- the support for global references is started ...
- some refactoring.
inductive item0: Type[0] ≝
| Sort: nat → item0 (* sort: starting at 0 *)
| LRef: nat → item0 (* reference by index: starting at 0 *)
+ | GRef: nat → item0 (* reference by position: starting at 0 *)
.
(* binary binding items *)
interpretation "local reference (term)" 'LRef i = (TAtom (LRef i)).
+interpretation "global reference (term)" 'GRef p = (TAtom (GRef p)).
+
interpretation "term construction (atomic)" 'SItem I = (TAtom I).
interpretation "term construction (binary)" 'SItem I T1 T2 = (TPair I T1 T2).
h : sort hierarchy parameter
i,j : local reference position index (de Bruijn's)
k : sort index
+p,q : global reference position
+
non associative with precedence 90
for @{ 'Star $k }.
-notation "hvbox( # term 90 k )"
+notation "hvbox( # term 90 i )"
non associative with precedence 90
- for @{ 'LRef $k }.
+ for @{ 'LRef $i }.
+
+notation "hvbox( § term 90 p )"
+ non associative with precedence 90
+ for @{ 'GRef $p }.
notation "hvbox( 𝕒 { I } )"
non associative with precedence 90
(* Advanced properties ******************************************************)
-lemma cpr_delta: ∀L,K,V1,W1,W2,i.
- ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫* W1 →
- ↑[0, i + 1] W1 ≡ W2 → L ⊢ #i ⇒ W2.
+lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
+ ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫* W1 →
+ ↑[0, i + 1] W1 ≡ W2 → L ⊢ #i ⇒ W2.
#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
@ex2_1_intro [2: // | skip | @tpss_subst /2 width=6/ ] (**) (* /4 width=6/ is too slow *)
qed.
(* Relocation properties ****************************************************)
(* Basic_1: was: pr2_lift *)
+lemma cpr_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
+ ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀T2,U2. ↑[d, e] T2 ≡ U2 →
+ K ⊢ T1 ⇒ T2 → L ⊢ U1 ⇒ U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
+elim (lt_or_ge (|K|) d) #HKd
+[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 T HLK [ /2/ | /3/ ]
+| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 T HLK // /3/
+]
+qed.
(* Basic_1: was: pr2_gen_lift *)
-
+lemma cpr_inv_lift: ∀L,K,d,e. ↓[d, e] L ≡ K →
+ ∀T1,U1. ↑[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ⇒ U2 →
+ ∃∃T2. ↑[d, e] T2 ≡ U2 & K ⊢ T1 ⇒ T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
+elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
+elim (lt_or_ge (|L|) d) #HLd
+[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U HLK [ /5/ | /2/ ]
+| elim (lt_or_ge (|L|) (d + e)) #HLde
+ [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U HLK // [ /5/ | /2/ ]
+ | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U HLK // /5/
+ ]
+]
+qed.
+++ /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/reduction/tpr_lift.ma".
-include "Basic_2/reduction/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Basic_1: was: wcpr0_drop *)
-lemma ltpr_drop_conf: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀L2. L1 ⇒ L2 →
- ∃∃K2. ↓[d, e] L2 ≡ K2 & K1 ⇒ K2.
-#L1 #K1 #d #e #H elim H -H L1 K1 d e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
-| #K1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
- elim (IHLK1 … HL12) -IHLK1 HL12 /3/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
- elim (tpr_inv_lift … HV12 … HWV1) -HV12 HWV1;
- elim (IHLK1 … HL12) -IHLK1 HL12 /3 width=5/
-]
-qed.
-
-(* Basic_1: was: wcpr0_drop_back *)
-lemma ltpr_drop_trans: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀K2. K1 ⇒ K2 →
- ∃∃L2. ↓[d, e] L2 ≡ K2 & L1 ⇒ L2.
-#L1 #K1 #d #e #H elim H -H L1 K1 d e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
-| #K1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
- elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct -X;
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (tpr_lift … HW12 … HWV1 … HWV2) -HW12 HWV1;
- elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
-]
-qed.
--- /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/reduction/tpr_lift.ma".
+include "Basic_2/reduction/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Basic_1: was: wcpr0_ldrop *)
+lemma ltpr_ldrop_conf: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀L2. L1 ⇒ L2 →
+ ∃∃K2. ↓[d, e] L2 ≡ K2 & K1 ⇒ K2.
+#L1 #K1 #d #e #H elim H -H L1 K1 d e
+[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
+| #K1 #I #V1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
+ elim (IHLK1 … HL12) -IHLK1 HL12 /3/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
+ elim (tpr_inv_lift … HV12 … HWV1) -HV12 HWV1;
+ elim (IHLK1 … HL12) -IHLK1 HL12 /3 width=5/
+]
+qed.
+
+(* Basic_1: was: wcpr0_ldrop_back *)
+lemma ltpr_ldrop_trans: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀K2. K1 ⇒ K2 →
+ ∃∃L2. ↓[d, e] L2 ≡ K2 & L1 ⇒ L2.
