(* *)
(**************************************************************************)
-include "Basic_2/substitution/ltps.ma".
-include "Basic_2/unfold/tpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+include "basic_2/unfold/tpss.ma".
+
+(* PARALLEL UNFOLD ON LOCAL ENVIRONMENTS ************************************)
+
+(* Basic_1: includes: csubst1_bind *)
+inductive ltpss: nat → nat → relation lenv ≝
+| ltpss_atom : ∀d,e. ltpss d e (⋆) (⋆)
+| ltpss_pair : ∀L,I,V. ltpss 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltpss_tpss2: ∀L1,L2,I,V1,V2,e.
+ ltpss 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
+ ltpss 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+| ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ ltpss d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
+ ltpss (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel unfold (local environment)"
+ 'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
-definition ltpss: nat → nat → relation lenv ≝
- λd,e. TC … (ltps d e).
+(* Basic properties *********************************************************)
-interpretation "partial unfold (local environment)"
- 'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
+lemma ltpss_tps2: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_tps1: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
+
+lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+qed.
-(* Basic eliminators ********************************************************)
+lemma ltpss_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+qed.
-lemma ltpss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 [d, e] ▶* L → L [d, e] ▶ L2 → R L → R L2) →
- ∀L2. L1 [d, e] ▶* L2 → R L2.
-#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
+lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
-(* Basic properties *********************************************************)
+lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
+/3 width=1/ qed.
-lemma ltpss_strap: ∀L1,L,L2,d,e.
- L1 [d, e] ▶ L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
-/2 width=3/ qed.
+(* Basic_1: was by definition: csubst1_refl *)
+lemma ltpss_refl: ∀L,d,e. L ▶* [d, e] L.
+#L elim L -L //
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
+qed.
-lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
-/2 width=1/ qed.
+lemma ltpss_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12 /3 width=5/
+ ]
+]
+qed.
-lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
-#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2
->(ltps_fwd_length … HL2) /3 width=5/
+lemma ltpss_weak_all: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
qed.
(* Basic forward lemmas *****************************************************)
-lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
-/2 width=3 by ltps_fwd_length/
+lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+normalize //
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
-#d #L1 #L2 #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 //
-qed-.
+fact ltpss_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
+]
+qed.
-lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
-#d #e #L2 #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL destruct
->(ltps_inv_atom1 … HL2) -HL2 //
-qed-.
+lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
+/2 width=4/ qed-.
-fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
-#L2 #L #_ #HL2 #IHL2 #H destruct
-lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
+fact ltpss_inv_atom1_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
qed.
-lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
+lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
/2 width=5/ qed-.
-(*
-fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
- K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
+
+fact ltpss_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #_ #K1 #I #V1 #H destruct
+| #L1 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpss_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=5/ qed-.
+
+fact ltpss_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K1 #V1 #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpss_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ltpss_inv_atom2_aux: ∀d,e,L1,L2.
+ L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma ltpss_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
+/2 width=5/ qed-.
+
+fact ltpss_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #_ #K1 #I #V1 #H destruct
| #L1 #I #V #_ #H elim (lt_refl_false … H)
| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
]
qed.
-lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=5/ qed.
-
-fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
+lemma ltpss_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=5/ qed-.
+
+fact ltpss_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K2 #V2 #H destruct
| #L #I #V #H elim (lt_refl_false … H)
| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
]
qed.
-lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=1/ qed.
+lemma ltpss_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic_1: removed theorems 28:
+ csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
+ csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
+ csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_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
+ csubst1_head csubst1_flat csubst1_gen_head
+ csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
+ fsubst0_gen_base
*)
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