(* Basic_1: includes: csubst1_bind *)
inductive ltps: nat → nat → relation lenv ≝
| ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
(* Basic_1: includes: csubst1_bind *)
inductive ltps: nat → nat → relation lenv ≝
| ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
| ltps_tps2: ∀L1,L2,I,V1,V2,e.
ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 →
| ltps_tps2: ∀L1,L2,I,V1,V2,e.
ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 →
| ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 →
| ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 →
lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ 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 →
#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 →
/2 width=5/ qed-.
fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
/2 width=5/ qed-.
fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ V2 &
∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ 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)
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #_ #K1 #I #V1 #H destruct
| #L1 #I #V #_ #H elim (lt_refl_false … H)
-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 &
∃∃K2,V2. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
/2 width=5/ qed-.
fact ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
/2 width=5/ qed-.
fact ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ 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)
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K1 #V1 #H destruct
| #L #I #V #H elim (lt_refl_false … H)
-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 &
∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
/2 width=3/ qed-.
fact ltps_inv_atom2_aux: ∀d,e,L1,L2.
/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 →
/2 width=5/ qed-.
fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ V2 &
∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
K2 ⊢ V1 [0, e - 1] ▶ 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)
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #_ #K1 #I #V1 #H destruct
| #L1 #I #V #_ #H elim (lt_refl_false … H)
-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 &
∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
/2 width=5/ qed-.
fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
/2 width=5/ qed-.
fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K2 #V2 #H destruct
| #L #I #V #H elim (lt_refl_false … H)
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K2 #V2 #H destruct
| #L #I #V #H elim (lt_refl_false … H)
-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 &
∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
K2 ⊢ V1 [d - 1, e] ▶ V2 &