Length of product

FiniteSets/kuratowski/length.v

@@ -1,7 +1,7 @@
 Require Import HoTT HitTactics prelude interfaces.lattice_interface interfaces.lattice_examples.
 Require Import kuratowski.operations kuratowski.properties kuratowski.kuratowski_sets isomorphism.
 
-Section Length.
+Section length.
   Context {A : Type} `{MerelyDecidablePaths A} `{Univalence}.
   
   Definition length : FSet A -> nat.
@@ -148,5 +148,83 @@ Section Length.
     rewrite plus_assoc.
     reflexivity.
   Defined.
+End length.
+
+Section length_product.
+  Context {A B : Type} `{MerelyDecidablePaths A} `{MerelyDecidablePaths B} `{Univalence}.
   
-End Length.
+  Theorem length_singleproduct (a : A) (X : FSet B)
+    : length (single_product a X) = length X.
+  Proof.
+    simple refine (FSet_cons_ind (fun Z => _) _ _ _ _ _ X)
+    ; try (intros ; apply path_ishprop) ; simpl.
+    - reflexivity.
+    - intros b X1 HX1.
+      rewrite ?length_compute, ?HX1.
+      enough(b ∈_d X1 = (a, b) ∈_d (single_product a X1)) as HE.
+      { rewrite HE ; reflexivity. }
+      rewrite singleproduct_isIn_d_aa ; try reflexivity.
+  Defined.  
+
+  Open Scope nat.
+
+  Lemma single_product_disjoint (a : A) (X1 : FSet A) (Y : FSet B)
+        : sum (prod (a ∈_d X1 = true)
+                        ((single_product a Y) ∪ (product X1 Y) = (product X1 Y)))
+                  (prod (a ∈_d X1 = false)
+                        (disjoint (single_product a Y) (product X1 Y))).
+  Proof.
+    pose (b := a ∈_d X1).
+    assert (a ∈_d X1 = b) as HaX1.
+    { reflexivity. }
+    destruct b.
+    * refine (inl(HaX1,_)).
+      apply ext.
+      intros [a1 b1].
+      rewrite ?union_isIn_d.
+      unfold member_dec, fset_member_bool in *.
+      destruct (dec ((a1, b1) ∈ (single_product a Y))) as [t | t]
+      ; destruct (dec ((a1, b1) ∈ (product X1 Y))) as [p | p]
+      ; try reflexivity.
+      rewrite singleproduct_isIn in t.
+      destruct t as [t1 t2].
+      rewrite product_isIn in p.
+      strip_truncations.
+      rewrite <- t1 in HaX1.
+      destruct (dec (a1 ∈ X1)) ; try (contradiction false_ne_true).
+      contradiction (p(t,t2)).
+    * refine (inr(HaX1,_)).
+      apply ext.
+      intros [a1 b1].
+      rewrite intersection_isIn_d, empty_isIn_d.
+      unfold member_dec, fset_member_bool in *.
+      destruct (dec ((a1, b1) ∈ (single_product a Y))) as [t | t]
+      ; destruct (dec ((a1, b1) ∈ (product X1 Y))) as [p | p]
+      ; try reflexivity.
+      rewrite singleproduct_isIn in t ; destruct t as [t1 t2].
+      rewrite product_isIn in p ; destruct p as [p1 p2].
+      strip_truncations.
+      rewrite t1 in p1.
+      destruct (dec (a ∈ X1)).
+      ** contradiction true_ne_false.
+      ** contradiction (n p1).
+  Defined.
+    
+  Theorem length_product (X : FSet A) (Y : FSet B)
+    : length (product X Y) = length X * length Y.
+  Proof.
+    simple refine (FSet_cons_ind (fun Z => _) _ _ _ _ _ X)
+    ; try (intros ; apply path_ishprop) ; simpl.
+    - reflexivity.
+    - intros a X1 HX1.
+      rewrite length_compute.
+      destruct (single_product_disjoint a X1 Y) as [[p1 p2] | [p1 p2]].
+      * rewrite p2.
+        destruct (a ∈_d X1).
+        ** apply HX1.
+        ** contradiction false_ne_true.
+      * rewrite p1, length_disjoint, HX1 ; try assumption.
+        rewrite length_singleproduct.
+        reflexivity.
+  Defined.
+End length_product.

FiniteSets/kuratowski/properties.v

@@ -52,7 +52,7 @@ Section properties_membership.
 
   Context {B : Type}.
 
-  Lemma isIn_singleproduct (a : A) (b : B) (c : A) : forall (Y : FSet B),
+  Lemma singleproduct_isIn (a : A) (b : B) (c : A) : forall (Y : FSet B),
       (a, b) ∈ (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (b ∈ Y).
   Proof.
     hinduction ; try (intros ; apply path_ishprop).
@@ -91,13 +91,13 @@ Section properties_membership.
            split ; try (apply (tr H1)) ; try (apply Hb2).
   Defined.
 
