(**************************************************************************)
include "basic_2/notation/relations/lrsubeqa_3.ma".
-include "basic_2/static/lsubr.ma".
include "basic_2/static/aaa.ma".
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
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_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
+| 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
+| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #A #_ #_ #_ #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: ∀G,L1,L2. G ⊢ L1 ⁝⫃ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (â\88\83â\88\83K2. G â\8a¢ K1 â\81\9dâ«\83 K2 & L2 = K2.â\93\91{I}X) ∨
+fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83â\81\9d K2 & L2 = K2.â\93\98{I}) ∨
∃∃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.â\93\9bW & X = â\93\9dW.V.
+ G â\8a¢ K1 â«\83â\81\9d K2 & I = BPair Abbr (â\93\9dW.V) & L2 = K2.â\93\9bW.
#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/
-| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by or_intror, ex6_4_intro/
+[ #J #K1 #H destruct
+| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #H destruct /3 width=9 by ex5_4_intro, or_intror/
]
qed-.
-lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⁝⫃ L2 →
- (â\88\83â\88\83K2. G â\8a¢ K1 â\81\9dâ«\83 K2 & L2 = K2.â\93\91{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 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=3 by lsuba_inv_pair1_aux/ qed-.
+lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
+ (â\88\83â\88\83K2. G â\8a¢ K1 â«\83â\81\9d K2 & L2 = K2.â\93\98{I}) ∨
+ 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 = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+/2 width=3 by lsuba_inv_bind1_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
+| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #A #_ #_ #_ #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: ∀G,L1,L2. G ⊢ L1 ⁝⫃ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
- (â\88\83â\88\83K1. G â\8a¢ K1 â\81\9dâ«\83 K2 & L1 = K1.â\93\91{I}W) ∨
- ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⁝⫃ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ (â\88\83â\88\83K1. G â\8a¢ K1 â«\83â\81\9d K2 & L1 = K1.â\93\98{I}) ∨
+ ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & 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/
-| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by or_intror, ex5_3_intro/
+[ #J #K2 #H destruct
+| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #H destruct /3 width=9 by ex5_4_intro, or_intror/
]
qed-.
-lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⁝⫃ K2.ⓑ{I}W →
- (∃∃K1. G ⊢ K1 ⁝⫃ K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⁝⫃ K2 &
- I = Abst & L1 = K1.ⓓⓝW.V.
-/2 width=3 by lsuba_inv_pair2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⁝⫃ L2 → L1 ⫃ L2.
-#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_abst/
-qed-.
+lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_bind2_aux/ qed-.
(* Basic properties *********************************************************)
-lemma lsuba_refl: â\88\80G,L. G â\8a¢ L â\81\9dâ«\83 L.
-#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
+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_bind/
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.
-#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
- [ 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/
- ]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (ldrop_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/
- ]
-]
-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.
-#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
- [ 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/
- ]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (ldrop_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/
- ]
-]
-qed-.
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