--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/relations/rcoafter_3.ma".
+include "ground_2/relocation/rtmap_sor.ma".
+include "ground_2/relocation/rtmap_istot.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+coinductive coafter: relation3 rtmap rtmap rtmap ≝
+| coafter_refl: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
+ ↑f1 = g1 → ↑f2 = g2 → ↑f = g → coafter g1 g2 g
+| coafter_push: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
+ ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → coafter g1 g2 g
+| coafter_next: ∀f1,f2,f,g1,g. coafter f1 f2 f →
+ ⫯f1 = g1 → ↑f = g → coafter g1 f2 g
+.
+
+interpretation "relational co-composition (rtmap)"
+ 'RCoAfter f1 f2 f = (coafter f1 f2 f).
+
+definition H_coafter_inj: predicate rtmap ≝
+ λf1. 𝐓⦃f1⦄ →
+ ∀f,f21,f22. f1 ~⊚ f21 ≡ f → f1 ~⊚ f22 ≡ f → f21 ≗ f22.
+
+definition H_coafter_fwd_isid2: predicate rtmap ≝
+ λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
+ ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
+ >(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
+ /2 width=3 by ex2_intro/
+| #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+ elim (discr_push_next … Hx2)
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+ elim (discr_push_next … Hx1)
+]
+qed-.
+
+lemma coafter_inv_pnx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
+ ∃∃f. f1 ~⊚ f2 ≡ f & ⫯f = g.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+ elim (discr_next_push … Hx2)
+| #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
+ >(injective_push … Hx1) >(injective_next … Hx2) -x2 -x1
+ /2 width=3 by ex2_intro/
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+ elim (discr_push_next … Hx1)
+]
+qed-.
+
+lemma coafter_inv_nxx: ∀g1,f2,g. g1 ~⊚ f2 ≡ g → ∀f1. ⫯f1 = g1 →
+ ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
+#g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+ elim (discr_next_push … Hx1)
+| #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+ elim (discr_next_push … Hx1)
+| #g #Hf #H1 #H #x1 #Hx1 destruct
+ >(injective_next … Hx1) -x1
+ /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma coafter_inv_ppp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ~⊚ f2 ≡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
+elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
+<(injective_push … Hx) -f //
+qed-.
+
+lemma coafter_inv_ppn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
+elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
+elim (discr_push_next … Hx)
+qed-.
+
+lemma coafter_inv_pnn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ~⊚ f2 ≡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
+elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
+<(injective_next … Hx) -f //
+qed-.
+
+lemma coafter_inv_pnp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
+elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
+elim (discr_next_push … Hx)
+qed-.
+
+lemma coafter_inv_nxp: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
+ ∀f1,f. ⫯f1 = g1 → ↑f = g → f1 ~⊚ f2 ≡ f.
+#g1 #f2 #g #Hg #f1 #f #H1 #H
+elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
+<(injective_push … Hx) -f //
+qed-.
+
+lemma coafter_inv_nxn: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
+ ∀f1,f. ⫯f1 = g1 → ⫯f = g → ⊥.
+#g1 #f2 #g #Hg #f1 #f #H1 #H
+elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
+elim (discr_push_next … Hx)
+qed-.
+
+lemma coafter_inv_pxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f. ↑f1 = g1 → ↑f = g →
+ ∃∃f2. f1 ~⊚ f2 ≡ f & ↑f2 = g2.
+#g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
+[ lapply (coafter_inv_ppp … Hg … H1 H2 H) -g1 -g /2 width=3 by ex2_intro/
+| elim (coafter_inv_pnp … Hg … H1 H2 H)
+]
+qed-.
+
+lemma coafter_inv_pxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ ∀f1,f. ↑f1 = g1 → ⫯f = g →
+ ∃∃f2. f1 ~⊚ f2 ≡ f & ⫯f2 = g2.
+#g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
+[ elim (coafter_inv_ppn … Hg … H1 H2 H)
+| lapply (coafter_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma coafter_inv_xxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ⫯f = g →
+ ∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2.
+#g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
+[ elim (coafter_inv_pxn … Hg … H1 H) -g /2 width=5 by ex3_2_intro/
+| elim (coafter_inv_nxn … Hg … H1 H)
+]
+qed-.
+
+lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ↑f = g →
+ (∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2) ∨
+ ∃∃f1. f1 ~⊚ g2 ≡ f & ⫯f1 = g1.