+#L1 #K1 #d #e #H elim H -H L1 K1 d e
+[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
+| #K1 #I #V1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
+ elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct -X;
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (tpr_lift … HW12 … HWV1 … HWV2) -HW12 HWV1;
+ elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
+]
+qed.
lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1
[ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct -U2 //
| lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct -U2 //
+ | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct -U2 //
]
| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
[ * #i #d #e #U1 #HU1
[ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/
| lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/
+ | lapply (lift_inv_gref2 … HU1) -HU1 #H destruct -U1 /2/
]
| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
(**************************************************************************)
include "Basic_2/unfold/ltpss_ltpss.ma".
-include "Basic_2/reduction/ltpr_drop.ma".
+include "Basic_2/reduction/ltpr_ldrop.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
elim (tps_inv_atom1 … H) -H
[ #H destruct -X /2/
| * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct -I;
- elim (ltpr_drop_conf … HLK1 … HL12) -HLK1 HL12 #X #HLK2 #H
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 HL12 #X #HLK2 #H
elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct -X;
elim (lift_total V2 0 (i+1)) #U2 #HVU2
lapply (tpr_lift … HV12 … HVU1 … HVU2) -HV12 HVU1 #HU12
+++ /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/grammar/lenv_weight.ma".
-include "Basic_2/grammar/lsubs.ma".
-include "Basic_2/substitution/lift.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Basic_1: includes: drop_skip_bind *)
-inductive drop: nat → nat → relation lenv ≝
-| drop_atom: ∀d,e. drop d e (⋆) (⋆)
-| drop_pair: ∀L,I,V. drop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
-| drop_drop: ∀L1,L2,I,V,e. drop 0 e L1 L2 → drop 0 (e + 1) (L1. 𝕓{I} V) L2
-| drop_skip: ∀L1,L2,I,V1,V2,d,e.
- drop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
- drop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
-.
-
-interpretation "dropping" 'RDrop d e L1 L2 = (drop d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
-#d #e #L1 #L2 * -d e L1 L2
-[ //
-| //
-| #L1 #L2 #I #V #e #_ #_ #H
- elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
- elim (plus_S_eq_O_false … H)
-]
-qed.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed.
-
-fact drop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
-#d #e #L1 #L2 * -d e L1 L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V #e #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: drop_gen_sort *)
-lemma drop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed.
-
-fact drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. 𝕓{I} V →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
- (0 < e ∧ ↓[d, e - 1] K ≡ L2).
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #K #I #V #H destruct
-| #L #I #V #_ #K #J #W #HX destruct -L I V /3/
-| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
-]
-qed.
-
-lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
- (0 < e ∧ ↓[0, e - 1] K ≡ L2).
-/2/ qed.
-
-(* Basic_1: was: drop_gen_drop *)
-lemma drop_inv_drop1: ∀e,K,I,V,L2.
- ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
-#e #K #I #V #L2 #H #He
-elim (drop_inv_O1 … H) -H * // #H destruct -e;
-elim (lt_refl_false … He)
-qed.
-
-fact drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
- ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #I #K #V #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
- /2 width=5/
-]
-qed.
-
-(* Basic_1: was: drop_gen_skip_l *)
-lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
-/2/ qed.
-
-fact drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
- ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #I #K #V #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
- /2 width=5/
-]
-qed.
-
-(* Basic_1: was: drop_gen_skip_r *)
-lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
- ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
-/2/ qed.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was by definition: drop_refl *)
-lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
-#L elim L -L //
-qed.
-
-lemma drop_drop_lt: ∀L1,L2,I,V,e.
- ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
-#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
-qed.
-
-lemma drop_lsubs_drop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V →
- d ≤ i → i < d + e →
- ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
- ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e
-[ #d #e #K1 #V #i #H
- lapply (drop_inv_atom1 … H) -H #H destruct
-| #L1 #L2 #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
- elim (drop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 Hie; destruct -i K1 W;
- <minus_n_O <minus_plus_m_m /2/
- | -HL12;
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
- ]
-| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
- elim (drop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 Hie Hi; destruct
- | elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
- ]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- lapply (plus_S_le_to_pos … Hdi) #Hi
- lapply (drop_inv_drop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
-]
-qed.
-
-(* Basic forvard lemmas *****************************************************)
-
-(* Basic_1: was: drop_S *)
-lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
- ↓[O, e + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e K2 I2 V2 /2/
- | @drop_drop >(plus_minus_m_m e 1) /2/
- ]
-]
-qed.
-
-lemma drop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
-#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
-[ /2/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
- >(tw_lift … HV21) -HV21 /2/
-]
-qed.
-
-lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
- ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e K2 I2 V2 //
- | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
- ]
-]
-qed.
-
-lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 elim L1 -L1
-[ #L2 #e #H >(drop_inv_atom1 … H) -H //
-| #K1 #I1 #V1 #IHL1 #L2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e L2 //
- | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
- >minus_le_minus_minus_comm //
- ]
-]
-qed.
-
-(* Basic_1: removed theorems 49:
- drop_skip_flat
- cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- drop_clear drop_clear_O drop_clear_S
- clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
- clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
- getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
- getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
- drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
- getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
- getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
- getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
-*)
+++ /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/lift_lift.ma".
-include "Basic_2/substitution/drop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop_mono *)
-theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
- ∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
-#d #e #L #L1 #H elim H -H d e L L1
-[ #d #e #L2 #H
- >(drop_inv_atom1 … H) -H L2 //
-| #K #I #V #L2 #HL12
- <(drop_inv_refl … HL12) -HL12 L2 //
-| #L #K #I #V #e #_ #IHLK #L2 #H
- lapply (drop_inv_drop1 … H ?) -H /2/
-| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
- elim (drop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
- >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
- >(IHLK1 … HLK2) -IHLK1 HLK2 //
-]
-qed.
-
-(* Basic_1: was: drop_conf_ge *)
-theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
- ↓[0, e2 - e1] L1 ≡ L2.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
-[ #d #e #e2 #L2 #H
- >(drop_inv_atom1 … H) -H L2 //
-| //
-| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
- lapply (drop_inv_drop1 … H ?) -H /2/ #HL2
- <minus_plus_comm /3/
-| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
- lapply (transitive_le 1 … Hdee2) // #He2
- lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2
- lapply (transitive_le (1 + e) … Hdee2) // #Hee2
- @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
-]
-qed.
-
-(* Basic_1: was: drop_conf_lt *)
-theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
- ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
- ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
-[ #d #e #e2 #K2 #I #V2 #H
- lapply (drop_inv_atom1 … H) -H #H destruct
-| #L #I #V #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
- elim (drop_inv_O1 … H) -H *
- [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
- | -HL12 -HV12 #He #HLK
- elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
- ]
-]
-qed.
-
-(* Basic_1: was: drop_trans_le *)
-theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
- ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
-[ #d #e #e2 #L2 #H
- >(drop_inv_atom1 … H) -H L2 /2/
-| #K #I #V #e2 #L2 #HL2 #H
- lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
-| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
- lapply (le_O_to_eq_O … H) -H #H destruct -e2;
- elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
- lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
- elim (drop_inv_O1 … H) -H *
- [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
- | -HL12 HV12 #He2 #HL2
- elim (IHL12 … HL2 ?) -IHL12 HL2 L2
- [ >minus_le_minus_minus_comm // /3/ | /2/ ]
- ]
-]
-qed.
-
-(* Basic_1: was: drop_trans_ge *)
-theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
-[ #d #e #e2 #L2 #H
- >(drop_inv_atom1 … H) -H L2 //
-| //
-| /3/
-| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
- lapply (drop_inv_drop1 … H ?) -H // #HL2
- @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
-]
-qed.
-
-theorem drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
- ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 →
- ↓[0, e2 + e1] L1 ≡ L2.
-#e1 #e1 #e2 >commutative_plus /2 width=5/
-qed.
-
-(* Basic_1: was: drop_conf_rev *)
-axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
- ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.
--- /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/grammar/lenv_weight.ma".
+include "Basic_2/grammar/lsubs.ma".
+include "Basic_2/substitution/lift.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Basic_1: includes: ldrop_skip_bind *)
+inductive ldrop: nat → nat → relation lenv ≝
+| ldrop_atom: ∀d,e. ldrop d e (⋆) (⋆)
+| ldrop_pair: ∀L,I,V. ldrop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
+| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. 𝕓{I} V) L2
+| ldrop_skip: ∀L1,L2,I,V1,V2,d,e.
+ ldrop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
+ ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+.
+
+interpretation "ldropping" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 * -d e L1 L2
+[ //
+| //
+| #L1 #L2 #I #V #e #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_refl *)
+lemma ldrop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5/ qed.
+
+fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆.
+#d #e #L1 #L2 * -d e L1 L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V #e #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_sort *)
+lemma ldrop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
+/2 width=5/ qed.
+
+fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #K #I #V #H destruct
+| #L #I #V #_ #K #J #W #HX destruct -L I V /3/
+| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma ldrop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[0, e - 1] K ≡ L2).
+/2/ qed.
+
+(* Basic_1: was: ldrop_gen_ldrop *)
+lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
+ ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
+#e #K #I #V #L2 #H #He
+elim (ldrop_inv_O1 … H) -H * // #H destruct -e;
+elim (lt_refl_false … He)
+qed.
+
+fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_skip_l *)
+lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+/2/ qed.
+
+fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_skip_r *)
+lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+/2/ qed.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was by definition: ldrop_refl *)
+lemma ldrop_refl: ∀L. ↓[0, 0] L ≡ L.
+#L elim L -L //
+qed.
+
+lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
+ ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
+qed.
