(* *)
(**************************************************************************)
-include "basic_2/reduction/lpr_lpr.ma".
-include "basic_2/computation/cprs_lift.ma".
+include "ground_2/xoa/ex_4_5.ma".
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/rt_computation/cprs_cpr.ma".
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
(* Main properties **********************************************************)
(* Basic_1: was: pr3_t *)
(* Basic_1: includes: pr1_t *)
-theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
-normalize /2 width=3 by trans_TC/ qed-.
+theorem cprs_trans (h) (G) (L): Transitive … (cpms h G L 0).
+/2 width=3 by cpms_trans/ qed-.
(* Basic_1: was: pr3_confluence *)
(* Basic_1: includes: pr1_confluence *)
-theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
-normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
-
-theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=5 by cprs_trans, cprs_bind_dx/
-qed.
+theorem cprs_conf (h) (G) (L): confluent2 … (cpms h G L 0) (cpms h G L 0).
+#h #G #L #T0 #T1 #HT01 #T2 #HT02
+elim (TC_confluent2 … T0 T1 … T2)
+[ /3 width=3 by cprs_CTC, ex2_intro/ |5,6: skip
+| /2 width=1 by cprs_inv_CTC/
+| /2 width=1 by cprs_inv_CTC/
+| /2 width=3 by cpr_conf/
+]
+qed-.
(* Basic_1: was: pr3_flat *)
-theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
-#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
-qed.
-
-theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
-#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
-@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
-qed.
-
-theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
-qed.
-
-theorem cprs_theta_rc: ∀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 #HT12 #H @(cprs_ind … H) -W2
-/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
-qed.
-
-theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⬆[0, 1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1
-/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
+theorem cprs_flat (h) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 →
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
+ ∀I. ❪G,L❫ ⊢ ⓕ[I]V1.T1 ➡*[h] ⓕ[I]V2.T2.
+#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
+[ /2 width=3 by cprs_flat_dx/
+| /3 width=3 by cpr_pair_sn, cprs_step_dx/
+]
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
- U2 = ⓐV2. T2
- | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
- ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
- | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⬆[0,1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
- ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
-#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
-#U #U2 #_ #HU2 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
- | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
- lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
- lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
- /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_beta, ex2_3_intro, or3_intro1/
- | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- /5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/
- ]
-| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
-| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
+(* Basic_2A1: was: cprs_inv_appl1 *)
+lemma cprs_inv_appl_sn (h) (G) (L):
+ ∀V1,T1,X2. ❪G,L❫ ⊢ ⓐV1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡*[h] V2 &
+ ❪G,L❫ ⊢ T1 ➡*[h] T2 &
+ X2 = ⓐV2. T2
+ | ∃∃p,W,T. ❪G,L❫ ⊢ T1 ➡*[h] ⓛ[p]W.T &
+ ❪G,L❫ ⊢ ⓓ[p]ⓝW.V1.T ➡*[h] X2
+ | ∃∃p,V0,V2,V,T. ❪G,L❫ ⊢ V1 ➡*[h] V0 & ⇧*[1] V0 ≘ V2 &
+ ❪G,L❫ ⊢ T1 ➡*[h] ⓓ[p]V.T &
+ ❪G,L❫ ⊢ ⓓ[p]V.ⓐV2.T ➡*[h] X2.
+#h #G #L #V1 #T1 #X2 #H elim (cpms_inv_appl_sn … H) -H *
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+| #n1 #n2 #p #V2 #T2 #HT12 #HTX2 #H
+ elim (plus_inv_O3 … H) -H #H1 #H2 destruct
+ /3 width=5 by or3_intro1, ex2_3_intro/
+| #n1 #n2 #p #V2 #W2 #V #T2 #HV12 #HVW2 #HT12 #HTX2 #H
+ elim (plus_inv_O3 … H) -H #H1 #H2 destruct
+ /3 width=9 by or3_intro2, ex4_5_intro/
]
qed-.
-
-(* Basic_1: was: pr3_strip *)
-(* Basic_1: includes: pr1_strip *)
-lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
-normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
-
-lemma cprs_lpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/
-#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01)
-#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0
-#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T
-/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
-qed-.
-
-lemma cprs_lpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
-/3 width=3 by lpr_cprs_trans, ex2_intro/
-qed-.
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