(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lrsubeq_3.ma".
+include "basic_2/notation/relations/lrsubeqc_4.ma".
+include "basic_2/static/lsubr.ma".
include "basic_2/static/aaa.ma".
-include "basic_2/computation/acp_cr.ma".
+include "basic_2/computation/gcp_cr.ma".
-(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-inductive lsubc (RP:lenv→predicate term): relation lenv ≝
-| lsubc_atom: lsubc RP (⋆) (⋆)
-| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ ϵ[RP] 〚A〛 → ⦃L1, W⦄ ϵ[RP] 〚A〛 → L2 ⊢ W ⁝ A →
- lsubc RP L1 L2 → lsubc RP (L1. ⓓⓝW.V) (L2.ⓛW)
+inductive lsubc (RP) (G): relation lenv ≝
+| lsubc_atom: lsubc RP G (⋆) (⋆)
+| lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
.
interpretation
- "local environment refinement (abstract candidates of reducibility)"
- 'LRSubEq RP L1 L2 = (lsubc RP L1 L2).
+ "local environment refinement (generic reducibility)"
+ 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L1 = ⋆ → L2 = ⋆.
-#RP #L1 #L2 * -L1 -L2
+fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
qed-.
(* Basic_1: was just: csubc_gen_sort_r *)
-lemma lsubc_inv_atom1: ∀RP,L2. ⋆ ⊑[RP] L2 → L2 = ⋆.
-/2 width=4 by lsubc_inv_atom1_aux/ qed-.
+lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
+/2 width=5 by lsubc_inv_atom1_aux/ qed-.
-fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. K1 ⊑[RP] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,V,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & ⦃K1, W⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
- K1 ⊑[RP] K2 &
+fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
L2 = K2. ⓛW & X = ⓝW.V & I = Abbr.
-#RP #L1 #L2 * -L1 -L2
+#RP #G #L1 #L2 * -L1 -L2
[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10/
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10 by ex7_4_intro, or_intror/
]
qed-.
(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_pair1: ∀RP,I,K1,L2,X. K1.ⓑ{I}X ⊑[RP] L2 →
- (∃∃K2. K1 ⊑[RP] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,V,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & ⦃K1, W⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
- K1 ⊑[RP] K2 &
+lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃[RP] L2 →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
L2 = K2.ⓛW & X = ⓝW.V & I = Abbr.
/2 width=3 by lsubc_inv_pair1_aux/ qed-.
-fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L2 = ⋆ → L1 = ⋆.
-#RP #L1 #L2 * -L1 -L2
+fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
qed-.
(* Basic_1: was just: csubc_gen_sort_l *)
-lemma lsubc_inv_atom2: ∀RP,L1. L1 ⊑[RP] ⋆ → L1 = ⋆.
-/2 width=4 by lsubc_inv_atom2_aux/ qed-.
+lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
+/2 width=5 by lsubc_inv_atom2_aux/ qed-.
-fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
- (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & ⦃K1, W⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
- K1 ⊑[RP] K2 &
+fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = Abst.
-#RP #L1 #L2 * -L1 -L2
+#RP #G #L1 #L2 * -L1 -L2
[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8/
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8 by ex6_3_intro, or_intror/
]
qed-.
(* Basic_1: was just: csubc_gen_head_l *)
-lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 ⊑[RP] K2.ⓑ{I} W →
- (∃∃K1. K1 ⊑[RP] K2 & L1 = K1.ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & ⦃K1, W⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
- K1 ⊑[RP] K2 &
+lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⫃[RP] K2.ⓑ{I} W →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = Abst.
/2 width=3 by lsubc_inv_pair2_aux/ qed-.
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
+#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
(* Basic properties *********************************************************)
(* Basic_1: was just: csubc_refl *)
-lemma lsubc_refl: ∀RP,L. L ⊑[RP] L.
-#RP #L elim L -L // /2 width=1/
+lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
+#RP #G #L elim L -L /2 width=1 by lsubc_pair/
qed.
(* Basic_1: removed theorems 3: