+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic-2/substitution/tps_lift.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic-1: was: subst1_confluence_eq *)
-theorem tps_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫ T1 →
- ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 →
- ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T2 [d1, e1] ≫ T.
-#L #T0 #T1 #d1 #e1 #H elim H -H L T0 T1 d1 e1
-[ /2/
-| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (tps_inv_lref1 … H) -H
- [ #HX destruct -T2 /4/
- | -Hd1 Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (drop_mono … HLK1 … HLK2) -HLK1 HLK2 #H destruct -V1 K1
- >(lift_mono … HVT1 … HVT2) -HVT1 HVT2 /2/
- ]
-| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V1) ?) -HT02 /2/ #HT02
- elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
- lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /2/
- lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V) ?) -HT2 /3 width=5/
-| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV01 … HV02) -IHV01 HV02;
- elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
-]
-qed.
-
-(* Basic-1: was: subst1_confluence_neq *)
-theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫ T1 →
- ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 →
- (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫ T.
-#L1 #T0 #T1 #d1 #e1 #H elim H -H L1 T0 T1 d1 e1
-[ /2/
-| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
- elim (tps_inv_lref1 … H1) -H1
- [ #H destruct -T2 /4/
- | -HLK1 HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
- [ -Hd1 Hde2;
- lapply (transitive_le … Hded Hd2) -Hded Hd2 #H
- lapply (lt_to_le_to_lt … Hde1 H) -Hde1 H #H
- elim (lt_refl_false … H)
- | -Hd2 Hde1;
- lapply (transitive_le … Hded Hd1) -Hded Hd1 #H
- lapply (lt_to_le_to_lt … Hde2 H) -Hde2 H #H
- elim (lt_refl_false … H)
- ]
- ]
-| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV01 … HV02 H) -IHV01 HV02 #V #HV1 #HV2
- elim (IHT01 … HT02 ?) -IHT01 HT02
- [ -H #T #HT1 #HT2
- lapply (tps_leq_repl_dx … HT1 (L2. 𝕓{I} V) ?) -HT1 /2/
- lapply (tps_leq_repl_dx … HT2 (L1. 𝕓{I} V) ?) -HT2 /3 width=5/
- | -HV1 HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %) elim H -H #H
- [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3/ is too slow *)
- ]
-| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV01 … HV02 H) -IHV01 HV02;
- elim (IHT01 … HT02 H) -IHT01 HT02 H /3 width=5/
-]
-qed.
-
-(* Note: the constant 1 comes from tps_subst *)
-(* Basic-1: was: subst1_trans *)
-theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 [d, e] ≫ T0 →
- ∀T2. L ⊢ T0 [d, 1] ≫ T2 → 1 ≤ e →
- L ⊢ T1 [d, e] ≫ T2.
-#L #T1 #T0 #d #e #H elim H -L T1 T0 d e
-[ #L #I #d #e #T2 #H #He
- elim (tps_inv_atom1 … H) -H
- [ #H destruct -T2 //
- | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct -I;
- lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2/
- ]
-| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2/ #HVT2
- <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2/
-| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct -X;
- lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V0) ?) -HT02 /2/ #HT02
- lapply (IHT10 … HT02 He) -IHT10 HT02 #HT12
- lapply (tps_leq_repl_dx … HT12 (L. 𝕓{I} V2) ?) -HT12 /3/
-| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct -X /3/
-]
-qed.
-
-theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
- ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫ T2.
-#L #T1 #T0 #d1 #e1 #H elim H -L T1 T0 d1 e1
-[ /2/
-| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
- lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2; normalize /2/ -Hde2i1 #HWT2
- <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4/
-| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V0) ?) -HT02 /2/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 HV02 // #V
- elim (IHT10 … HT02 ?) -IHT10 HT02 [2: /2/ ] #T #HT1 #HT2
- lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1;
- lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V2) ?) -HT2 /3 width=6/
-| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV10 … HV02 ?) -IHV10 HV02 //
- elim (IHT10 … HT02 ?) -IHT10 HT02 // /3 width=6/
-]
-qed.
projects / helm.git / blobdiff