-lemma lsubf_inv_push1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- (∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃g,g2,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ g & g2 ⋓ g ≡ g1 &
- ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & I = Abbr & f2 = ↑g2 & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=5 by lsubf_inv_push1_aux/ qed-.
-
-fact lsubf_inv_next1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I,K1,X. f1 = ⫯g1 → L1 = K1.ⓑ{I}X →
- (∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃g,g2,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ g & g2 ⋓ g ≡ g1 &
- ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & I = Abbr & f2 = ⫯g2 & L2 = K2.ⓛW & X = ⓝW.V.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f #g1 #J #K1 #X #_ #H destruct
-| #f1 #f2 #I #L1 #L2 #V #_ #g1 #J #K1 #X #H elim (discr_push_next … H)
-| #f1 #f2 #I #L1 #L2 #V #HL12 #g1 #J #K1 #X #H1 #H2 destruct
- /3 width=5 by injective_next, ex3_2_intro, or_introl/
-| #f1 #f2 #f #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #X #H elim (discr_push_next … H)
-| #f1 #f2 #f #L1 #L2 #W #V #Hf2 #Hf1 #HL12 #g1 #J #K1 #X #H1 #H2 destruct
- /3 width=11 by injective_next, ex7_5_intro, or_intror/
-]
-qed-.
-
-lemma lsubf_inv_next1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- (∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃g,g2,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ g & g2 ⋓ g ≡ g1 &
- ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & I = Abbr & f2 = ⫯g2 & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=5 by lsubf_inv_next1_aux/ qed-.
+lemma lsubf_inv_pair1: ∀f1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ (∃∃K2. f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & L2 = K2.ⓑ{I}X) ∨
+ ∃∃f,K2,W,V. K1 ⊢ 𝐅*⦃V⦄ ≡ f & f ⋓ ⫱f2 ≡ ⫱f1 &
+ f2 ⊆ f1 & ⦃K1, ⫱f1⦄ ⫃𝐅* ⦃K2, ⫱f2⦄ & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsubf_inv_pair1_aux/ qed-.