-(* Properties with at *******************************************************)
-
-lemma after_uni_dx: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
- ∀f. f2 ⊚ 𝐔❨i1❩ ≘ f → 𝐔❨i2❩ ⊚ ⫱*[i2] f2 ≘ f.
-#i2 elim i2 -i2
-[ #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
- lapply (after_isid_inv_dx … Hf ?) -Hf
- /3 width=3 by after_isid_sn, after_eq_repl_back0/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
- [ #g2 #j1 #Hg2 #H1 #H2 destruct
- elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
- <tls_xn /3 width=5 by after_next/
- | #g2 #Hg2 #H2 destruct
- elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
- <tls_xn /3 width=5 by after_next/
- ]
-]
-qed.
-
-lemma after_uni_sn: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
- ∀f. 𝐔❨i2❩ ⊚ ⫱*[i2] f2 ≘ f → f2 ⊚ 𝐔❨i1❩ ≘ f.
-#i2 elim i2 -i2
-[ #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
- lapply (after_isid_inv_sn … Hf ?) -Hf
- /3 width=3 by after_isid_dx, after_eq_repl_back0/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
- elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
- elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
- [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by after_push/
- | #g2 #Hg2 #H2 destruct /3 width=5 by after_next/
- ]
-]
-qed-.
-
-lemma after_uni_succ_dx: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
- ∀f. f2 ⊚ 𝐔❨↑i1❩ ≘ f → 𝐔❨↑i2❩ ⊚ ⫱*[↑i2] f2 ≘ f.
-#i2 elim i2 -i2
-[ #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
- elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
- lapply (after_isid_inv_dx … Hg ?) -Hg
- /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
- [ #g2 #j1 #Hg2 #H1 #H2 destruct
- elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
- <tls_xn /3 width=5 by after_next/
- | #g2 #Hg2 #H2 destruct
- elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
- <tls_xn /3 width=5 by after_next/
- ]
-]
-qed.
-
-lemma after_uni_succ_sn: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
- ∀f. 𝐔❨↑i2❩ ⊚ ⫱*[↑i2] f2 ≘ f → f2 ⊚ 𝐔❨↑i1❩ ≘ f.
-#i2 elim i2 -i2
-[ #i1 #f2 #Hf2 #f #Hf
- elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
- elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
- lapply (after_isid_inv_sn … Hg ?) -Hg
- /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
- elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
- elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
- [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
- | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
- ]
-]
-qed-.
-
-lemma after_uni_one_dx: ∀f2,f. ⫯f2 ⊚ 𝐔❨↑O❩ ≘ f → 𝐔❨↑O❩ ⊚ f2 ≘ f.
-#f2 #f #H @(after_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
-qed.
-
-lemma after_uni_one_sn: ∀f1,f. 𝐔❨↑O❩ ⊚ f1 ≘ f → ⫯f1 ⊚ 𝐔❨↑O❩ ≘ f.
-/3 width=3 by after_uni_succ_sn, at_refl/ qed-.
-