milestone update in ground, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / lib / exteq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/lib/exteq.ma b/matita/matita/contribs/lambdadelta/ground/lib/exteq.ma
index d62edd8ede2d6c83f218507176b313f7f874fe64..6e39281188722a81545cf2b47572a1e0003f8f2f 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/lib/exteq.ma
+++ b/matita/matita/contribs/lambdadelta/ground/lib/exteq.ma
@@ -18,12 +18,13 @@ include "ground/lib/relations.ma".
 (* EXTENSIONAL EQUIVALENCE **************************************************)
 
 definition exteq (A,B:Type[0]): relation (A → B) ≝
-                                λf1,f2. ∀a. f1 a = f2 a.
+           λf1,f2. ∀a. f1 a = f2 a.
 
-interpretation "extensional equivalence"
-   'DotEq A B f1 f2 = (exteq A B f1 f2).
+interpretation
+  "extensional equivalence"
+  'DotEq A B f1 f2 = (exteq A B f1 f2).
 
-(* Basic_properties *********************************************************)
+(* Basic Constructions ******************************************************)
 
 lemma exteq_refl (A) (B): reflexive … (exteq A B).
 // qed.
@@ -36,3 +37,37 @@ lemma exteq_sym (A) (B): symmetric … (exteq A B).
 
 lemma exteq_trans (A) (B): Transitive … (exteq A B).
 /2 width=1 by exteq_repl/ qed-.
+
+lemma exteq_canc_sn (A) (B): left_cancellable … (exteq A B).
+/2 width=1 by exteq_repl/ qed-.
+
+lemma exteq_canc_dx (A) (B): right_cancellable … (exteq A B).
+/2 width=1 by exteq_repl/ qed-.
+
+(* Constructions with function composition **********************************)
+
+lemma compose_repl_fwd_dx (A) (B) (C) (g) (f1) (f2):
+      f1 ≐{A,B} f2 → g ∘ f1 ≐{A,C} g ∘ f2.
+#A #B #C #g #f1 #f2 #Hf12 #a
+whd in ⊢ (??%%); //
+qed.
+
+lemma compose_repl_bak_dx (A) (B) (C) (g) (f1) (f2):
+      f1 ≐{A,B} f2 → g ∘ f2 ≐{A,C} g ∘ f1.
+/3 width=1 by compose_repl_fwd_dx, exteq_sym/
+qed.
+
+lemma compose_repl_fwd_sn (A) (B) (C) (g1) (g2) (f):
+      g1 ≐{B,C} g2 → g1 ∘ f ≐{A,C} g2 ∘ f.
+#A #B #C #g1 #g2 #f #Hg12 #a
+whd in ⊢ (??%%); //
+qed.
+
+lemma compose_repl_bak_sn (A) (B) (C) (g1) (g2) (f):
+      g1 ≐{B,C} g2 → g2 ∘ f ≐{A,C} g1 ∘ f.
+/3 width=1 by compose_repl_fwd_sn, exteq_sym/
+qed.
+
+lemma compose_assoc (A1) (A2) (A3) (A4) (f3:A3→A4) (f2:A2→A3) (f1:A1→A2):
+      f3 ∘ (f2 ∘ f1) ≐ f3 ∘ f2 ∘ f1.
+// qed.