+
+lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
+ ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V →
+ d ≤ i → i < d + e →
+ ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
+ ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e
+[ #d #e #K1 #V #i #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #L2 #K1 #V #i #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 Hie; destruct -i K1 W;
+ <minus_n_O <minus_plus_m_m /2/
+ | -HL12;
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+ ]
+| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 Hie Hi; destruct
+ | elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+ ]
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
+ lapply (plus_S_le_to_pos … Hdi) #Hi
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
+]
+qed.
+
+(* Basic forvard lemmas *****************************************************)
+
+(* Basic_1: was: ldrop_S *)
+lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
+ ↓[O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 /2/
+ | @ldrop_ldrop >(plus_minus_m_m e 1) /2/
+ ]
+]
+qed.
+
+lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
+[ /2/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(tw_lift … HV21) -HV21 /2/
+]
+qed.
+
+lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
+ ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 //
+ | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
+ ]
+]
+qed.
+
+lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 elim L1 -L1
+[ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
+| #K1 #I1 #V1 #IHL1 #L2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e L2 //
+ | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
+ >minus_le_minus_minus_comm //
+ ]
+]
+qed.
+
+(* Basic_1: removed theorems 49:
+ ldrop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ ldrop_clear ldrop_clear_O ldrop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_ldrop getl_ldrop_conf_lt
+ getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
+ ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
+ getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
--- /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/lift_lift.ma".
+include "Basic_2/substitution/ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: ldrop_mono *)
+theorem ldrop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
+ ∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
+#d #e #L #L1 #H elim H -H d e L L1
+[ #d #e #L2 #H
+ >(ldrop_inv_atom1 … H) -H L2 //
+| #K #I #V #L2 #HL12
+ <(ldrop_inv_refl … HL12) -HL12 L2 //
+| #L #K #I #V #e #_ #IHLK #L2 #H
+ lapply (ldrop_inv_ldrop1 … H ?) -H /2/
+| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
+ elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
+ >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
+ >(IHLK1 … HLK2) -IHLK1 HLK2 //
+]
+qed.
+
+(* Basic_1: was: ldrop_conf_ge *)
+theorem ldrop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
+ ↓[0, e2 - e1] L1 ≡ L2.
+#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -H L2 //
+| //
+| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H /2/ #HL2
+ <minus_plus_comm /3/
+| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
+ lapply (transitive_le 1 … Hdee2) // #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2
+ lapply (transitive_le (1 + e) … Hdee2) // #Hee2
+ @ldrop_ldrop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
+]
+qed.
+
+(* Basic_1: was: ldrop_conf_lt *)
+theorem ldrop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
+ ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
+ ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
+#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+[ #d #e #e2 #K2 #I #V2 #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L #I #V #e2 #K2 #J #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
+ elim (ldrop_inv_O1 … H) -H *
+ [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
+ | -HL12 -HV12 #He #HLK
+ elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
+ ]
+]
+qed.
+
+(* Basic_1: was: ldrop_trans_le *)
+theorem ldrop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
+ ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
+#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -H L2 /2/
+| #K #I #V #e2 #L2 #HL2 #H
+ lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
+| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
+ lapply (le_O_to_eq_O … H) -H #H destruct -e2;
+ elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
+ lapply (ldrop_inv_refl … H) -H #H destruct -L1 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
+ elim (ldrop_inv_O1 … H) -H *
+ [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
+ | -HL12 HV12 #He2 #HL2
+ elim (IHL12 … HL2 ?) -IHL12 HL2 L2
+ [ >minus_le_minus_minus_comm // /3/ | /2/ ]
+ ]
+]
+qed.
+
+(* Basic_1: was: ldrop_trans_ge *)
+theorem ldrop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
+#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -H L2 //
+| //
+| /3/
+| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HL2
+ @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
+]
+qed.
+
+theorem ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
+ ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 →
+ ↓[0, e2 + e1] L1 ≡ L2.
+#e1 #e1 #e2 >commutative_plus /2 width=5/
+qed.
+
+(* Basic_1: was: ldrop_conf_rev *)
+axiom ldrop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
+ ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.
| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k)
| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
+| lift_gref : ∀p,d,e. lift d e (§p) (§p)
| lift_bind : ∀I,V1,V2,T1,T2,d,e.
lift d e V1 V2 → lift (d + 1) e T1 T2 →
lift d e (𝕓{I} V1. T1) (𝕓{I} V2. T2)
| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
<(arith_d1 i e2 e1) // /3/
+| /3/
| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
elim (IHT (d2+1) … ? ? He12) /3 width = 5/
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
| #j #d #e #Hj #i #Hi destruct /3/
+| #p #d #e #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
elim (lt_refl_false … Hdd)
qed.
+fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#d #e #T1 #T2 * -d e T1 T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed.
+
+lemma lift_inv_gref1: ∀d,e,T2,p. ↑[d,e] §p ≡ T2 → T2 = §p.
+/2 width=5/ qed.
+
fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #d #e #I #V1 #U1 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
]
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #d #e #I #V1 #U1 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
]
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
+| #p #d #e #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
elim (plus_lt_false … Hdd)
qed.
+fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#d #e #T1 #T2 * -d e T1 T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed.