+#g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
+[ elim (coafter_inv_pxp … Hg … H1 H) -g
+ /3 width=5 by or_introl, ex3_2_intro/
+| /4 width=5 by coafter_inv_nxp, or_intror, ex2_intro/
+]
+qed-.
+
+lemma coafter_inv_pxx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1. ↑f1 = g1 →
+ (∃∃f2,f. f1 ~⊚ f2 ≡ f & ↑f2 = g2 & ↑f = g) ∨
+ (∃∃f2,f. f1 ~⊚ f2 ≡ f & ⫯f2 = g2 & ⫯f = g).
+#g1 #g2 #g #Hg #f1 #H1 elim (pn_split g2) * #f2 #H2
+[ elim (coafter_inv_ppx … Hg … H1 H2) -g1
+ /3 width=5 by or_introl, ex3_2_intro/
+| elim (coafter_inv_pnx … Hg … H1 H2) -g1
+ /3 width=5 by or_intror, ex3_2_intro/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≡ f).
+#f1 #f #f2 * -f2 -f1 -f
+#f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
+[ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_refl/
+| cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_push/
+| cases (eq_inv_nx … H0 … H21) -g21 /3 width=5 by coafter_next/
+]
+qed-.
+
+lemma coafter_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ~⊚ f1 ≡ f).
+#f1 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back2/
+qed-.
+
+corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≡ f).
+#f2 #f #f1 * -f2 -f1 -f
+#f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
+[ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by coafter_refl/
+| cases (eq_inv_nx … H0 … H11) -g11 /3 width=7 by coafter_push/
+| @(coafter_next … H2 H) /2 width=5 by/
+]
+qed-.
+
+lemma coafter_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ~⊚ f1 ≡ f).
+#f2 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back1/
+qed-.
+
+corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≡ f).
+#f2 #f1 #f * -f2 -f1 -f
+#f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
+[ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by coafter_refl/
+| cases (eq_inv_nx … H0 … H01) -g01 /3 width=7 by coafter_push/
+| cases (eq_inv_px … H0 … H01) -g01 /3 width=5 by coafter_next/
+]
+qed-.
+
+lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≡ f).
+#f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
+qed-.
+
+(* Main properties **********************************************************)
+(*
+corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
+ ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
+ ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by coafter_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+ [ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_nxx … Hg … H2) -g2
+ #f #Hf #H /3 width=7 by coafter_push/
+ | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
+ ]
+]
+qed-.
+
+corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
+ ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
+ ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f2 #f3 #Hf0 #H2 #H3
+ cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+ [ #f2 #f3 #Hf0 #H2 #H3
+ cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by coafter_push/
+ | #f2 #Hf0 #H2
+ cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=6 by coafter_next/
+ ]
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+ cases (coafter_inv_nxx … Hg … H1) -g1
+ #f #Hg #H /3 width=6 by coafter_next/
+]
+qed-.
+*)
+(* Main inversion lemmas ****************************************************)
+
+corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
+#f1 #f2 #x #y * -f1 -f2 -x
+#f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
+[ cases (coafter_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
+| cases (coafter_inv_pnx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_next/
+| cases (coafter_inv_nxx … Hy … H1) -g1 /3 width=8 by eq_push/
+]
+qed-.
+
+lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
+ f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
+/4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
+
+(* Properties on tls ********************************************************)
+
+lemma coafter_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
+ f1 ~⊚ f2 ≡ f → ⫱*[n]f1 ~⊚ f2 ≡ ⫱*[n]f.
+#n elim n -n //
+#n #IH #f1 #f2 #f #Hf1 #Hf
+cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
+cases (coafter_inv_nxx … Hf … H1) -Hf /2 width=1 by/
+qed.
+
+(* Properties on isid *******************************************************)
+
+corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≡ f2.
+#f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
+/3 width=7 by coafter_push, coafter_refl/
+qed.
+
+corec lemma coafter_isid_dx: ∀f2,f. 𝐈⦃f2⦄ → 𝐈⦃f⦄ → ∀f1. f1 ~⊚ f2 ≡ f.
+#f2 #f * -f2 #f2 #g2 #Hf2 #H2 * -f #f #g #Hf #H #f1 cases (pn_split f1) * #g1 #H1
+[ /3 width=7 by coafter_refl/
+| @(coafter_next … H1 … H) /3 width=3 by isid_push/
+]
+qed.
+
+(* Inversion lemmas on isid *************************************************)
+
+lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
+/3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
+
+lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f2⦄ → 𝐈⦃f⦄.
