(* *)
(**************************************************************************)
-include "Basic-2/substitution/tps_lift.ma".
+include "Basic_2/substitution/tps_lift.ma".
(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
(* Main properties **********************************************************)
-(* Basic-1: was: subst1_confluence_eq *)
+(* 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 #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
+[ /2 width=3/
| #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/
+ [ #HX destruct /4 width=4/
+ | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
]
| #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/
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubs_conf … HT02 (L. 𝕓{I} V1) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L. 𝕓{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … 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/
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5/
]
qed.
-(* Basic-1: was: subst1_confluence_neq *)
+(* 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
+#L1 #T0 #T1 #d1 #e1 #H elim H -L1 -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #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
+ [ #H destruct /4 width=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
+ | -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
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02 ?) -T0
[ -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 *)
+ lapply (tps_lsubs_conf … HT1 (L2. 𝕓{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … HT2 (L1. 𝕓{I} V) ?) -HT2 /2 width=1/ /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/
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5/
]
qed.
(* Note: the constant 1 comes from tps_subst *)
-(* Basic-1: was: subst1_trans *)
+(* 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 #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/
+ [ #H destruct //
+ | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
]
| #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/
+ lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
+ <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
| #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/
+ elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (tps_lsubs_conf … HT02 (L. 𝕓{I} V0) ?) -HT02 /2 width=1/ #HT02
+ lapply (IHT10 … HT02 He) -T0 #HT12
+ lapply (tps_lsubs_conf … HT12 (L. 𝕓{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
| #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/
+ elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1/
]
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 #T1 #T0 #d1 #e1 #H elim H -L -T1 -T0 -d1 -e1
+[ /2 width=3/
| #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/
+ lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=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/
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (tps_lsubs_conf … HT02 (L. 𝕓{I} V0) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L. 𝕓{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … HT2 (L. 𝕓{I} V2) ?) -HT2 /2 width=1/ /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/
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02 ?) -V0 //
+ elim (IHT10 … HT02 ?) -T0 // /3 width=6/
]
qed.
projects / helm.git / blobdiff