more results on sor ...

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_sor.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
index e76c1ca3c3bdd2f6f2656c709624597d1be417c6..8f38b806273e7ac3a4b9b4dd1e8adf0b31e8abc4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
@@ -72,6 +72,141 @@ try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
 /2 width=3 by ex2_intro/
 qed-.
 
+(* Advanced inversion lemmas ************************************************)
+
+lemma sor_inv_ppn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_push_next/
+qed-.
+
+lemma sor_inv_nxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f. ⫯f1 = g1 → ↑f = g → ⊥.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ elim (sor_inv_npx … H … H1 H2)
+| elim (sor_inv_nnx … H … H1 H2)
+] -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_next_push/
+qed-.
+
+lemma sor_inv_xnp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f2,f. ⫯f2 = g2 → ↑f = g → ⊥.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pnx … H … H1 H2)
+| elim (sor_inv_nnx … H … H1 H2)
+] -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_next_push/
+qed-.
+
+lemma sor_inv_ppp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_push … H) -f //
+qed-.
+
+lemma sor_inv_npn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_npx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next … H) -f //
+qed-.
+
+lemma sor_inv_pnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_pnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next … H) -f //
+qed-.
+
+lemma sor_inv_nnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_nnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next … H) -f //
+qed-.
+
+lemma sor_inv_pxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f. ↑f1 = g1 → ↑f = g →
+                   ∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ /3 width=7 by sor_inv_ppp, ex2_intro/
+| elim (sor_inv_xnp … H … H2 H0)
+]
+qed-.
+
+lemma sor_inv_xpp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f2,f. ↑f2 = g2 → ↑f = g →
+                   ∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ /3 width=7 by sor_inv_ppp, ex2_intro/
+| elim (sor_inv_nxp … H … H1 H0)
+]
+qed-.
+
+lemma sor_inv_pxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f. ↑f1 = g1 → ⫯f = g →
+                   ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ elim (sor_inv_ppn … H … H1 H2 H0)
+| /3 width=7 by sor_inv_pnn, ex2_intro/
+]
+qed-.
+
+lemma sor_inv_xpn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f2,f. ↑f2 = g2 → ⫯f = g →
+                   ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_ppn … H … H1 H2 H0)
+| /3 width=7 by sor_inv_npn, ex2_intro/
+]
+qed-.
+
+lemma sor_inv_xxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ↑f = g →
+                   ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2.
+#g1 #g2 #g #H #f #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pxp … H … H1 H0) -g /2 width=5 by ex3_2_intro/
+| elim (sor_inv_nxp … H … H1 H0)
+]
+qed-.
+
+lemma sor_inv_nxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f1,f. ⫯f1 = g1 → ⫯f = g →
+                   (∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2) ∨
+                   ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
+#g1 #g2 elim (pn_split g2) *
+/4 width=7 by sor_inv_npn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
+qed-.
+
+lemma sor_inv_xnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
+                   ∀f2,f. ⫯f2 = g2 → ⫯f = g →
+                   (∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1) ∨
+                   ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
+#g1 elim (pn_split g1) *
+/4 width=7 by sor_inv_pnn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
+qed-.
+
+lemma sor_inv_xxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ⫯f = g →
+                   ∨∨ ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ↑f2 = g2
+                    | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2
+                    | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ⫯f2 = g2.
+#g1 #g2 #g #H #f #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pxn … H … H1 H0) -g
+  /3 width=5 by or3_intro1, ex3_2_intro/
+| elim (sor_inv_nxn … H … H1 H0) -g *
+  /3 width=5 by or3_intro0, or3_intro2, ex3_2_intro/
+]
+qed-.
+
 (* Main inversion lemmas ****************************************************)
 
 corec theorem sor_mono: ∀f1,f2,x,y. f1 ⋓ f2 ≡ x → f1 ⋓ f2 ≡ y → x ≗ y.
@@ -131,7 +266,7 @@ corec lemma sor_sym: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f2 ⋓ f1 ≡ f.
 [ @sor_pp | @sor_pn | @sor_np | @sor_nn ] /2 width=7 by/
 qed-.
 
-(* Properies on identity ****************************************************)
+(* Properies on test for identity *******************************************)
 
 corec lemma sor_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⋓ f2 ≡ f2.
 #f1 * -f1
@@ -145,6 +280,33 @@ corec lemma sor_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⋓ f2 ≡ f1.
 /3 width=7 by sor_pp, sor_np/
 qed.
 
+(* Inversion lemmas on test for identity ************************************)
+
+lemma sor_isid_inv_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
+/3 width=4 by sor_isid_sn, sor_mono/
+qed-.
+
+lemma sor_isid_inv_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≗ f.
+/3 width=4 by sor_isid_dx, sor_mono/
+qed-.
+
+corec lemma sor_fwd_isid1: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
+[ /4 width=6 by isid_inv_push, isid_push/ ]
+cases (isid_inv_next … Hg … H)
+qed-.
+
+corec lemma sor_fwd_isid2: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
+[ /4 width=6 by isid_inv_push, isid_push/ ]
+cases (isid_inv_next … Hg … H)
+qed-.
+
+lemma sor_inv_isid3: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
+/3 width=4 by sor_fwd_isid2, sor_fwd_isid1, conj/ qed-.
+
 (* Properties on finite colength assignment *********************************)
 
 lemma sor_fcla_ex: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡ n2 →