+
+lemma lift_inv_gref2: ∀d,e,T1,p. ↑[d,e] T1 ≡ §p → T1 = §p.
+/2 width=5/ qed.
+
fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
]
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
]
lapply (lift_inv_lref2_lt … HX ?) -HX //
| #i #d #e #Hdi #X #HX
lapply (lift_inv_lref2_ge … HX ?) -HX /2/
+| #p #d #e #X #HX
+ lapply (lift_inv_gref2 … HX) -HX //
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /2/ /3/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
[ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
| -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
@(ex2_1_intro … #(i - e2))
[ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
- | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
+ | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /2/ /3/
]
]
+| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct -T2 /3/
| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
+ >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /2/ /3 width = 5/
| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- elim (IHU … HU2 ?) /3 width = 5/
+ elim (IHU … HU2 ?) // /3 width = 5/
]
qed.
lapply (lift_inv_lref1_lt … HX ?) -HX //
| #i #d #e #Hdi #X #HX
lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #p #d #e #X #HX
+ lapply (lift_inv_gref1 … HX) -HX //
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
[ @(transitive_le … Hd21 ?) -Hd21 /2/
| -Hd21 /2/
]
+| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_gref1 … HT2) -HT2 //
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2/
+ lapply (IHT12 … HT20 ? ?) // /2/
]
qed.
>(lift_inv_sort1 … HX) -HX /2/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
- elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
+ elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /3/ /4/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
>plus_plus_comm_23 /4/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ /3 width=5/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/
]
qed.
lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
- [2: >plus_plus_comm_23] /4/
+ [ /4/ | >plus_plus_comm_23 /4/ ]
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
elim (IHV12 … HV20 ?) -IHV12 HV20 //
elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
- <plus_minus /3 width=5/
+ <plus_minus /2/ /3 width=5/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/ (**) (* just /3 width=5/ crashes *)
]
qed.
/2/ qed.
(* Basic_1: removed theorems 27:
- csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
+ csubst0_clear_O csubst0_ldrop_lt csubst0_ldrop_gt csubst0_ldrop_eq
csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
- csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
+ csubst0_ldrop_gt_back csubst0_ldrop_eq_back csubst0_ldrop_lt_back
csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
+++ /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/ltps.ma".
-
-(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
-
-lemma ltps_drop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- lapply (plus_le_weak … He12) #He2
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- lapply (plus_le_weak … Hd1e2) #He2
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/
-]
-qed.
-
-lemma ltps_drop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- lapply (plus_le_weak … He12) #He2
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- lapply (plus_le_weak … Hd1e2) #He2
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/
-]
-qed.
-
-lemma ltps_drop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 [0, d1 + e1 - e2] ≫ L & ↓[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2;
- lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2
- [ destruct -IHK01 He21 e2 L2 <minus_n_O /3/
- | -HK01 HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -IHK01 HK0L2 /3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- lapply (plus_le_weak … Hd1e2) #He2
- <minus_le_minus_minus_comm //
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -IHK01 HK0L2 /3/
-]
-qed.
-
-lemma ltps_drop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L [0, d1 + e1 - e2] ≫ L2 & ↓[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2;
- lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2
- [ destruct -IHK10 He21 e2 L2 <minus_n_O /3/
- | -HK10 HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -IHK10 HK0L2 /3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- lapply (plus_le_weak … Hd1e2) #He2
- <minus_le_minus_minus_comm //
- lapply (drop_inv_drop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -IHK10 HK0L2 /3/
-]
-qed.
-
-lemma ltps_drop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 [d1 - e2, e1] ≫ L & ↓[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/
-| /2/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2;
- lapply (drop_inv_refl … H) -H #H destruct -L2 /3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2
- [ destruct -IHK01 He2d1 e2 L2 <minus_n_O /3/
- | -HK01 HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -IHK01 HK0L2 /3/
- ]
-]
-qed.
-
-lemma ltps_drop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L [d1 - e2, e1] ≫ L2 & ↓[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
-[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/
-| /2/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2;
- lapply (drop_inv_refl … H) -H #H destruct -L2 /3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2
- [ destruct -IHK10 He2d1 e2 L2 <minus_n_O /3/
- | -HK10 HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -IHK10 HK0L2 /3/
- ]
-]
-qed.
--- /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/ltps.ma".
+
+(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
+
+lemma ltps_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ lapply (plus_le_weak … He12) #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ lapply (plus_le_weak … Hd1e2) #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/
+]
+qed.
+
+lemma ltps_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ lapply (plus_le_weak … He12) #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ lapply (plus_le_weak … Hd1e2) #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/
+]
+qed.
+
+lemma ltps_ldrop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 [0, d1 + e1 - e2] ≫ L & ↓[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2;
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct -L2 /2/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ destruct -IHK01 He21 e2 L2 <minus_n_O /3/
+ | -HK01 HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -IHK01 HK0L2 /3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ lapply (plus_le_weak … Hd1e2) #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -IHK01 HK0L2 /3/
+]
+qed.
