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 ]
-[4,11,18,25: @sand_sym // |1,2,3,8,9,10,15,16,17,22,23,24: skip |*: // ]
+[ @sand_pp | @sand_pn | @sand_np | @sand_nn ] /2 width=7 by/
qed-.