inductive lsuba (G:genv): relation lenv ≝
| lsuba_atom: lsuba G (⋆) (⋆)
| lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsuba_abbr: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
- "local environment refinement (atomic arity assigment)"
+ "local environment refinement (atomic arity assignment)"
'LRSubEqA G L1 L2 = (lsuba G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsuba_inv_atom1_aux: â\88\80G,L1,L2. G â\8a¢ L1 â\81\9dâ«\83 L2 → L1 = ⋆ → L2 = ⋆.
+fact lsuba_inv_atom1_aux: â\88\80G,L1,L2. G â\8a¢ L1 â«\83â\81\9d L2 → L1 = ⋆ → L2 = ⋆.
#G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
]
qed-.
-lemma lsuba_inv_atom1: â\88\80G,L2. G â\8a¢ â\8b\86 â\81\9dâ«\83 L2 → L2 = ⋆.
+lemma lsuba_inv_atom1: â\88\80G,L2. G â\8a¢ â\8b\86 â«\83â\81\9d L2 → L2 = ⋆.
/2 width=4 by lsuba_inv_atom1_aux/ qed-.
-fact lsuba_inv_pair1_aux: â\88\80G,L1,L2. G â\8a¢ L1 â\81\9dâ«\83 L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (â\88\83â\88\83K2. G â\8a¢ K1 â\81\9dâ«\83 K2 & L2 = K2.ⓑ{I}X) ∨
+fact lsuba_inv_pair1_aux: â\88\80G,L1,L2. G â\8a¢ L1 â«\83â\81\9d L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83â\81\9d K2 & L2 = K2.ⓑ{I}X) ∨
∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
- G â\8a¢ K1 â\81\9dâ«\83 K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+ G â\8a¢ K1 â«\83â\81\9d K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
#G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
]
qed-.
-lemma lsuba_inv_pair1: â\88\80I,G,K1,L2,X. G â\8a¢ K1.â\93\91{I}X â\81\9dâ«\83 L2 →
- (â\88\83â\88\83K2. G â\8a¢ K1 â\81\9dâ«\83 K2 & L2 = K2.ⓑ{I}X) ∨
- â\88\83â\88\83K2,W,V,A. â¦\83G, K1â¦\84 â\8a¢ â\93\9dW.V â\81\9d A & â¦\83G, K2â¦\84 â\8a¢ W â\81\9d A & G â\8a¢ K1 â\81\9dâ«\83 K2 &
+lemma lsuba_inv_pair1: â\88\80I,G,K1,L2,X. G â\8a¢ K1.â\93\91{I}X â«\83â\81\9d L2 →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83â\81\9d K2 & L2 = K2.ⓑ{I}X) ∨
+ â\88\83â\88\83K2,W,V,A. â¦\83G, K1â¦\84 â\8a¢ â\93\9dW.V â\81\9d A & â¦\83G, K2â¦\84 â\8a¢ W â\81\9d A & G â\8a¢ K1 â«\83â\81\9d K2 &
I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
/2 width=3 by lsuba_inv_pair1_aux/ qed-.
-fact lsuba_inv_atom2_aux: â\88\80G,L1,L2. G â\8a¢ L1 â\81\9dâ«\83 L2 → L2 = ⋆ → L1 = ⋆.
+fact lsuba_inv_atom2_aux: â\88\80G,L1,L2. G â\8a¢ L1 â«\83â\81\9d L2 → L2 = ⋆ → L1 = ⋆.
#G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
]
qed-.
-lemma lsubc_inv_atom2: â\88\80G,L1. G â\8a¢ L1 â\81\9dâ«\83 ⋆ → L1 = ⋆.
+lemma lsubc_inv_atom2: â\88\80G,L1. G â\8a¢ L1 â«\83â\81\9d ⋆ → L1 = ⋆.
/2 width=4 by lsuba_inv_atom2_aux/ qed-.
-fact lsuba_inv_pair2_aux: â\88\80G,L1,L2. G â\8a¢ L1 â\81\9dâ«\83 L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
- (â\88\83â\88\83K1. G â\8a¢ K1 â\81\9dâ«\83 K2 & L1 = K1.ⓑ{I}W) ∨
+fact lsuba_inv_pair2_aux: â\88\80G,L1,L2. G â\8a¢ L1 â«\83â\81\9d L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
+ (â\88\83â\88\83K1. G â\8a¢ K1 â«\83â\81\9d K2 & L1 = K1.ⓑ{I}W) ∨
∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
- G â\8a¢ K1 â\81\9dâ«\83 K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+ G â\8a¢ K1 â«\83â\81\9d K2 & I = Abst & L1 = K1.ⓓⓝW.V.
#G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
]
qed-.
-lemma lsuba_inv_pair2: â\88\80I,G,L1,K2,W. G â\8a¢ L1 â\81\9dâ«\83 K2.ⓑ{I}W →
- (â\88\83â\88\83K1. G â\8a¢ K1 â\81\9dâ«\83 K2 & L1 = K1.ⓑ{I}W) ∨
- â\88\83â\88\83K1,V,A. â¦\83G, K1â¦\84 â\8a¢ â\93\9dW.V â\81\9d A & â¦\83G, K2â¦\84 â\8a¢ W â\81\9d A & G â\8a¢ K1 â\81\9dâ«\83 K2 &
+lemma lsuba_inv_pair2: â\88\80I,G,L1,K2,W. G â\8a¢ L1 â«\83â\81\9d K2.ⓑ{I}W →
+ (â\88\83â\88\83K1. G â\8a¢ K1 â«\83â\81\9d K2 & L1 = K1.ⓑ{I}W) ∨
+ â\88\83â\88\83K1,V,A. â¦\83G, K1â¦\84 â\8a¢ â\93\9dW.V â\81\9d A & â¦\83G, K2â¦\84 â\8a¢ W â\81\9d A & G â\8a¢ K1 â«\83â\81\9d K2 &
I = Abst & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsuba_inv_pair2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lsuba_fwd_lsubr: â\88\80G,L1,L2. G â\8a¢ L1 â\81\9dâ«\83 L2 → L1 ⫃ L2.
-#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_abst/
+lemma lsuba_fwd_lsubr: â\88\80G,L1,L2. G â\8a¢ L1 â«\83â\81\9d L2 → L1 ⫃ L2.
+#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
qed-.
(* Basic properties *********************************************************)
-lemma lsuba_refl: â\88\80G,L. G â\8a¢ L â\81\9dâ«\83 L.
+lemma lsuba_refl: â\88\80G,L. G â\8a¢ L â«\83â\81\9d L.
#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
qed.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⫃ L2 → ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ⁝⫃ K2 & ⇩[s, 0, e] L2 ≡ K2.
+lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,s,e. ⬇[s, 0, e] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, e] L2 ≡ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_abbr, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
]
qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⫃ L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ⁝⫃ K2 & ⇩[s, 0, e] L1 ≡ K1.
+lemma lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,s,e. ⬇[s, 0, e] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, e] L1 ≡ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_abbr, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
]
qed-.