+
+lemma ltps_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L [0, d1 + e1 - e2] ≫ L2 & ↓[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2;
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct -L2 /2/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ destruct -IHK10 He21 e2 L2 <minus_n_O /3/
+ | -HK10 HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -IHK10 HK0L2 /3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ lapply (plus_le_weak … Hd1e2) #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -IHK10 HK0L2 /3/
+]
+qed.
+
+lemma ltps_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 [d1 - e2, e1] ≫ L & ↓[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2/
+| /2/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2;
+ lapply (ldrop_inv_refl … H) -H #H destruct -L2 /3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ destruct -IHK01 He2d1 e2 L2 <minus_n_O /3/
+ | -HK01 HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -IHK01 HK0L2 /3/
+ ]
+]
+qed.
+
+lemma ltps_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L [d1 - e2, e1] ≫ L2 & ↓[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2/
+| /2/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2;
+ lapply (ldrop_inv_refl … H) -H #H destruct -L2 /3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ destruct -IHK10 He2d1 e2 L2 <minus_n_O /3/
+ | -HK10 HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -IHK10 HK0L2 /3/
+ ]
+]
+qed.
(**************************************************************************)
include "Basic_2/substitution/tps_lift.ma".
-include "Basic_2/substitution/ltps_drop.ma".
+include "Basic_2/substitution/ltps_ldrop.ma".
(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
[ //
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltps_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /2/
+ lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /2/
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
@tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *)
| /3/
[ /2/
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltps_drop_conf_le … HL01 … HLK0 ?) -HL01 HLK0 /2/ #X #H #HLK1
+ [ elim (ltps_ldrop_conf_le … HL01 … HLK0 ?) -HL01 HLK0 /2/ #X #H #HLK1
elim (ltps_inv_tps11 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
- lapply (drop_fwd_drop2 … HLK1) #H
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // >arith_a2 /3/
| elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltps_drop_conf_be … HL01 … HLK0 ? ?) -HL01 HLK0 [2,3: /2/ ] #X #H #HLK1
+ [ elim (ltps_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 HLK0 [2,3: /2/ ] #X #H #HLK1
elim (ltps_inv_tps21 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
- lapply (drop_fwd_drop2 … HLK1) #H
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // normalize #HW01
lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 //
- | lapply (ltps_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/
+ | lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/
]
]
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
[ //
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltps_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /2/
+ lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /2/
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
@tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *)
| /3/
[ /2/
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltps_drop_trans_le … HL10 … HLK0 ?) -HL10 /2/ #X #H #HLK1
+ [ elim (ltps_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2/ #X #H #HLK1
elim (ltps_inv_tps12 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
- lapply (drop_fwd_drop2 … HLK0) -HLK0 #H
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // >arith_a2 /3/
| elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltps_drop_trans_be … HL10 … HLK0 ? ?) -HL10 [2,3: /2/ ] #X #H #HLK1
+ [ elim (ltps_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 [2,3: /2/ ] #X #H #HLK1
elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
- lapply (drop_fwd_drop2 … HLK0) -HLK0 #H
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // normalize #HW01
lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 //
- | lapply (ltps_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /3/
+ | lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /3/
]
]
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
(**************************************************************************)
include "Basic_2/grammar/cl_weight.ma".
-include "Basic_2/substitution/drop.ma".
+include "Basic_2/substitution/ldrop.ma".
(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
[ //
| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (drop_lsubs_drop1_abbr … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
+ elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
| /4/
| /3/
]
#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
[ //
| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
- lapply (drop_fwd_drop2_length … HLK) #Hi
+ lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
lapply (le_to_lt_to_lt … Hdi Hi) #Hd
lapply (plus_minus_m_m_comm (|L|) d ?) /2/
| normalize /2/
(* *)
(**************************************************************************)
-include "Basic_2/substitution/drop_drop.ma".
+include "Basic_2/substitution/ldrop_ldrop.ma".
include "Basic_2/substitution/tps.ma".
(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
[ //
| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct -e;
>(le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK
- lapply (drop_mono … HLK0 … HLK) #H destruct
+ lapply (ldrop_mono … HLK0 … HLK) #H destruct
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
- >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 /2/
+ >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 // /2/
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
>(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
]
#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
>(lift_mono … H1 … H2) -H1 H2 //
-| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
- elim (lift_trans_ge … HVW … HVU2 ?) -HVW HVU2 W // <minus_plus #W #HVW #HWU2
- elim (drop_trans_le … HLK … HKV ?) -HLK HKV K [2: /2/] #X #HLK #H
- elim (drop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
+ elim (lift_trans_ge … HVW … HWU2 ?) -HVW HWU2 W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -HLK HKV K [2: /2/] #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
>(lift_mono … HVY … HVW) -HVY HVW Y #H destruct -X /2/
| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+ @tps_bind [ /2 width=6/ | @IHT12 [4,5: // |1,2: skip | /2/ | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ /3 width=6/
+]
+qed.
+
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+ ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
+ ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+ dt ≤ d → d ≤ dt + et →
+ L ⊢ U1 [dt, et + e] ≫ U2.
