Simplify the syntax by combining proj1 and proj2
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3 changed files with 109 additions and 145 deletions
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@ -5,13 +5,26 @@ Require Import Setoid Morphisms Relation_Definitions.
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Module Core.
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Inductive PTag : Type :=
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| PL : PTag
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| PR : PTag.
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Lemma congr_PL : PL = PL.
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Proof.
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exact (eq_refl).
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Qed.
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Lemma congr_PR : PR = PR.
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Proof.
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exact (eq_refl).
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Qed.
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Inductive Tm (n_Tm : nat) : Type :=
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| VarTm : fin n_Tm -> Tm n_Tm
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| Abs : Tm (S n_Tm) -> Tm n_Tm
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| App : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
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| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
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| Proj1 : Tm n_Tm -> Tm n_Tm
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| Proj2 : Tm n_Tm -> Tm n_Tm.
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| Proj : PTag -> Tm n_Tm -> Tm n_Tm.
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Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
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(H0 : s0 = t0) : Abs m_Tm s0 = Abs m_Tm t0.
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@ -35,16 +48,12 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair m_Tm x s1) H0))
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(ap (fun x => Pair m_Tm t0 x) H1)).
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Qed.
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Lemma congr_Proj1 {m_Tm : nat} {s0 : Tm m_Tm} {t0 : Tm m_Tm} (H0 : s0 = t0) :
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Proj1 m_Tm s0 = Proj1 m_Tm t0.
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Lemma congr_Proj {m_Tm : nat} {s0 : PTag} {s1 : Tm m_Tm} {t0 : PTag}
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{t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
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Proj m_Tm s0 s1 = Proj m_Tm t0 t1.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => Proj1 m_Tm x) H0)).
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Qed.
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Lemma congr_Proj2 {m_Tm : nat} {s0 : Tm m_Tm} {t0 : Tm m_Tm} (H0 : s0 = t0) :
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Proj2 m_Tm s0 = Proj2 m_Tm t0.
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Proof.
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exact (eq_trans eq_refl (ap (fun x => Proj2 m_Tm x) H0)).
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exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj m_Tm x s1) H0))
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(ap (fun x => Proj m_Tm t0 x) H1)).
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Qed.
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Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
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@ -66,8 +75,7 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
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| Abs _ s0 => Abs n_Tm (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
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| App _ s0 s1 => App n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
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| Pair _ s0 s1 => Pair n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
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| Proj1 _ s0 => Proj1 n_Tm (ren_Tm xi_Tm s0)
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| Proj2 _ s0 => Proj2 n_Tm (ren_Tm xi_Tm s0)
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| Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
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end.
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Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
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@ -90,8 +98,7 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
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| Abs _ s0 => Abs n_Tm (subst_Tm (up_Tm_Tm sigma_Tm) s0)
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| App _ s0 s1 => App n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
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| Pair _ s0 s1 => Pair n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
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| Proj1 _ s0 => Proj1 n_Tm (subst_Tm sigma_Tm s0)
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| Proj2 _ s0 => Proj2 n_Tm (subst_Tm sigma_Tm s0)
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| Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
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end.
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Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
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@ -126,8 +133,7 @@ subst_Tm sigma_Tm s = s :=
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| Pair _ s0 s1 =>
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congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
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(idSubst_Tm sigma_Tm Eq_Tm s1)
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| Proj1 _ s0 => congr_Proj1 (idSubst_Tm sigma_Tm Eq_Tm s0)
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| Proj2 _ s0 => congr_Proj2 (idSubst_Tm sigma_Tm Eq_Tm s0)
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| Proj _ s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
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end.
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Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
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@ -164,8 +170,8 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
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| Pair _ s0 s1 =>
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congr_Pair (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
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(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
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| Proj1 _ s0 => congr_Proj1 (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
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| Proj2 _ s0 => congr_Proj2 (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
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end.
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Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
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@ -204,8 +210,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
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| Pair _ s0 s1 =>
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congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
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(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
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| Proj1 _ s0 => congr_Proj1 (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
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| Proj2 _ s0 => congr_Proj2 (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
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| Proj _ s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
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end.
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Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
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@ -243,8 +248,8 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
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| Pair _ s0 s1 =>
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congr_Pair (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
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(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
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| Proj1 _ s0 => congr_Proj1 (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
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| Proj2 _ s0 => congr_Proj2 (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
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end.
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Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
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@ -291,10 +296,9 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
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| Pair _ s0 s1 =>
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congr_Pair (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
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(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
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| Proj1 _ s0 =>
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congr_Proj1 (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
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| Proj2 _ s0 =>
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congr_Proj2 (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0)
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(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
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@ -362,10 +366,9 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
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| Pair _ s0 s1 =>
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congr_Pair (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
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(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
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| Proj1 _ s0 =>
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congr_Proj1 (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
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| Proj2 _ s0 =>
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congr_Proj2 (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0)
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(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
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@ -434,10 +437,9 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
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| Pair _ s0 s1 =>
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congr_Pair (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
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(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
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| Proj1 _ s0 =>
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congr_Proj1 (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
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| Proj2 _ s0 =>
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congr_Proj2 (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0)
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(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
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end.
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Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
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@ -546,8 +548,8 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
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| Pair _ s0 s1 =>
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congr_Pair (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
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(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
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| Proj1 _ s0 => congr_Proj1 (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
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| Proj2 _ s0 => congr_Proj2 (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
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| Proj _ s0 s1 =>
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congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
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end.
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Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
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@ -746,9 +748,7 @@ Core.
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Arguments VarTm {n_Tm}.
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Arguments Proj2 {n_Tm}.
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Arguments Proj1 {n_Tm}.
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Arguments Proj {n_Tm}.
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Arguments Pair {n_Tm}.
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