+++ /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/relocation/gr_after_eq.ma".
-
-(* RELATIONAL COMPOSITION FOR GENERIC RELOCATION MAPS ***********************)
-
-(* Main constructions *******************************************************)
-
-(*** after_trans1 *)
-corec theorem gr_after_trans_sn:
- ∀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 (gr_after_inv_push … Hg0 … H0) -g0
- #f1 #f2 #Hf0 #H1 #H2
- cases (gr_after_inv_push_bi … Hg … H2 H3) -g2 -g3
- #f #Hf #H /3 width=7 by gr_after_refl/
-| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
- cases (gr_after_inv_push … Hg0 … H0) -g0
- #f1 #f2 #Hf0 #H1 #H2
- cases (gr_after_inv_push_next … Hg … H2 H3) -g2 -g3
- #f #Hf #H /3 width=7 by gr_after_push/
-| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
- cases (gr_after_inv_next … Hg0 … H0) -g0 *
- [ #f1 #f2 #Hf0 #H1 #H2
- cases (gr_after_inv_next_sn … Hg … H2) -g2
- #f #Hf #H /3 width=7 by gr_after_push/
- | #f1 #Hf0 #H1 /3 width=6 by gr_after_next/
- ]
-]
-qed-.
-
-(*** after_trans2 *)
-corec theorem gr_after_trans_dx:
- ∀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 (gr_after_inv_push … Hg0 … H0) -g0
- #f2 #f3 #Hf0 #H2 #H3
- cases (gr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=7 by gr_after_refl/
-| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
- cases (gr_after_inv_next … Hg0 … H0) -g0 *
- [ #f2 #f3 #Hf0 #H2 #H3
- cases (gr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=7 by gr_after_push/
- | #f2 #Hf0 #H2
- cases (gr_after_inv_push_next … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=6 by gr_after_next/
- ]
-| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
- cases (gr_after_inv_next_sn … Hg … H1) -g1
- #f #Hg #H /3 width=6 by gr_after_next/
-]
-qed-.
-
-(* Main inversions **********************************************************)
-
-(*** after_mono *)
-corec theorem gr_after_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 (gr_after_inv_push_bi … Hy … H1 H2) -g1 -g2 /3 width=8 by gr_eq_push/
-| cases (gr_after_inv_push_next … Hy … H1 H2) -g1 -g2 /3 width=8 by gr_eq_next/
-| cases (gr_after_inv_next_sn … Hy … H1) -g1 /3 width=8 by gr_eq_next/
-]
-qed-.
-
-(*** after_mono_eq *)
-lemma gr_after_mono_eq:
- ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
- f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
-/4 width=4 by gr_after_mono, gr_after_eq_repl_back_dx, gr_after_eq_repl_back_sn/ qed-.