+#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 H2 //
+| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct -U1;
+ [ -Hdtd;
+ lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K [2: /2/] #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct -X /2/
+ | -Hdti;
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // [ /2/ ] >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3/
+ ]
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2/ | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
lapply (lift_trans_be … HVW … HWU2 ? ?) -HVW HWU2 W // [ /2/ ] >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid /3/
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid /3/
| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
[ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
| elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
]
| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
- elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #U #HKL #_ #HUV
+ elim (ldrop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #U #HKL #_ #HUV
elim (lift_trans_le … HUV … HVW ?) -HUV HVW V // >arith_a2 // -Hid /3/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK Hdetd (**) (* /3 width=5/ is too slow *)
+ elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @ldrop_skip // |2: skip ] -HLK Hdetd (**) (* /3 width=5/ is too slow *)
/3 width=5/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
]
qed.
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1
+ [ -Hdtd Hidet;
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >arith_a2 // -Hid /3/
+ | -Hdti Hdedet;
+ lapply (transitive_le … (i - e) Hdtd ?) [ /2/ ] -Hdtd #Hdtie
+ lapply (plus_le_weak … Hid) #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [4: // |2,3: /2/ ] -Hid >arith_e2 // /4/
+ ]
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ elim (IHV12 … HLK … HWV1 ? ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -IHU12 HTU1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2/ ]
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ elim (IHV12 … HLK … HWV1 ? ?) -IHV12 HWV1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -IHU12 HLK HTU1 // /3 width=5/
+]
+qed.
+
(* Basic_1: was: subst1_gen_lift_ge *)
lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
[ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
| elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
]
| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
lapply (transitive_le … Hdedt … Hdti) #Hdei
lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
lapply (plus_le_weak … Hdei) #Hei
lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
- lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [4: // | 2,3: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
@ex2_1_intro
[2: @tps_subst [3: /2/ |5,6: // |1,2: skip |4: @arith5 // ]
|1: skip
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
lapply (plus_le_weak … Hdetd) #Hedt
elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
+ elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @ldrop_skip // |2: skip |3: /2/ ]
<plus_minus // /3 width=5/
| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 [dt, d - dt] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // [ /2/ ] -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 L /2/
+qed.
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
+ lapply (ldrop_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
]
qed.
-lemma drop_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆.
+lemma ldrop_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆.
/2 width=5/ qed.
fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e →
+++ /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/ltps_drop.ma".
-include "Basic_2/unfold/ltpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
-
-lemma ltpss_drop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
-#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1 /3 width=6/
-qed.
-
-lemma ltpss_drop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
- d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0 /3 width=6/
-qed.
-
-lemma ltpss_drop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L2 [0, d1 + e1 - e2] ≫* L & ↓[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
-[ /2/
-| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
- elim (IHL … HL02 Hd1e2 He2de1) -L0 #L0 #HL20 #HL0
- elim (ltps_drop_conf_be … HL1 … HL0 Hd1e2 He2de1) -L /3/
-]
-qed.
-
-lemma ltpss_drop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L [0, d1 + e1 - e2] ≫* L2 & ↓[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
-[ /2/
-| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
- elim (ltps_drop_trans_be … HL0 … HL02 Hd1e2 He2de1) -L0 #L0 #HL02 #HL0
- elim (IHL … HL0 Hd1e2 He2de1) -L /3/
-]
-qed.
-
-lemma ltpss_drop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 [d1 - e2, e1] ≫* L & ↓[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
-[ /2/
-| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #He2d1
- elim (IHL … HL02 He2d1) -L0 #L0 #HL20 #HL0
- elim (ltps_drop_conf_le … HL1 … HL0 He2d1) -L /3/
-]
-qed.
-
-lemma ltpss_drop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
- ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L [d1 - e2, e1] ≫* L2 & ↓[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
-[ /2/
-| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #He2d1
- elim (ltps_drop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL02 #HL0
- elim (IHL … HL0 He2d1) -L /3/
-]
-qed.
--- /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/ltps_ldrop.ma".
+include "Basic_2/unfold/ltpss.ma".
+
+(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+
+lemma ltpss_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1 /3 width=6/
+qed.
+
+lemma ltpss_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 →
+ d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0 /3 width=6/
+qed.
+
+lemma ltpss_ldrop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L2 [0, d1 + e1 - e2] ≫* L & ↓[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
+[ /2/
+| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
+ elim (IHL … HL02 Hd1e2 He2de1) -L0 #L0 #HL20 #HL0
+ elim (ltps_ldrop_conf_be … HL1 … HL0 Hd1e2 He2de1) -L /3/
+]
+qed.
+
+lemma ltpss_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L [0, d1 + e1 - e2] ≫* L2 & ↓[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
+[ /2/
+| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
+ elim (ltps_ldrop_trans_be … HL0 … HL02 Hd1e2 He2de1) -L0 #L0 #HL02 #HL0
+ elim (IHL … HL0 Hd1e2 He2de1) -L /3/
+]
+qed.