+/4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
+(*
+(* Properties on isuni ******************************************************)
+
+lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ⫯f2 ≡ ⫯f1.
+#f1 #f2 #Hf2 #H elim H -H
+/5 width=7 by coafter_isid_dx, coafter_eq_repl_back2, coafter_next, coafter_push, eq_push_inv_isid/
+qed.
+
+lemma coafter_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ~⊚ f1 ≡ f → f2 ~⊚ ⫯f1 ≡ f.
+#f2 #H elim H -f2
+[ #f2 #Hf2 #f1 #f #Hf
+ elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+ /4 width=7 by coafter_isid_inv_sn, coafter_isid_sn, coafter_eq_repl_back0, eq_next/
+| #f2 #_ #g2 #H2 #IH #f1 #f #Hf
+ elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+ /3 width=5 by coafter_next/
+]
+qed.
+
+(* Properties on uni ********************************************************)
+
+lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+@nat_elim2
+/4 width=5 by coafter_uni_next2, coafter_isid_sn, coafter_isid_dx, coafter_next/
+qed.
+
+(* Forward lemmas on at *****************************************************)
+
+lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡ f →
+ ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
+#i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
+[ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
+ [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
+| elim (at_inv_xxp … Hf) -Hf //
+ #g #H1 #H
+]
+[2: elim (coafter_inv_xxn … Hf21 … H) -f *
+ [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
+|*: elim (coafter_inv_xxp … Hf21 … H) -f
+ #g2 #g1 #Hg21 #H2 #H1
+]
+[4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
+/3 width=9 by at_refl, at_push, at_next, ex2_intro/
+qed-.
+
+lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
+ ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
+#i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
+[ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
+ [ elim (at_inv_xxn … Hf1 … H22) -i2 *
+ #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
+ [ elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
+ | elim (coafter_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
+ ]
+ | elim (coafter_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
+ ]
+| elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
+ elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
+ elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
+]
+qed-.
+
+lemma coafter_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
+ ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
+#f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (coafter_at_fwd … Hf … H) -f
+#j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
+qed-.
+
+lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
+ ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
+#i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
+[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ #g [ #j1 ] #Hg [ #H01 ] #H00
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
+ #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
+ [ elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
+ | elim (coafter_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
+ | elim (coafter_inv_nxp … Hf1 … H20 H00)
+ | /4 width=9 by coafter_inv_nxn, at_next/
+ ]
+| elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
+ elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
+]
+qed-.
+
+(* Properties with at *******************************************************)
+
+lemma coafter_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 (coafter_isid_inv_dx … Hf ?) -Hf
+ /3 width=3 by coafter_isid_sn, coafter_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 (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ /3 width=5 by coafter_next/
+ | #g2 #Hg2 #H2 destruct
+ elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ /3 width=5 by coafter_next/
+ ]
+]
+qed.
+
+lemma coafter_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 (coafter_isid_inv_sn … Hf ?) -Hf
+ /3 width=3 by coafter_isid_dx, coafter_eq_repl_back0/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (coafter_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 coafter_push/
+ | #g2 #Hg2 #H2 destruct /3 width=5 by coafter_next/
+ ]
+]
+qed-.
+
+lemma coafter_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 (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
+ lapply (coafter_isid_inv_dx … Hg ?) -Hg
+ /4 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct
+ elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ /3 width=5 by coafter_next/
+ | #g2 #Hg2 #H2 destruct
+ elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ /3 width=5 by coafter_next/
+ ]
+]
+qed.
+
+lemma coafter_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 (coafter_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ lapply (coafter_isid_inv_sn … Hg ?) -Hg
+ /4 width=7 by coafter_isid_dx, coafter_eq_repl_back0, coafter_push/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (coafter_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 coafter_push/
+ | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by coafter_next/
+ ]
+]
+qed-.
+
+lemma coafter_uni_one_dx: ∀f2,f. ↑f2 ~⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ~⊚ f2 ≡ f.
+#f2 #f #H @(coafter_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
+qed.
+
+lemma coafter_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ~⊚ f1 ≡ f → ↑f1 ~⊚ 𝐔❴⫯O❵ ≡ f.
+/3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
+*)
+(* Forward lemmas with istot ************************************************)
+(*
+lemma coafter_istot_fwd: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
+#f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
+#i2 #Hf1 elim (Hf2 i2) -Hf2
+/3 width=7 by coafter_fwd_at, ex_intro/
+qed-.
