(* Forward lemmas with generic slicing for local environments ***************)
(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
-lemma lsubr_fwd_drops2_pair: ∀L1,L2. L1 ⫃ L2 →
- ∀I,K2,W,c,f. 𝐔⦃f⦄ → ⬇*[c, f] L2 ≡ K2.ⓑ{I}W →
- (∃∃K1. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓑ{I}W) ∨
- ∃∃K1,V. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓓⓝW.V & I = Abst.
+lemma lsubr_fwd_drops2_bind: ∀L1,L2. L1 ⫃ L2 →
+ ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
#L1 #L2 #H elim H -L1 -L2
-[ #L #I #K2 #W #c #f #_ #H
+[ #b #f #I #K2 #_ #H
elim (drops_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #c #f #Hf #H
- elim (drops_inv_pair1_isuni … Hf H) -H * #g #HLK2 try #H destruct [ -IHL12 | -HL12 ]
- [ /4 width=4 by drops_refl, ex2_intro, or_introl/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 /2 width=3 by isuni_inv_next/ *
- /4 width=4 by drops_drop, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
-| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #c #f #Hf #H
- elim (drops_inv_pair1_isuni … Hf H) -H * #g #HLK2 try #H destruct [ -IHL12 | -HL12 ]
- [ /4 width=4 by drops_refl, ex3_2_intro, or_intror/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 /2 width=3 by isuni_inv_next/ *
- /4 width=4 by drops_drop, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
+| #J #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V1
+]
+#HL12 #IH #b #f #I #K2 #Hf #H
+elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+[1,3,5: #Hf #H destruct -IH
+ /4 width=6 by drops_refl, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
+|2,4,6: #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -IH -Hg -HLK2 *
+ /4 width=6 by drops_drop, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
]
qed-.
(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
- ∀K2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] L2 ≡ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓓV.
-#L1 #L2 #HL12 #K2 #V #c #f #Hf #HLK2
-elim (lsubr_fwd_drops2_pair … HL12 … Hf HLK2) -L2 -Hf // *
-#K1 #W #_ #_ #H destruct
+ ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓV.
+#L1 #L2 #HL12 #b #f #K2 #V #Hf #HLK2
+elim (lsubr_fwd_drops2_bind … HL12 … Hf HLK2) -L2 -Hf // *
+[ #K1 #W #V #_ #_ #H destruct
+| #I1 #I2 #K1 #V #_ #_ #H destruct
+]
qed-.