(* 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 → L2 ⊢ V1 [0, e] ▶ V2 →
- ltps 0 (e + 1) (L1. 𝕓{I} V1) L2. 𝕓{I} 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 → L2 ⊢ V1 [d, e] ▶ V2 →
- ltps (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+ ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
.
interpretation "parallel substritution (local environment)"
lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
- 0 < e → L1. 𝕓{I} V1 [0, e] ▶ L2. 𝕓{I} 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.
lemma ltps_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
L1 [d - 1, e] ▶ L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 →
- 0 < d → L1. 𝕓{I} V1 [d, e] ▶ L2. 𝕓{I} 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.
#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.
/2 width=5/ qed-.
fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
- ∀K1,I,V1. L1 = K1. 𝕓{I} V1 →
+ ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L2 = K2. 𝕓{I} 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)
]
qed.
-lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. 𝕓{I} V1 [0, e] ▶ L2 → 0 < e →
+lemma ltps_inv_tps21: ∀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. 𝕓{I} V2.
+ L2 = K2. ⓑ{I} V2.
/2 width=5/ qed-.
fact ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L2 = K2. 𝕓{I} 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)
]
qed.
-lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. 𝕓{I} V1 [d, e] ▶ L2 → 0 < d →
+lemma ltps_inv_tps11: ∀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. 𝕓{I} V2.
+ L2 = K2. ⓑ{I} V2.
/2 width=3/ qed-.
fact ltps_inv_atom2_aux: ∀d,e,L1,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. 𝕓{I} V2 →
+ ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. 𝕓{I} V1.
+ 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)
]
qed.
-lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. 𝕓{I} V2 → 0 < e →
+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. 𝕓{I} V1.
+ 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. 𝕓{I} V2 →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. 𝕓{I} V1.
+ 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)
]
qed.
-lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. 𝕓{I} V2 → 0 < d →
+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. 𝕓{I} V1.
+ L1 = K1. ⓑ{I} V1.
/2 width=3/ qed-.
(* Basic_1: removed theorems 27: