update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / pr_nat.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_nat.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_nat.ma
index 13154a1bb7249d1775c4c65771216de4f896d5c8..afe21c5e5ee896e3c025687a186702ea23a7cfdb 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/relocation/pr_nat.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/pr_nat.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/notation/relations/ratsucc_3.ma".
+include "ground/notation/relations/ratsection_3.ma".
 include "ground/arith/nat_pred_succ.ma".
 include "ground/relocation/pr_pat.ma".
 
@@ -23,30 +23,30 @@ definition pr_nat: relation3 pr_map nat nat ≝
 
 interpretation
   "relational non-negative application (partial relocation maps)"
-  'RAtSucc l1 f l2 = (pr_nat f l1 l2).
+  'RAtSection l1 f l2 = (pr_nat f l1 l2).
 
 (* Basic constructions ******************************************************)
 
 lemma pr_nat_refl (f) (g) (k1) (k2):
-      (⫯f) = g → 𝟎 = k1 → 𝟎 = k2 → @↑❨k1,g❩ ≘ k2.
+      (⫯f) = g → 𝟎 = k1 → 𝟎 = k2 → ＠§❨k1,g❩ ≘ k2.
 #f #g #k1 #k2 #H1 #H2 #H3 destruct
 /2 width=2 by pr_pat_refl/
 qed.
 
 lemma pr_nat_push (f) (l1) (l2) (g) (k1) (k2):
-      @↑❨l1,f❩ ≘ l2 → ⫯f = g → ↑l1 = k1 → ↑l2 = k2 → @↑❨k1,g❩ ≘ k2.
+      ＠§❨l1,f❩ ≘ l2 → ⫯f = g → ↑l1 = k1 → ↑l2 = k2 → ＠§❨k1,g❩ ≘ k2.
 #f #l1 #l2 #g #k1 #k2 #Hf #H1 #H2 #H3 destruct
 /2 width=7 by pr_pat_push/
 qed.
 
 lemma pr_nat_next (f) (l1) (l2) (g) (k2):
-      @↑❨l1,f❩ ≘ l2 → ↑f = g → ↑l2 = k2 → @↑❨l1,g❩ ≘ k2.
+      ＠§❨l1,f❩ ≘ l2 → ↑f = g → ↑l2 = k2 → ＠§❨l1,g❩ ≘ k2.
 #f #l1 #l2 #g #k2 #Hf #H1 #H2 destruct
 /2 width=5 by pr_pat_next/
 qed.
 
 lemma pr_nat_pred_bi (f) (i1) (i2):
-      ＠⧣❨i1,f❩ ≘ i2 → @↑❨↓i1,f❩ ≘ ↓i2.
+      ＠⧣❨i1,f❩ ≘ i2 → ＠§❨↓i1,f❩ ≘ ↓i2.
 #f #i1 #i2
 >(npsucc_pred i1) in ⊢ (%→?); >(npsucc_pred i2) in ⊢ (%→?);
 //
@@ -56,7 +56,7 @@ qed.
 
 (*** pr_nat_inv_ppx *)
 lemma pr_nat_inv_zero_push (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g. 𝟎 = l1 → ⫯g = f → 𝟎 = l2.
+      ＠§❨l1,f❩ ≘ l2 → ∀g. 𝟎 = l1 → ⫯g = f → 𝟎 = l2.
 #f #l1 #l2 #H #g #H1 #H2 destruct
 lapply (pr_pat_inv_unit_push … H ???) -H
 /2 width=2 by eq_inv_npsucc_bi/
@@ -64,8 +64,8 @@ qed-.
 
 (*** pr_nat_inv_npx *)
 lemma pr_nat_inv_succ_push (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k1. ↑k1 = l1 → ⫯g = f →
-      ∃∃k2. @↑❨k1,g❩ ≘ k2 & ↑k2 = l2.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k1. ↑k1 = l1 → ⫯g = f →
+      ∃∃k2. ＠§❨k1,g❩ ≘ k2 & ↑k2 = l2.
 #f #l1 #l2 #H #g #k1 #H1 #H2 destruct
 elim (pr_pat_inv_succ_push … H) -H [|*: // ] #k2 #Hg
 >(npsucc_pred (↑l2)) #H
@@ -74,8 +74,8 @@ qed-.
 
 (*** pr_nat_inv_xnx *)
 lemma pr_nat_inv_next (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g. ↑g = f →
-      ∃∃k2. @↑❨l1,g❩ ≘ k2 & ↑k2 = l2.
+      ＠§❨l1,f❩ ≘ l2 → ∀g. ↑g = f →
+      ∃∃k2. ＠§❨l1,g❩ ≘ k2 & ↑k2 = l2.
 #f #l1 #l2 #H #g #H1 destruct
 elim (pr_pat_inv_next … H) -H [|*: // ] #k2
 >(npsucc_pred (k2)) in ⊢ (%→?→?); #Hg #H
@@ -86,50 +86,50 @@ qed-.
 
 (*** pr_nat_inv_ppn *)
 lemma pr_nat_inv_zero_push_succ (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k2. 𝟎 = l1 → ⫯g = f → ↑k2 = l2 → ⊥.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k2. 𝟎 = l1 → ⫯g = f → ↑k2 = l2 → ⊥.
 #f #l1 #l2 #Hf #g #k2 #H1 #H <(pr_nat_inv_zero_push … Hf … H1 H) -f -g -l1 -l2
 /2 width=3 by eq_inv_nsucc_zero/
 qed-.
 
 (*** pr_nat_inv_npp *)
 lemma pr_nat_inv_succ_push_zero (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k1. ↑k1 = l1 → ⫯g = f → 𝟎 = l2 → ⊥.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k1. ↑k1 = l1 → ⫯g = f → 𝟎 = l2 → ⊥.
 #f #l1 #l2 #Hf #g #k1 #H1 #H elim (pr_nat_inv_succ_push … Hf … H1 H) -f -l1
 #x2 #Hg * -l2 /2 width=3 by eq_inv_zero_nsucc/
 qed-.
 
 (*** pr_nat_inv_npn *)
 lemma pr_nat_inv_succ_push_succ (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k1,k2. ↑k1 = l1 → ⫯g = f → ↑k2 = l2 → @↑❨k1,g❩ ≘ k2.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k1,k2. ↑k1 = l1 → ⫯g = f → ↑k2 = l2 → ＠§❨k1,g❩ ≘ k2.
 #f #l1 #l2 #Hf #g #k1 #k2 #H1 #H elim (pr_nat_inv_succ_push … Hf … H1 H) -f -l1
 #x2 #Hg * -l2 #H >(eq_inv_nsucc_bi … H) -k2 //
 qed-.
 
 (*** pr_nat_inv_xnp *)
 lemma pr_nat_inv_next_zero (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g. ↑g = f → 𝟎 = l2 → ⊥.
+      ＠§❨l1,f❩ ≘ l2 → ∀g. ↑g = f → 𝟎 = l2 → ⊥.
 #f #l1 #l2 #Hf #g #H elim (pr_nat_inv_next … Hf … H) -f
 #x2 #Hg * -l2 /2 width=3 by eq_inv_zero_nsucc/
 qed-.
 
 (*** pr_nat_inv_xnn *)
 lemma pr_nat_inv_next_succ (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k2. ↑g = f → ↑k2 = l2 → @↑❨l1,g❩ ≘ k2.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k2. ↑g = f → ↑k2 = l2 → ＠§❨l1,g❩ ≘ k2.
 #f #l1 #l2 #Hf #g #k2 #H elim (pr_nat_inv_next … Hf … H) -f
 #x2 #Hg * -l2 #H >(eq_inv_nsucc_bi … H) -k2 //
 qed-.
 
 (*** pr_nat_inv_pxp *)
 lemma pr_nat_inv_zero_bi (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → 𝟎 = l1 → 𝟎 = l2 → ∃g. ⫯g = f.
+      ＠§❨l1,f❩ ≘ l2 → 𝟎 = l1 → 𝟎 = l2 → ∃g. ⫯g = f.
 #f elim (pr_map_split_tl … f) /2 width=2 by ex_intro/
 #H #l1 #l2 #Hf #H1 #H2 cases (pr_nat_inv_next_zero … Hf … H H2)
 qed-.
 
 (*** pr_nat_inv_pxn *)
 lemma pr_nat_inv_zero_succ (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀k2. 𝟎 = l1 → ↑k2 = l2 →
-      ∃∃g. @↑❨l1,g❩ ≘ k2 & ↑g = f.
+      ＠§❨l1,f❩ ≘ l2 → ∀k2. 𝟎 = l1 → ↑k2 = l2 →
+      ∃∃g. ＠§❨l1,g❩ ≘ k2 & ↑g = f.
 #f elim (pr_map_split_tl … f)
 #H #l1 #l2 #Hf #k2 #H1 #H2
 [ elim (pr_nat_inv_zero_push_succ … Hf … H1 H H2)
@@ -139,7 +139,7 @@ qed-.
 
 (*** pr_nat_inv_nxp *)
 lemma pr_nat_inv_succ_zero (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀k1. ↑k1 = l1 → 𝟎 = l2 → ⊥.
+      ＠§❨l1,f❩ ≘ l2 → ∀k1. ↑k1 = l1 → 𝟎 = l2 → ⊥.
 #f elim (pr_map_split_tl f)
 #H #l1 #l2 #Hf #k1 #H1 #H2
 [ elim (pr_nat_inv_succ_push_zero … Hf … H1 H H2)
@@ -149,9 +149,9 @@ qed-.
 