+
+lemma coafter_fwd_istot_dx: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i2 #Hf elim (coafter_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
+qed-.
+
+lemma coafter_fwd_istot_sn: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i #Hf elim (coafter_at_fwd … Hf … H) -f
+#i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
+#Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
+qed-.
+
+lemma coafter_inv_istot: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
+/3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
+
+lemma coafter_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ~⊚ f1 ≡ f →
+ ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
+#f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
+/3 width=8 by coafter_fwd_at, ex2_intro/
+qed-.
+
+lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f1 ≗ f → 𝐈⦃f2⦄.
+#f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
+#Hf2 #Hf1 #H @isid_at_total // -Hf2
+#i2 #i #Hf2 elim (Hf1 i2) -Hf1
+#i0 #Hf1 lapply (at_increasing … Hf1)
+#Hi20 lapply (coafter_fwd_at2 … i0 … Hf1 … Hf) -Hf
+/3 width=7 by at_eq_repl_back, at_mono, at_id_le/
+qed-.
+
+lemma coafter_fwd_isid_dx: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f2 ≗ f → 𝐈⦃f1⦄.
+#f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
+#Hf2 #Hf1 #H2 @isid_at_total // -Hf1
+#i1 #i2 #Hi12 elim (coafter_at1_fwd … Hi12 … Hf) -f1
+/3 width=8 by at_inj, at_eq_repl_back/
+qed-.
+*)
+corec fact coafter_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1.
+#f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
+lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
+cases (H2g1 0) #n #Hn
+cases (coafter_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
+[ cases (coafter_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
+ @(eq_push … H21 H22) -f21 -f22
+| cases (coafter_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
+ @(eq_next … H21 H22) -f21 -f22
+]
+@(coafter_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -coafter_inj_O_aux
+/2 width=1 by coafter_tls, istot_tls, at_pxx_tls/
+qed-.
+
+fact coafter_inj_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1) →
+ ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_inj f1.
+#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
+#i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
+elim (coafter_inv_nxx … H1f … H1) -H1f #g #H1g #H
+lapply (coafter_inv_nxp … H2f … H1 H) -f #H2g
+/3 width=6 by istot_inv_next/
+qed-.
+
+theorem coafter_inj: ∀f1. H_coafter_inj f1.
+#f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
+qed-.
+
+corec fact coafter_fwd_isid2_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
+ H_coafter_fwd_isid2 f1.
+#f1 #H1f1 #f2 #f #H #H2f1 #Hf
+cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
+lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
+cases (H2g1 0) #n #Hn
+cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
+[ lapply (isid_inv_push … Hf … H0) -Hf #Hg
+ @(isid_push … H2)
+ /3 width=7 by coafter_tls, istot_tls, at_pxx_tls, isid_tls/
+| cases (isid_inv_next … Hf … H0)
+]
+qed-.
+
+fact coafter_fwd_isid2_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_fwd_isid2 f1) →
+ ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_fwd_isid2 f1.
+#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
+#i2 #IH #f1 #H1f1 #f2 #f #H #H2f1 #Hf
+elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
+elim (coafter_inv_nxx … H … H1) -H #g #Hg #H0
+@(IH … Hg1 … Hg) /2 width=3 by istot_inv_next, isid_inv_push/ (**) (* full auto fails *)
+qed-.
+
+lemma coafter_fwd_isid2: ∀f1. H_coafter_fwd_isid2 f1.
+#f1 #f2 #f #Hf #H cases (H 0)
+/3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
+qed-.
+
+lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
+ ∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
+@isfin_ind
+[ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #H2f
+ elim (sor_inv_isid3 … H2f) -H2f //
+ lapply (coafter_fwd_isid2 … H1f ??) -H1f //
+ /3 width=5 by ex3_2_intro, coafter_isid_dx, sor_isid/
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
+ elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #H2f
+ elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ] #g2 [ #g1 ] #H1f #Hgf2
+ [ lapply (istot_inv_push … Hf2 … Hgf2) | lapply (istot_inv_next … Hf2 … Hgf2) ] -Hf2 #Hg2
+ elim (IH … Hg2 … H1f … H2f) -f -Hg2
+ /3 width=11 by sor_pp, ex3_2_intro, coafter_refl, coafter_next/
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #fa #fb #H2
+ elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #H1f #Hgf2
+ lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #H2f
+ elim (IH … Hg2 … H1f … H2f) -f -Hg2
+ /3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
+]
+qed-.
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