(* Basic_1: includes: csubst1_bind *)
inductive ltps: nat → nat → relation lenv ≝
| ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
-| ltps_pair: ∀L,I,V. ltps 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
+| ltps_pair: ∀L,I,V. ltps 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
| ltps_tps2: ∀L1,L2,I,V1,V2,e.
- ltps 0 e L1 L2 â\86\92 L2 â\8a¢ V1 [0, e] â\89« V2 →
- ltps 0 (e + 1) (L1. 𝕓{I} V1) L2. 𝕓{I} V2
+ ltps 0 e L1 L2 â\86\92 L2 â\8a¢ V1 [0, e] â\96¶ V2 →
+ ltps 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2
| ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
- ltps d e L1 L2 â\86\92 L2 â\8a¢ V1 [d, e] â\89« V2 →
- ltps (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+ ltps d e L1 L2 â\86\92 L2 â\8a¢ V1 [d, e] â\96¶ V2 →
+ ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
.
interpretation "parallel substritution (local environment)"
(* Basic properties *********************************************************)
lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] â\89« L2 â\86\92 L2 â\8a¢ V1 [0, e - 1] â\89« V2 →
- 0 < e → L1. 𝕓{I} V1 [0, e] ≫ L2. 𝕓{I} V2.
+ L1 [0, e - 1] â\96¶ L2 â\86\92 L2 â\8a¢ V1 [0, e - 1] â\96¶ 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/
+>(plus_minus_m_m e 1) /2 width=1/
qed.
lemma ltps_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 [d - 1, e] â\89« L2 â\86\92 L2 â\8a¢ V1 [d - 1, e] â\89« V2 →
- 0 < d → L1. 𝕓{I} V1 [d, e] ≫ L2. 𝕓{I} V2.
+ L1 [d - 1, e] â\96¶ L2 â\86\92 L2 â\8a¢ V1 [d - 1, e] â\96¶ 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/
+>(plus_minus_m_m d 1) /2 width=1/
qed.
(* Basic_1: was by definition: csubst1_refl *)
-lemma ltps_refl: â\88\80L,d,e. L [d, e] â\89« L.
+lemma ltps_refl: â\88\80L,d,e. L [d, e] â\96¶ L.
#L elim L -L //
-#L #I #V #IHL * /2/ * /2/
+#L #I #V #IHL * /2 width=1/ * /2 width=1/
qed.
+lemma ltps_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 ltps_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 ***************************************************)
-fact ltps_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -H d e L1 L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ #H
- elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct -e
+fact ltps_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 // >(tps_inv_refl_O2 … HV12) //
]
qed.
-lemma ltps_inv_refl_O2: â\88\80d,L1,L2. L1 [d, 0] â\89« L2 → L1 = L2.
-/2/ qed-.
+lemma ltps_inv_refl_O2: â\88\80d,L1,L2. L1 [d, 0] â\96¶ L2 → L1 = L2.
+/2 width=4/ qed-.
fact ltps_inv_atom1_aux: ∀d,e,L1,L2.
- L1 [d, e] â\89« L2 → L1 = ⋆ → L2 = ⋆.
-#d #e #L1 #L2 * -d e L1 L2
+ L1 [d, e] â\96¶ L2 → L1 = ⋆ → L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
[ //
| #L #I #V #H destruct
| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
]
qed.
-lemma ltps_inv_atom1: â\88\80d,e,L2. â\8b\86 [d, e] â\89« L2 → L2 = ⋆.
+lemma ltps_inv_atom1: â\88\80d,e,L2. â\8b\86 [d, e] â\96¶ L2 → L2 = ⋆.
/2 width=5/ qed-.
-fact ltps_inv_tps21_aux: â\88\80d,e,L1,L2. L1 [d, e] â\89« L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. 𝕓{I} V1 →
- â\88\83â\88\83K2,V2. K1 [0, e - 1] â\89« K2 &
- K2 â\8a¢ V1 [0, e - 1] â\89« V2 &
- L2 = K2. 𝕓{I} V2.
-#d #e #L1 #L2 * -d e L1 L2
+fact ltps_inv_tps21_aux: â\88\80d,e,L1,L2. L1 [d, e] â\96¶ L2 → d = 0 → 0 < e →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ â\88\83â\88\83K2,V2. K1 [0, e - 1] â\96¶ K2 &
+ K2 â\8a¢ V1 [0, e - 1] â\96¶ 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 -L1 I V1 /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_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 ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. 𝕓{I} V1 [0, e] ≫ L2 → 0 < e →
- â\88\83â\88\83K2,V2. K1 [0, e - 1] â\89« K2 & K2 â\8a¢ V1 [0, e - 1] â\89« V2 &
- L2 = K2. 𝕓{I} V2.
+lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶ L2 → 0 < e →
+ â\88\83â\88\83K2,V2. K1 [0, e - 1] â\96¶ K2 & K2 â\8a¢ V1 [0, e - 1] â\96¶ V2 &
+ L2 = K2. ⓑ{I} V2.
/2 width=5/ qed-.
