X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_sand.ma;h=2a2058c5942ad6573a5d240992280918fc05b781;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=85ad506a26f5582c2b84737104f65691fdeac26c;hpb=859c5cbb8ebffeddd1dd9cbc462e046b0709b4e4;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sand.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sand.ma
index 85ad506a2..2a2058c59 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sand.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sand.ma
@@ -16,23 +16,102 @@ include "ground_2/notation/relations/rintersection_3.ma".
 include "ground_2/relocation/rtmap_sle.ma".
 
 coinductive sand: relation3 rtmap rtmap rtmap ≝
-| sand_pp: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → sand g1 g2 g
-| sand_np: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → sand g1 g2 g
-| sand_pn: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → sand g1 g2 g
-| sand_nn: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → sand g1 g2 g
+| sand_pp: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → sand g1 g2 g
+| sand_np: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → sand g1 g2 g
+| sand_pn: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → sand g1 g2 g
+| sand_nn: ∀f1,f2,f,g1,g2,g. sand f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → sand g1 g2 g
 .
 
 interpretation "intersection (rtmap)"
    'RIntersection f1 f2 f = (sand f1 f2 f).
 
+(* Basic inversion lemmas ***************************************************)
+
+lemma sand_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 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
+try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
+try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
+try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma sand_inv_npx: ∀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 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
+try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
+try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
+try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma sand_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 #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
+try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
+try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
+try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma sand_inv_nnx: ∀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 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
+try elim (discr_push_next … Hx1) try elim (discr_next_push … Hx1)
+try (>(injective_push … Hx2) -x2) try (>(injective_next … Hx2) -x2)
+try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
 (* Basic properties *********************************************************)
 
-let corec sand_refl: ∀f. f ⋒ f ≡ f ≝ ?.
+corec lemma sand_eq_repl_back1: ∀f2,f. eq_repl_back … (λf1. f1 ⋒ f2 ≘ f).
+#f2 #f #f1 * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x #Hx
+try cases (eq_inv_px … Hx … H1) try cases (eq_inv_nx … Hx … H1) -g1
+/3 width=7 by sand_pp, sand_np, sand_pn, sand_nn/
+qed-.
+
+lemma sand_eq_repl_fwd1: ∀f2,f. eq_repl_fwd … (λf1. f1 ⋒ f2 ≘ f).
+#f2 #f @eq_repl_sym /2 width=3 by sand_eq_repl_back1/
+qed-.
+
+corec lemma sand_eq_repl_back2: ∀f1,f. eq_repl_back … (λf2. f1 ⋒ f2 ≘ f).
+#f1 #f #f2 * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
+try cases (eq_inv_px … Hx … H2) try cases (eq_inv_nx … Hx … H2) -g2
+/3 width=7 by sand_pp, sand_np, sand_pn, sand_nn/
+qed-.
+
+lemma sand_eq_repl_fwd2: ∀f1,f. eq_repl_fwd … (λf2. f1 ⋒ f2 ≘ f).
+#f1 #f @eq_repl_sym /2 width=3 by sand_eq_repl_back2/
+qed-.
+
+corec lemma sand_eq_repl_back3: ∀f1,f2. eq_repl_back … (λf. f1 ⋒ f2 ≘ f).
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
+try cases (eq_inv_px … Hx … H0) try cases (eq_inv_nx … Hx … H0) -g
+/3 width=7 by sand_pp, sand_np, sand_pn, sand_nn/
+qed-.
+
+lemma sand_eq_repl_fwd3: ∀f1,f2. eq_repl_fwd … (λf. f1 ⋒ f2 ≘ f).
+#f1 #f2 @eq_repl_sym /2 width=3 by sand_eq_repl_back3/
+qed-.
+
+corec lemma sand_refl: ∀f. f ⋒ f ≘ f.
 #f cases (pn_split f) * #g #H
 [ @(sand_pp … H H H) | @(sand_nn … H H H) ] -H //
 qed.
 
-let corec sand_sym: ∀f1,f2,f. f1 ⋒ f2 ≡ f → f2 ⋒ f1 ≡ f ≝ ?.
+corec lemma sand_sym: ∀f1,f2,f. f1 ⋒ f2 ≘ f → f2 ⋒ f1 ≘ f.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf * * * -g1 -g2 -g
 [ @sand_pp | @sand_pn | @sand_np | @sand_nn ] /2 width=7 by/