := (SemLEq Γ a b A) (at level 70).
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Definition SemLEq {n} Γ (A B : PTm n) := Sub.R A B /\ exists i, forall ρ, ρ_ok Γ ρ -> exists S0 S1, ⟦ subst_PTm ρ A ⟧ i ↘ S0 /\ ⟦ subst_PTm ρ B ⟧ i ↘ S1.
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Notation "Γ ⊨ a ≲ b" := (SemLEq Γ a b) (at level 70).
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Lemma SemWt_Univ n Γ (A : PTm n) i :
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Γ ⊨ A ∈ PUniv i <->
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forall ρ, ρ_ok Γ ρ -> exists S, ⟦ subst_PTm ρ A ⟧ i ↘ S.
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Proof.
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rewrite /SemWt.
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split.
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- hauto lq:on rew:off use:InterpUniv_Univ_inv.
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- move => /[swap] ρ /[apply].
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move => [PA hPA].
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exists (S i). eexists.
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split.
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+ simp InterpUniv. apply InterpExt_Univ. lia.
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+ simpl. eauto.
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Qed.
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Lemma SemEq_SemWt n Γ (a b A : PTm n) : Γ ⊨ a ≡ b ∈ A -> Γ ⊨ a ∈ A /\ Γ ⊨ b ∈ A /\ DJoin.R a b.
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Proof. hauto lq:on rew:off unfold:SemEq, SemWt. Qed.
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Lemma SemLEq_SemWt n Γ (A B : PTm n) : Γ ⊨ A ≲ B -> Sub.R A B /\ exists i, Γ ⊨ A ∈ PUniv i /\ Γ ⊨ B ∈ PUniv i.
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Proof. hauto q:on use:SemWt_Univ. Qed.
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Lemma SemWt_SemEq n Γ (a b A : PTm n) : Γ ⊨ a ∈ A -> Γ ⊨ b ∈ A -> (DJoin.R a b) -> Γ ⊨ a ≡ b ∈ A.
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Proof.
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move => ha hb heq. split => //= ρ hρ.
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@ -524,7 +555,22 @@ Proof.
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hauto lq:on.
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Qed.
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Lemma SemEq_SemLEq n Γ
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Lemma SemWt_SemLEq n Γ (A B : PTm n) i j :
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Γ ⊨ A ∈ PUniv i -> Γ ⊨ B ∈ PUniv j -> Sub.R A B -> Γ ⊨ A ≲ B.
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Proof.
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move => ha hb heq. split => //.
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exists (Nat.max i j).
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have [? ?] : i <= Nat.max i j /\ j <= Nat.max i j by lia.
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move => ρ hρ.
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have {}/ha := hρ.
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have {}/hb := hρ.
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move => [k][PA][/= /InterpUniv_Univ_inv [? hPA]]hpb.
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move => [k0][PA0][/= /InterpUniv_Univ_inv [? hPA0]]hpa. subst.
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move : hpb => [PA]hPA'.
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move : hpa => [PB]hPB'.
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exists PB, PA.
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split; apply : InterpUniv_cumulative; eauto.
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Qed.
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(* Semantic context wellformedness *)
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Definition SemWff {n} Γ := forall (i : fin n), exists j, Γ ⊨ Γ i ∈ PUniv j.
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@ -614,21 +660,6 @@ Lemma SemEq_SN_Join n Γ (a b A : PTm n) :
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SN a /\ SN b /\ SN A /\ DJoin.R a b.
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Proof. hauto l:on use:SemEq_SemWt, SemWt_SN. Qed.
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Lemma SemWt_Univ n Γ (A : PTm n) i :
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Γ ⊨ A ∈ PUniv i <->
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forall ρ, ρ_ok Γ ρ -> exists S, ⟦ subst_PTm ρ A ⟧ i ↘ S.
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Proof.
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rewrite /SemWt.
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split.
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- hauto lq:on rew:off use:InterpUniv_Univ_inv.
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- move => /[swap] ρ /[apply].
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move => [PA hPA].
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exists (S i). eexists.
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split.
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+ simp InterpUniv. apply InterpExt_Univ. lia.
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+ simpl. eauto.
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Qed.
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(* Structural laws for Semantic context wellformedness *)
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Lemma SemWff_nil : SemWff null.
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Proof. case. Qed.
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@ -861,6 +892,20 @@ Proof.
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hauto l:on use:DJoin.transitive.
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Qed.
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Lemma SLEq_Transitive n Γ (A B C : PTm n) :
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Γ ⊨ A ≲ B ->
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Γ ⊨ B ≲ C ->
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Γ ⊨ A ≲ C.
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Proof.
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move => h0 h1.
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rewrite /SemLEq in h0 h1.
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move : h0 => [hAB]h0.
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move : h1 => [hBC]h1.
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rewrite /SemLEq.
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split. eauto using Sub.transitive.
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Lemma Γ_eq_ρ_ok n Γ Δ (ρ : fin n -> PTm 0) : Γ_eq Γ Δ -> ⊨ Γ -> ρ_ok Γ ρ -> ρ_ok Δ ρ.
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Proof.
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move => hΓΔ hΓ h.
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