(* *)
(**************************************************************************)
-include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/substitution/lpx_sn.ma".
(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
]
qed-.
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
(* Inversion lemmas on transitive closure ***********************************)
lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
]
qed-.
-lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_trans … R (lpx_sn R) →
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
]
qed-.
-lemma TC_lpx_sn_ind: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
#R #L2 #H elim H -L2
[ /2 width=2 by lpx_sn_inv_atom1/
]
qed-.
-fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_trans … R (lpx_sn R) →
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀L1,L2. TC … (lpx_sn R) L1 L2 →
∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
]
qed-.
-lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_trans … R (lpx_sn R) →
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_trans … R (lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
(* Forward lemmas on transitive closure *************************************)
lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.