 (*** pr_nat_inv_nxn *)
 lemma pr_nat_inv_succ_bi (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀k1,k2. ↑k1 = l1 → ↑k2 = l2 →
-      ∨∨ ∃∃g. @↑❨k1,g❩ ≘ k2 & ⫯g = f
-       | ∃∃g. @↑❨l1,g❩ ≘ k2 & ↑g = f.
+      ＠§❨l1,f❩ ≘ l2 → ∀k1,k2. ↑k1 = l1 → ↑k2 = l2 →
+      ∨∨ ∃∃g. ＠§❨k1,g❩ ≘ k2 & ⫯g = f
+       | ∃∃g. ＠§❨l1,g❩ ≘ k2 & ↑g = f.
 #f elim (pr_map_split_tl f) *
 /4 width=7 by pr_nat_inv_next_succ, pr_nat_inv_succ_push_succ, ex2_intro, or_intror, or_introl/
 qed-.
@@ -159,9 +159,9 @@ qed-.
 (* Note: the following inversion lemmas must be checked *)
 (*** pr_nat_inv_xpx *)
 lemma pr_nat_inv_push (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g. ⫯g = f →
+      ＠§❨l1,f❩ ≘ l2 → ∀g. ⫯g = f →
       ∨∨ ∧∧ 𝟎 = l1 & 𝟎 = l2
-       | ∃∃k1,k2. @↑❨k1,g❩ ≘ k2 & ↑k1 = l1 & ↑k2 = l2.
+       | ∃∃k1,k2. ＠§❨k1,g❩ ≘ k2 & ↑k1 = l1 & ↑k2 = l2.
 #f * [2: #l1 ] #l2 #Hf #g #H
 [ elim (pr_nat_inv_succ_push … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/
 | >(pr_nat_inv_zero_push … Hf … H) -f /3 width=1 by conj, or_introl/
@@ -170,15 +170,15 @@ qed-.
 
 (*** pr_nat_inv_xpp *)
 lemma pr_nat_inv_push_zero (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g. ⫯g = f → 𝟎 = l2 → 𝟎 = l1.
+      ＠§❨l1,f❩ ≘ l2 → ∀g. ⫯g = f → 𝟎 = l2 → 𝟎 = l1.
 #f #l1 #l2 #Hf #g #H elim (pr_nat_inv_push … Hf … H) -f * //
 #k1 #k2 #_ #_ * -l2 #H elim (eq_inv_zero_nsucc … H)
 qed-.
 
 (*** pr_nat_inv_xpn *)
 lemma pr_nat_inv_push_succ (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → ∀g,k2. ⫯g = f → ↑k2 = l2 →
-      ∃∃k1. @↑❨k1,g❩ ≘ k2 & ↑k1 = l1.
+      ＠§❨l1,f❩ ≘ l2 → ∀g,k2. ⫯g = f → ↑k2 = l2 →
+      ∃∃k1. ＠§❨k1,g❩ ≘ k2 & ↑k1 = l1.
 #f #l1 #l2 #Hf #g #k2 #H elim (pr_nat_inv_push … Hf … H) -f *
 [ #_ * -l2 #H elim (eq_inv_nsucc_zero … H)
 | #x1 #x2 #Hg #H1 * -l2 #H
@@ -189,7 +189,7 @@ qed-.
 
 (*** pr_nat_inv_xxp *)
 lemma pr_nat_inv_zero_dx (f) (l1) (l2):
-      @↑❨l1,f❩ ≘ l2 → 𝟎 = l2 → ∃∃g. 𝟎 = l1 & ⫯g = f.
+      ＠§❨l1,f❩ ≘ l2 → 𝟎 = l2 → ∃∃g. 𝟎 = l1 & ⫯g = f.
 #f elim (pr_map_split_tl f)
 #H #l1 #l2 #Hf #H2
 [ /3 width=6 by pr_nat_inv_push_zero, ex2_intro/
@@ -198,9 +198,9 @@ lemma pr_nat_inv_zero_dx (f) (l1) (l2):
 qed-.
 
 (*** pr_nat_inv_xxn *)
-lemma pr_nat_inv_succ_dx (f) (l1) (l2): @↑❨l1,f❩ ≘ l2 → ∀k2.  ↑k2 = l2 →
-      ∨∨ ∃∃g,k1. @↑❨k1,g❩ ≘ k2 & ↑k1 = l1 & ⫯g = f
-       | ∃∃g. @↑❨l1,g❩ ≘ k2 & ↑g = f.
+lemma pr_nat_inv_succ_dx (f) (l1) (l2): ＠§❨l1,f❩ ≘ l2 → ∀k2.  ↑k2 = l2 →
+      ∨∨ ∃∃g,k1. ＠§❨k1,g❩ ≘ k2 & ↑k1 = l1 & ⫯g = f
+       | ∃∃g. ＠§❨l1,g❩ ≘ k2 & ↑g = f.
 #f elim (pr_map_split_tl f)
 #H #l1 #l2 #Hf #k2 #H2
 [ elim (pr_nat_inv_push_succ … Hf … H H2) -l2 /3 width=5 by or_introl, ex3_2_intro/