include "ground_2/relocation/rtmap_sle.ma".
coinductive sand: relation3 rtmap rtmap rtmap ≝
-| sand_pp: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sand g1 g2 g
-| sand_np: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sand g1 g2 g
-| sand_pn: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → sand g1 g2 g
-| sand_nn: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sand g1 g2 g
+| sand_pp: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sand g1 g2 g
+| sand_np: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sand g1 g2 g
+| sand_pn: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → sand g1 g2 g
+| sand_nn: â\88\80f1,f2,f,g1,g2,g. sand f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sand g1 g2 g
.
interpretation "intersection (rtmap)"
(* Basic inversion lemmas ***************************************************)
-lemma sand_inv_ppx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
- â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 f & â\86\91f = g.
+lemma sand_inv_ppx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
+ â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 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)
/2 width=3 by ex2_intro/
qed-.
-lemma sand_inv_npx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
- â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 f & â\86\91f = g.
+lemma sand_inv_npx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
+ â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 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)
/2 width=3 by ex2_intro/
qed-.
-lemma sand_inv_pnx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
- â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 f & â\86\91f = g.
+lemma sand_inv_pnx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
+ â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 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)
/2 width=3 by ex2_intro/
qed-.
-lemma sand_inv_nnx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
- â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 f & ⫯f = g.
+lemma sand_inv_nnx: â\88\80g1,g2,g. g1 â\8b\92 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
+ â\88\83â\88\83f. f1 â\8b\92 f2 â\89\98 f & â\86\91f = 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)