-  Definition isIn_product (a : A) (b : B) (X : FSet A) (Y : FSet B) :
+  Definition product_isIn (a : A) (b : B) (X : FSet A) (Y : FSet B) :
     (a,b) ∈ (product X Y) = land (a ∈ X) (b ∈ Y).
   Proof.
     hinduction X ; try (intros ; apply path_ishprop).
     - apply path_hprop ; symmetry ; apply prod_empty_l.
     - intros.
-      apply isIn_singleproduct.
+      apply singleproduct_isIn.
     - intros X1 X2 HX1 HX2.
       cbn.
       rewrite HX1, HX2.
@@ -324,7 +324,7 @@ Section properties_membership_decidable.
     a ∈_d (difference X Y) = andb (a ∈_d X) (negb (a ∈_d Y)).
   Proof.
     apply comprehension_isIn_d.
-  Defined.
+  Defined.  
   
   Lemma singleton_isIn_d `{IsHSet A} (a b : A) :
     a ∈ {|b|} -> a = b.
@@ -335,6 +335,53 @@ Section properties_membership_decidable.
   Defined.
 End properties_membership_decidable.
 
+Section product_membership.
+  Context {A B : Type} `{MerelyDecidablePaths A} `{MerelyDecidablePaths B} `{Univalence}.
+  
+  Lemma singleproduct_isIn_d_aa (a : A) (b : B) (c : A) (p : c = a) (Y : FSet B) :
+      (a, b) ∈_d (single_product c Y) = b ∈_d Y.
+  Proof.
+    unfold member_dec, fset_member_bool, dec ; simpl.
+    destruct (isIn_decidable (a, b) (single_product c Y)) as [t | t]
+    ; destruct (isIn_decidable b Y) as [n | n]
+    ; try reflexivity.
+    * rewrite singleproduct_isIn in t.
+      destruct t as [t1 t2].
+      contradiction (n t2).
+    * rewrite singleproduct_isIn in t.
+      contradiction (t (tr(p^),n)).
+  Defined.
+
+  Lemma singleproduct_isIn_d_false (a : A) (b : B) (c : A) (p : c = a -> Empty) (Y : FSet B) :
+    (a, b) ∈_d (single_product c Y) = false.
+  Proof.
+    unfold member_dec, fset_member_bool, dec ; simpl.
+    destruct (isIn_decidable (a, b) (single_product c Y)) as [t | t]
+    ; destruct (isIn_decidable b Y) as [n | n]
+    ; try reflexivity.
+    * rewrite singleproduct_isIn in t.
+      destruct t as [t1 t2].
+      strip_truncations.
+      contradiction (p t1^).
+    * rewrite singleproduct_isIn in t.
+      contradiction (n (snd t)).
+  Defined.
+
+  Lemma product_isIn_d (a : A) (b : B) (X : FSet A) (Y : FSet B) :
+    (a, b) ∈_d (product X Y) = andb (a ∈_d X) (b ∈_d Y).
+  Proof.
+    unfold member_dec, fset_member_bool, dec ; simpl.
+    destruct (isIn_decidable (a, b) (product X Y)) as [t | t]
+    ; destruct (isIn_decidable a X) as [n1 | n1]
+    ; destruct (isIn_decidable b Y) as [n2 | n2]
+    ; try reflexivity
+    ; rewrite ?product_isIn in t
+    ; try (destruct t as [t1 t2]
+           ; contradiction (n1 t1) || contradiction (n2 t2)).
+    contradiction (t (n1, n2)).
+  Defined.
+End product_membership.  
+  
 (* Some suporting tactics *)
 Ltac simplify_isIn_d :=
   repeat (rewrite union_isIn_d

FiniteSets/prelude.v

@@ -35,4 +35,29 @@ Proof.
   - refine (inr(fun p => _)).
     strip_truncations.
     apply (n p).
-Defined.
+Defined.
+
+Global Instance merely_decidable_paths_prod (A B : Type)
+       `{MerelyDecidablePaths A} `{MerelyDecidablePaths B}
+  : MerelyDecidablePaths(A * B).
+Proof.
+  intros x y.
+  destruct (m_dec_path (fst x) (fst y)) as [p1 | n1],
+                                           (m_dec_path (snd x) (snd y)) as [p2 | n2].
+  - apply inl.
+    strip_truncations.
+    apply tr.
+    apply path_prod ; assumption.
+  - apply inr.
+    intros pp.
+    strip_truncations.
+    apply (n2 (tr (ap snd pp))).
+  - apply inr.
+    intros pp.
+    strip_truncations.
+    apply (n1 (tr (ap fst pp))).
+  - apply inr.
+    intros pp.
+    strip_truncations.
+    apply (n1 (tr (ap fst pp))).
+Defined.