+
+lemma ltpss_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ≫* L1 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 [d1 - e2, e1] ≫* L & ↓[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
+[ /2/
+| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #He2d1
+ elim (IHL … HL02 He2d1) -L0 #L0 #HL20 #HL0
+ elim (ltps_ldrop_conf_le … HL1 … HL0 He2d1) -L /3/
+]
+qed.
+
+lemma ltpss_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 →
+ ∀L2,e2. ↓[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L [d1 - e2, e1] ≫* L2 & ↓[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
+[ /2/
+| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #He2d1
+ elim (ltps_ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL02 #HL0
+ elim (IHL … HL0 He2d1) -L /3/
+]
+qed.
| #U #U1 #_ #HU1 #IHU #U2 #HU12
elim (lift_total U 0 (i+1)) #U0 #HU0
lapply (IHU … HU0) -IHU #H
- lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 HLK HU0 HU12 // normalize #HU02
lapply (tps_weak … HU02 d e ? ?) -HU02 [ >arith_i2 // | /2/ | /2/ ]
]
[ #H destruct -T;
elim (tps_inv_atom1 … HT2) -HT2 [ /2/ | * /3 width=10/ ]
| * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (drop_fwd_drop2 … HLK) #H
+ lapply (ldrop_fwd_ldrop2 … HLK) #H
elim (tps_inv_lift1_up … HT2 … H … HVT ? ? ?) normalize -HT2 H HVT [2,3,4: /2/ ] #V2 <minus_plus #HV2 #HVT2
@or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2/ (**) (* /4 width=10/ is too slow *)
]
∀L,U1,d,e. dt + et ≤ d → ↓[d, e] L ≡ K →
↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 →
L ⊢ U1 [dt, et] ≫* U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … H) -H T2
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … H) -T2
[ #U2 #H >(lift_mono … HTU1 … H) -H //
| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
elim (lift_total T d e) #U #HTU
]
qed.
+lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫* T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ↓[d, e] L ≡ K → ↑[d, e] T1 ≡ U1 →
+ ∀U2. ↑[d, e] T2 ≡ U2 → L ⊢ U1 [dt, et + e] ≫* U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT … HTU) -IHT #HU1
+ lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 HLK HTU HTU2 /2/
+]
+qed.
+
lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫* T2 →
∀L,U1,d,e. d ≤ dt → ↓[d, e] L ≡ K →
↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 →
L ⊢ U1 [dt + e, et] ≫* U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -H T2
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2
[ #U2 #H >(lift_mono … HTU1 … H) -H //
| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
elim (lift_total T d e) #U #HTU
∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
dt + et ≤ d →
∃∃T2. K ⊢ T1 [dt, et] ≫* T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -H U2
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
+[ /2/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 HLK HTU // /3/
+]
+qed.
+
+lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ≫* T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
[ /2/
| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 HLK HTU /3/
+ elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 HLK HTU // /3/
]
qed.
∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
d + e ≤ dt →
∃∃T2. K ⊢ T1 [dt - e, et] ≫* T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -H U2
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
[ /2/
| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 HLK HTU /3/
+ elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 HLK HTU // /3/
]
qed.
lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
L ⊢ U1 [d, e] ≫* U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -H U2 //
+#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
#U #U2 #_ #HU2 #IHU destruct -U1
<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 HTU1 //
qed.
+
+lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 [dt, d - dt] ≫* T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
+[ /2/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 HLK HTU // /3/
+]
+qed.
#L1 #T2 #U2 #d #e * -L1 T2 U2 d e
[ //
| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L0 #HL10 #H1 #H2 destruct -Y1 X2;
- lapply (drop_fwd_lw … HLK1) normalize #H1
- elim (ltps_drop_trans_be … HL10 … HLK1 ? ?) -HL10 HLK1 [2,3: /2/ ] #X #H #HLK0
+ lapply (ldrop_fwd_lw … HLK1) normalize #H1
+ elim (ltps_ldrop_trans_be … HL10 … HLK1 ? ?) -HL10 HLK1 [2,3: /2/ ] #X #H #HLK0
elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K0 #V0 #HK01 #HV01 #H destruct -X;
lapply (tps_fwd_tw … HV01) #H2
lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 [ /2/ ] -H2 #H
| #m #IHm * [ /2/ ]
#n elim (IHm n) -IHm #H
[ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *)
- qed.
+ qed.
lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n.
-#m #n * -n /3/
-qed.
+/2/ qed. (**) (* REMOVE: this is le_to_or_lt_eq *)
lemma plus_le_weak: ∀m,n,p. m + n ≤ p → n ≤ p.
/2/ qed.
elim (lt_refl_false … H)
qed.
+lemma le_fwd_plus_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m1 #m2 #H elim H -H m1
+[ /2/
+| #m1 #_ #IHm1 #n1 #n2 #H @IHm1 /2/
+]
+qed.
+
lemma monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
#p #q #n #H1 #H2
@lt_plus_to_minus_r <plus_minus_m_m //.
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