-fact ltps_inv_tps11_aux: â\88\80d,e,L1,L2. L1 [d, e] â\89« L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
- â\88\83â\88\83K2,V2. K1 [d - 1, e] â\89« K2 &
- K2 â\8a¢ V1 [d - 1, e] â\89« V2 &
- L2 = K2. 𝕓{I} V2.
-#d #e #L1 #L2 * -d e L1 L2
+fact ltps_inv_tps11_aux: â\88\80d,e,L1,L2. L1 [d, e] â\96¶ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ â\88\83â\88\83K2,V2. K1 [d - 1, e] â\96¶ K2 &
+ K2 â\8a¢ V1 [d - 1, e] â\96¶ 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 -L1 I V1
- /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
]
qed.
-lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. 𝕓{I} V1 [d, e] ≫ L2 → 0 < d →
- â\88\83â\88\83K2,V2. K1 [d - 1, e] â\89« K2 &
- K2 â\8a¢ V1 [d - 1, e] â\89« V2 &
- L2 = K2. 𝕓{I} V2.
-/2/ qed-.
+lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶ L2 → 0 < d →
+ â\88\83â\88\83K2,V2. K1 [d - 1, e] â\96¶ K2 &
+ K2 â\8a¢ V1 [d - 1, e] â\96¶ V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
fact ltps_inv_atom2_aux: ∀d,e,L1,L2.
- L1 [d, e] â\89« L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 * -d e L1 L2
+ L1 [d, e] â\96¶ L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
[ //
| #L #I #V #H destruct
| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
]
qed.
-lemma ltps_inv_atom2: â\88\80d,e,L1. L1 [d, e] â\89« ⋆ → L1 = ⋆.
+lemma ltps_inv_atom2: â\88\80d,e,L1. L1 [d, e] â\96¶ ⋆ → L1 = ⋆.
/2 width=5/ qed-.
-fact ltps_inv_tps22_aux: â\88\80d,e,L1,L2. L1 [d, e] â\89« L2 → d = 0 → 0 < e →
- ∀K2,I,V2. L2 = K2. 𝕓{I} V2 →
- â\88\83â\88\83K1,V1. K1 [0, e - 1] â\89« K2 &
- K2 â\8a¢ V1 [0, e - 1] â\89« V2 &
- L1 = K1. 𝕓{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
+fact ltps_inv_tps22_aux: â\88\80d,e,L1,L2. L1 [d, e] â\96¶ L2 → d = 0 → 0 < e →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ â\88\83â\88\83K1,V1. K1 [0, e - 1] â\96¶ K2 &
+ K2 â\8a¢ V1 [0, e - 1] â\96¶ 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 -L2 I V2 /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_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 #_ #_ >commutative_plus normalize #H destruct
]
qed.
-lemma ltps_inv_tps22: â\88\80e,L1,K2,I,V2. L1 [0, e] â\89« K2. ð\9d\95\93{I} V2 → 0 < e →
- â\88\83â\88\83K1,V1. K1 [0, e - 1] â\89« K2 & K2 â\8a¢ V1 [0, e - 1] â\89« V2 &
- L1 = K1. 𝕓{I} V1.
+lemma ltps_inv_tps22: â\88\80e,L1,K2,I,V2. L1 [0, e] â\96¶ K2. â\93\91{I} V2 → 0 < e →
+ â\88\83â\88\83K1,V1. K1 [0, e - 1] â\96¶ K2 & K2 â\8a¢ V1 [0, e - 1] â\96¶ V2 &
+ L1 = K1. ⓑ{I} V1.
/2 width=5/ qed-.
-fact ltps_inv_tps12_aux: â\88\80d,e,L1,L2. L1 [d, e] â\89« L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
- â\88\83â\88\83K1,V1. K1 [d - 1, e] â\89« K2 &
- K2 â\8a¢ V1 [d - 1, e] â\89« V2 &
- L1 = K1. 𝕓{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
+fact ltps_inv_tps12_aux: â\88\80d,e,L1,L2. L1 [d, e] â\96¶ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ â\88\83â\88\83K1,V1. K1 [d - 1, e] â\96¶ K2 &
+ K2 â\8a¢ V1 [d - 1, e] â\96¶ 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)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct -L2 I V2
- /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
]
qed.
-lemma ltps_inv_tps12: â\88\80L1,K2,I,V2,d,e. L1 [d, e] â\89« K2. ð\9d\95\93{I} V2 → 0 < d →
- â\88\83â\88\83K1,V1. K1 [d - 1, e] â\89« K2 &
- K2 â\8a¢ V1 [d - 1, e] â\89« V2 &
- L1 = K1. 𝕓{I} V1.
-/2/ qed-.
+lemma ltps_inv_tps12: â\88\80L1,K2,I,V2,d,e. L1 [d, e] â\96¶ K2. â\93\91{I} V2 → 0 < d →
+ â\88\83â\88\83K1,V1. K1 [d - 1, e] â\96¶ K2 &
+ K2 â\8a¢ V1 [d - 1, e] â\96¶ V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
(* Basic_1: removed theorems 27:
csubst0_clear_O csubst0_ldrop_lt csubst0_ldrop_gt csubst0_ldrop_eq
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