Fix preservation and broken cases in logrel
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4 changed files with 33 additions and 57 deletions
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@ -30,12 +30,12 @@ Proof.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma Proj1_Inv n Γ (a : PTm n) U :
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Lemma Proj1_Inv Γ (a : PTm ) U :
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Γ ⊢ PProj PL a ∈ U ->
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exists A B, Γ ⊢ a ∈ PBind PSig A B /\ Γ ⊢ A ≲ U.
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Proof.
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move E : (PProj PL a) => u hu.
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move :a E. elim : n Γ u U / hu => n //=.
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move :a E. elim : Γ u U / hu => //=.
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- move => Γ a A B ha _ a0 [*]. subst.
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exists A, B. split => //=.
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eapply regularity in ha.
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@ -45,12 +45,12 @@ Proof.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma Proj2_Inv n Γ (a : PTm n) U :
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Lemma Proj2_Inv Γ (a : PTm) U :
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Γ ⊢ PProj PR a ∈ U ->
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exists A B, Γ ⊢ a ∈ PBind PSig A B /\ Γ ⊢ subst_PTm (scons (PProj PL a) VarPTm) B ≲ U.
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Proof.
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move E : (PProj PR a) => u hu.
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move :a E. elim : n Γ u U / hu => n //=.
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move :a E. elim : Γ u U / hu => //=.
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- move => Γ a A B ha _ a0 [*]. subst.
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exists A, B. split => //=.
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have ha' := ha.
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@ -62,30 +62,30 @@ Proof.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma Pair_Inv n Γ (a b : PTm n) U :
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Lemma Pair_Inv Γ (a b : PTm ) U :
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Γ ⊢ PPair a b ∈ U ->
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exists A B, Γ ⊢ a ∈ A /\
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Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B /\
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Γ ⊢ PBind PSig A B ≲ U.
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Proof.
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move E : (PPair a b) => u hu.
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move : a b E. elim : n Γ u U / hu => n //=.
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move : a b E. elim : Γ u U / hu => //=.
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- move => Γ a b A B i hS _ ha _ hb _ a0 b0 [*]. subst.
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exists A,B. repeat split => //=.
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move /E_Refl /Su_Eq : hS. apply.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma Ind_Inv n Γ P (a : PTm n) b c U :
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Lemma Ind_Inv Γ P (a : PTm) b c U :
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Γ ⊢ PInd P a b c ∈ U ->
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exists i, funcomp (ren_PTm shift) (scons PNat Γ) ⊢ P ∈ PUniv i /\
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exists i, (cons PNat Γ) ⊢ P ∈ PUniv i /\
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Γ ⊢ a ∈ PNat /\
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Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P /\
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funcomp (ren_PTm shift)(scons P (funcomp (ren_PTm shift) (scons PNat Γ))) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) /\
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(cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) /\
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Γ ⊢ subst_PTm (scons a VarPTm) P ≲ U.
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Proof.
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move E : (PInd P a b c)=> u hu.
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move : P a b c E. elim : n Γ u U / hu => n //=.
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move : P a b c E. elim : Γ u U / hu => //=.
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- move => Γ P a b c i hP _ ha _ hb _ hc _ P0 a0 b0 c0 [*]. subst.
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exists i. repeat split => //=.
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have : Γ ⊢ subst_PTm (scons a VarPTm) P ∈ subst_PTm (scons a VarPTm) (PUniv i) by hauto l:on use:substing_wt.
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@ -93,10 +93,10 @@ Proof.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma E_AppAbs : forall n (a : PTm (S n)) (b : PTm n) (Γ : fin n -> PTm n) (A : PTm n),
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Lemma E_AppAbs : forall (a : PTm) (b : PTm) Γ (A : PTm),
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Γ ⊢ PApp (PAbs a) b ∈ A -> Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ A.
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Proof.
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move => n a b Γ A ha.
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move => a b Γ A ha.
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move /App_Inv : ha.
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move => [A0][B0][ha][hb]hS.
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move /Abs_Inv : ha => [A1][B1][ha]hS0.
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@ -112,10 +112,10 @@ Proof.
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eauto using T_Conv.
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Qed.
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Lemma E_ProjPair1 : forall n (a b : PTm n) (Γ : fin n -> PTm n) (A : PTm n),
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Lemma E_ProjPair1 : forall (a b : PTm) Γ (A : PTm),
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Γ ⊢ PProj PL (PPair a b) ∈ A -> Γ ⊢ PProj PL (PPair a b) ≡ a ∈ A.
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Proof.
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move => n a b Γ A.
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move => a b Γ A.
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move /Proj1_Inv. move => [A0][B0][hab]hA0.
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move /Pair_Inv : hab => [A1][B1][ha][hb]hS.
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have [i ?] : exists i, Γ ⊢ PBind PSig A1 B1 ∈ PUniv i by sfirstorder use:regularity_sub0.
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@ -125,25 +125,25 @@ Proof.
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apply : E_ProjPair1; eauto.
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Qed.
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Lemma Suc_Inv n Γ (a : PTm n) A :
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Lemma Suc_Inv Γ (a : PTm) A :
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Γ ⊢ PSuc a ∈ A -> Γ ⊢ a ∈ PNat /\ Γ ⊢ PNat ≲ A.
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Proof.
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move E : (PSuc a) => u hu.
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move : a E.
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elim : n Γ u A /hu => //=.
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- move => n Γ a ha iha a0 [*]. subst.
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elim : Γ u A /hu => //=.
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- move => Γ a ha iha a0 [*]. subst.
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split => //=. eapply wff_mutual in ha.
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apply : Su_Eq.
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apply E_Refl. by apply T_Nat'.
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- hauto lq:on rew:off ctrs:LEq.
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Qed.
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Lemma RRed_Eq n Γ (a b : PTm n) A :
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Lemma RRed_Eq Γ (a b : PTm) A :
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Γ ⊢ a ∈ A ->
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RRed.R a b ->
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Γ ⊢ a ≡ b ∈ A.
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Proof.
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move => + h. move : Γ A. elim : n a b /h => n.
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move => + h. move : Γ A. elim : a b /h.
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- apply E_AppAbs.
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- move => p a b Γ A.
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case : p => //=.
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@ -214,7 +214,7 @@ Proof.
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- hauto lq:on use:Suc_Inv, E_Conv, E_SucCong.
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Qed.
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Theorem subject_reduction n Γ (a b A : PTm n) :
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Theorem subject_reduction Γ (a b A : PTm) :
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Γ ⊢ a ∈ A ->
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RRed.R a b ->
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Γ ⊢ b ∈ A.
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