include "basic_2/rt_transition/cpr.ma".
include "basic_2/rt_computation/cpms.ma".
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION FOR TERMS *************************)
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
(* Basic eliminators ********************************************************)
(* Basic_2A1: was: cprs_ind_dx *)
-lemma cprs_ind_sn (h) (G) (L) (T2) (R:predicate …):
- R T2 →
- (â\88\80T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h] T2 â\86\92 R T â\86\92 R T1) →
- â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h] T2 â\86\92 R T1.
-#h #G #L #T2 #R #IH1 #IH2 #T1
+lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
+ Q T2 →
+ (â\88\80T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h] T2 â\86\92 Q T â\86\92 Q T1) →
+ â\88\80T1. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h] T2 â\86\92 Q T1.
+#h #G #L #T2 #Q #IH1 #IH2 #T1
@(insert_eq_0 … 0) #n #H
@(cpms_ind_sn … H) -n -T1 //
#n1 #n2 #T1 #T #HT1 #HT2 #IH #H
qed-.
(* Basic_2A1: was: cprs_ind *)
-lemma cprs_ind_dx (h) (G) (L) (T1) (R:predicate …):
- R T1 →
- (â\88\80T,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[h] T2 â\86\92 R T â\86\92 R T2) →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h] T2 â\86\92 R T2.
-#h #G #L #T1 #R #IH1 #IH2 #T2
+lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
+ Q T1 →
+ (â\88\80T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h] T2 â\86\92 Q T â\86\92 Q T2) →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h] T2 â\86\92 Q T2.
+#h #G #L #T1 #Q #IH1 #IH2 #T2
@(insert_eq_0 … 0) #n #H
@(cpms_ind_dx … H) -n -T2 //
#n1 #n2 #T #T2 #HT1 #IH #HT2 #H
(* Basic_1: was: pr3_step *)
(* Basic_2A1: was: cprs_strap2 *)
lemma cprs_step_sn (h) (G) (L):
- â\88\80T1,T. â¦\83G, Lâ¦\84 ⊢ T1 ➡[h] T →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ T1 ➡*[h] T2.
+ â\88\80T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_sn/ qed-.
(* Basic_2A1: was: cprs_strap1 *)
lemma cprs_step_dx (h) (G) (L):
- â\88\80T1,T. â¦\83G, Lâ¦\84 ⊢ T1 ➡*[h] T →
- â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[h] T2 â\86\92 â¦\83G, Lâ¦\84 ⊢ T1 ➡*[h] T2.
+ â\88\80T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_dx/ qed-.
(* Basic_1: was only: pr3_thin_dx *)
lemma cprs_flat_dx (h) (I) (G) (L):
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80T1,T2. â¦\83G, Lâ¦\84 ⊢ T1 ➡*[h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡*[h] â\93\95{I}V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\95[I]V1.T1 â\9e¡*[h] â\93\95[I]V2.T2.
#h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
-(*
-lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
-#I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2
-/3 width=3 by cprs_strap1, cpr_cprs, cpr_pair_sn, cpr_flat/
-qed.
-
-lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⬆[0, 1] T2 ≘ T →
- ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h] T2.
-#G #L #V #T1 #T #T2 #HT2 #H @(cprs_ind_dx … H) -T1
-/3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/
-qed.
-lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h] T2.
-#G #L #T1 #T2 #H @(cprs_ind … H) -T2
-/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
-qed.
-
-lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h] ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2
-/4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/
-qed.
-
-lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆[0, 1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h] ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
-/4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/
+lemma cprs_flat_sn (h) (I) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
+ ❪G,L❫ ⊢ ⓕ[I] V1. T1 ➡*[h] ⓕ[I] V2. T2.
+#h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1
+/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] U2 → U2 = ⋆s.
-#G #L #U2 #s #H @(cprs_ind … H) -U2 //
-#U2 #U #_ #HU2 #IHU2 destruct
->(cpr_inv_sort1 … HU2) -HU2 //
-qed-.
+lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ❪G,L❫ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
+/2 width=4 by cpms_inv_sort1/ qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] U2 → ⦃G, L⦄ ⊢ T1 ➡*[h] U2 ∨
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & U2 = ⓝW2.T2.
-#G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_intror/
-#U2 #U #_ #HU2 * /3 width=3 by cprs_strap1, or_introl/ *
-#W #T #HW1 #HT1 #H destruct
-elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3 by cprs_strap1, or_introl/ *
-#W2 #T2 #HW2 #HT2 #H destruct /4 width=5 by cprs_strap1, ex3_2_intro, or_intror/
+lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ❪G,L❫ ⊢ ⓝW1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃W2,T2. ❪G,L❫ ⊢ W1 ➡*[h] W2 & ❪G,L❫ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
+ | ❪G,L❫ ⊢ T1 ➡*[h] X2.
+#h #G #L #W1 #T1 #X2 #H
+elim (cpms_inv_cast1 … H) -H
+[ /2 width=1 by or_introl/
+| /2 width=1 by or_intror/
+| * #m #_ #H destruct
+]
qed-.
-*)
+
(* Basic_1: removed theorems 13:
pr1_head_1 pr1_head_2 pr1_comp
clear_pr3_trans pr3_cflat pr3_gen_bind
projects / helm.git / blobdiff