(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predstar_4.ma".
-include "basic_2/reduction/cnr.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_computation/cpms.ma".
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Basic_1: includes: pr1_pr0 *)
-definition cprs: relation4 genv lenv term term ≝
- λG. LTC … (cpr G).
-
-interpretation "context-sensitive parallel computation (term)"
- 'PRedStar G L T1 T2 = (cprs G L T1 T2).
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
(* Basic eliminators ********************************************************)
-lemma cprs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2.
-#G #L #T1 #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind … HT1 IHT1 … HT12) //
+(* Basic_2A1: was: cprs_ind_dx *)
+lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
+ Q T2 →
+ (∀T1,T. ❪G,L❫ ⊢ T1 ➡[h] T → ❪G,L❫ ⊢ T ➡*[h] T2 → Q T → Q T1) →
+ ∀T1. ❪G,L❫ ⊢ T1 ➡*[h] T2 → 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
+elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct
+/3 width=4 by/
qed-.
-lemma cprs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1.
-#G #L #T2 #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+(* Basic_2A1: was: cprs_ind *)
+lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
+ Q T1 →
+ (∀T,T2. ❪G,L❫ ⊢ T1 ➡*[h] T → ❪G,L❫ ⊢ T ➡[h] T2 → Q T → Q T2) →
+ ∀T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 → 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
+elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct
+/3 width=4 by/
qed-.
(* Basic properties *********************************************************)
-(* Basic_1: was: pr3_pr2 *)
-lemma cpr_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
-/2 width=1 by inj/ qed.
-
-(* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀G,L,T. ⦃G, L⦄ ⊢ T ➡* T.
-/2 width=1 by cpr_cprs/ qed.
-
-lemma cprs_strap1: ∀G,L,T1,T,T2.
- ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
-normalize /2 width=3 by step/ qed-.
-
(* Basic_1: was: pr3_step *)
-lemma cprs_strap2: ∀G,L,T1,T,T2.
- ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr.
-/3 width=5 by lsubr_cpr_trans, LTC_lsub_trans/
-qed-.
-
-(* Basic_1: was: pr3_pr1 *)
-lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
-/2 width=3 by lsubr_cprs_trans/ qed.
-
-lemma cprs_bind_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
-#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
-/3 width=3 by cprs_strap2, cpr_cprs, cpr_pair_sn, cpr_bind/ qed.
+(* Basic_2A1: was: cprs_strap2 *)
+lemma cprs_step_sn (h) (G) (L):
+ ∀T1,T. ❪G,L❫ ⊢ T1 ➡[h] T →
+ ∀T2. ❪G,L❫ ⊢ T ➡*[h] T2 → ❪G,L❫ ⊢ T1 ➡*[h] T2.
+/2 width=3 by cpms_step_sn/ qed-.
+
+(* Basic_2A1: was: cprs_strap1 *)
+lemma cprs_step_dx (h) (G) (L):
+ ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h] T →
+ ∀T2. ❪G,L❫ ⊢ T ➡[h] T2 → ❪G,L❫ ⊢ T1 ➡*[h] T2.
+/2 width=3 by cpms_step_dx/ qed-.
(* Basic_1: was only: pr3_thin_dx *)
-lemma cprs_flat_dx: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2
-/3 width=5 by cprs_strap1, cpr_flat, cpr_cprs, cpr_pair_sn/
-qed.
-
-lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{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 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* 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 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
-#G #L #T1 #T2 #H @(cprs_ind … H) -T2
-/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
+lemma cprs_flat_dx (h) (I) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 →
+ ❪G,L❫ ⊢ ⓕ[I]V1.T1 ➡*[h] ⓕ[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_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{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 ➡ V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{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 ➡* 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 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* 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/
-qed-.
-
-(* Basic_1: was: nf2_pr3_unfold *)
-lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U.
-#G #L #T #U #H @(cprs_ind_dx … H) -T //
-#T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
+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:
projects / helm.git / blobdiff