Refactor the impossible case proof
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3 changed files with 294 additions and 178 deletions
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@ -5,6 +5,20 @@ Require Import Setoid Morphisms Relation_Definitions.
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Module Core.
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Inductive BTag : Type :=
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| PPi : BTag
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| PSig : BTag.
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Lemma congr_PPi : PPi = PPi.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma congr_PSig : PSig = PSig.
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Proof.
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exact (eq_refl).
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Qed.
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Inductive PTag : Type :=
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| PL : PTag
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| PR : PTag.
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@ -24,7 +38,9 @@ Inductive PTm (n_PTm : nat) : Type :=
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| PAbs : PTm (S n_PTm) -> PTm n_PTm
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| PApp : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
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| PPair : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
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| PProj : PTag -> PTm n_PTm -> PTm n_PTm.
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| PProj : PTag -> PTm n_PTm -> PTm n_PTm
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| PBind : BTag -> PTm n_PTm -> PTm (S n_PTm) -> PTm n_PTm
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| PUniv : nat -> PTm n_PTm.
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Lemma congr_PAbs {m_PTm : nat} {s0 : PTm (S m_PTm)} {t0 : PTm (S m_PTm)}
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(H0 : s0 = t0) : PAbs m_PTm s0 = PAbs m_PTm t0.
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@ -56,6 +72,23 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj m_PTm x s1) H0))
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(ap (fun x => PProj m_PTm t0 x) H1)).
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Qed.
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Lemma congr_PBind {m_PTm : nat} {s0 : BTag} {s1 : PTm m_PTm}
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{s2 : PTm (S m_PTm)} {t0 : BTag} {t1 : PTm m_PTm} {t2 : PTm (S m_PTm)}
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(H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) :
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PBind m_PTm s0 s1 s2 = PBind m_PTm t0 t1 t2.
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Proof.
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exact (eq_trans
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(eq_trans (eq_trans eq_refl (ap (fun x => PBind m_PTm x s1 s2) H0))
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(ap (fun x => PBind m_PTm t0 x s2) H1))
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(ap (fun x => PBind m_PTm t0 t1 x) H2)).
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Qed.
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Lemma congr_PUniv {m_PTm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
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PUniv m_PTm s0 = PUniv m_PTm t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => PUniv m_PTm x) H0)).
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Qed.
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Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
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fin (S m) -> fin (S n).
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Proof.
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@ -76,6 +109,9 @@ Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
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| PApp _ s0 s1 => PApp n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PPair _ s0 s1 => PPair n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
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| PProj _ s0 s1 => PProj n_PTm s0 (ren_PTm xi_PTm s1)
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| PBind _ s0 s1 s2 =>
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PBind n_PTm s0 (ren_PTm xi_PTm s1) (ren_PTm (upRen_PTm_PTm xi_PTm) s2)
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| PUniv _ s0 => PUniv n_PTm s0
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end.
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Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
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@ -102,6 +138,10 @@ Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
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| PPair _ s0 s1 =>
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PPair n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
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| PProj _ s0 s1 => PProj n_PTm s0 (subst_PTm sigma_PTm s1)
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| PBind _ s0 s1 s2 =>
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PBind n_PTm s0 (subst_PTm sigma_PTm s1)
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(subst_PTm (up_PTm_PTm sigma_PTm) s2)
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| PUniv _ s0 => PUniv n_PTm s0
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end.
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Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
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@ -140,6 +180,10 @@ Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
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(idSubst_PTm sigma_PTm Eq_PTm s1)
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
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(idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
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@ -180,6 +224,11 @@ ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
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(extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
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(extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
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(upExtRen_PTm_PTm _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
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@ -221,6 +270,11 @@ subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
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(ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
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(ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
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(upExt_PTm_PTm _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
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@ -262,6 +316,12 @@ Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0)
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(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0)
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(compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
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(compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
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(upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
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@ -312,6 +372,12 @@ Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0)
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(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0)
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(compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
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(compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
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(up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
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@ -382,6 +448,12 @@ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0)
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(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0)
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(compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
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(compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
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(up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
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@ -454,6 +526,12 @@ subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0)
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(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0)
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(compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
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(compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
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(up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
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@ -566,6 +644,11 @@ Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
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(rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
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| PProj _ s0 s1 =>
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congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
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| PBind _ s0 s1 s2 =>
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congr_PBind (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
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(rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
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(rinstInst_up_PTm_PTm _ _ Eq_PTm) s2)
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| PUniv _ s0 => congr_PUniv (eq_refl s0)
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end.
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Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
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@ -637,30 +720,6 @@ Proof.
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exact (fun x => eq_refl).
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Qed.
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Inductive Ty : Type :=
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| Fun : Ty -> Ty -> Ty
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| Prod : Ty -> Ty -> Ty
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| Void : Ty.
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Lemma congr_Fun {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
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(H1 : s1 = t1) : Fun s0 s1 = Fun t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => Fun x s1) H0))
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(ap (fun x => Fun t0 x) H1)).
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Qed.
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Lemma congr_Prod {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
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(H1 : s1 = t1) : Prod s0 s1 = Prod t0 t1.
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Proof.
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exact (eq_trans (eq_trans eq_refl (ap (fun x => Prod x s1) H0))
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(ap (fun x => Prod t0 x) H1)).
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Qed.
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Lemma congr_Void : Void = Void.
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Proof.
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exact (eq_refl).
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Qed.
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Class Up_PTm X Y :=
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up_PTm : X -> Y.
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@ -796,6 +855,10 @@ Core.
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Arguments VarPTm {n_PTm}.
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Arguments PUniv {n_PTm}.
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Arguments PBind {n_PTm}.
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Arguments PProj {n_PTm}.
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Arguments PPair {n_PTm}.
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@ -804,9 +867,9 @@ Arguments PApp {n_PTm}.
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Arguments PAbs {n_PTm}.
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#[global] Hint Opaque subst_PTm: rewrite.
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#[global]Hint Opaque subst_PTm: rewrite.
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#[global] Hint Opaque ren_PTm: rewrite.
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#[global]Hint Opaque ren_PTm: rewrite.
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End Extra.
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