From 7c41d5c35418ea7129d71188f646475c2057f36e Mon Sep 17 00:00:00 2001
From: Thomas Baruchel <baruchel@gmx.com>
Date: Mon, 30 Oct 2023 17:07:08 +0100
Subject: [PATCH] Update

---
 src/subsequences.v | 64 ++++++++++++++++++++++++----------------------
 1 file changed, 34 insertions(+), 30 deletions(-)

diff --git a/src/subsequences.v b/src/subsequences.v
index 3bf0798..9a65c7c 100644
--- a/src/subsequences.v
+++ b/src/subsequences.v
@@ -23,13 +23,13 @@ Definition subsequence3 (l s : list Type) :=
   exists f, s = fst (partition f l).
  *)
 
-Theorem subsequence_nil_r : forall (l : list Type), subsequence l nil.
+Theorem subsequence_nil_r {X: Type}: forall (l : list X), subsequence l nil.
 Proof.
   intro l. exists l. exists nil. rewrite app_nil_r. split; easy.
 Qed.
 
 
-Theorem subsequence_nil_cons_r : forall (l: list Type) (a:Type),
+Theorem subsequence_nil_cons_r {X: Type}: forall (l: list X) (a:X),
   ~ subsequence nil (a::l).
 Proof.
   intros l a. unfold not. intro H.
@@ -41,7 +41,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence2_nil_r : forall (l : list Type), subsequence2 l nil.
+Theorem subsequence2_nil_r {X: Type} : forall (l : list X), subsequence2 l nil.
 Proof.
   intro l. 
   exists (repeat false (length l)). rewrite repeat_length.
@@ -50,7 +50,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence2_nil_cons_r : forall (l: list Type) (a:Type),
+Theorem subsequence2_nil_cons_r {X: Type}: forall (l: list X) (a:X),
   ~ subsequence2 nil (a::l).
 Proof.
   intros l a. unfold not. intro H. destruct H.
@@ -60,7 +60,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence_cons_l : forall (l s: list Type) (a: Type),
+Theorem subsequence_cons_l {X: Type}: forall (l s: list X) (a: X),
   subsequence l s -> subsequence (a::l) s.
 Proof.
   intros l s a. intro H.
@@ -70,7 +70,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence2_cons_l : forall (l s: list Type) (a: Type),
+Theorem subsequence2_cons_l {X: Type} : forall (l s: list X) (a: X),
   subsequence2 l s -> subsequence2 (a::l) s.
 Proof.
   intros l s a. intro H.
@@ -80,7 +80,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence_cons_r : forall (l s: list Type) (a: Type),
+Theorem subsequence_cons_r {X: Type} : forall (l s: list X) (a: X),
   subsequence l (a::s) -> subsequence l s.
 Proof.
   intros l s a. intro H. destruct H. destruct H. destruct H.
@@ -90,7 +90,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence2_cons_r : forall (l s: list Type) (a: Type),
+Theorem subsequence2_cons_r {X: Type} : forall (l s: list X) (a: X),
   subsequence2 l (a::s) -> subsequence2 l s.
 Proof.
   intro l. induction l. intros. apply subsequence2_nil_cons_r in H.
@@ -104,7 +104,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence_cons_eq : forall (l1 l2: list Type) (a: Type),
+Theorem subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
   subsequence (a::l1) (a::l2) <-> subsequence l1 l2.
 Proof.
   intros l s a. split. intro H.
@@ -115,7 +115,7 @@ Proof.
   inversion H. rewrite H3. split; reflexivity.
   destruct x0. simpl in H0.
   apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
-  exists (x ++ (a::l0)). exists x0. inversion H. rewrite H2. split. reflexivity.
+  exists (x1 ++ (a::l0)). exists x0. inversion H. rewrite H2. split. reflexivity.
   inversion H0. rewrite <- app_assoc. apply app_inv_head_iff.
   rewrite app_comm_cons. reflexivity.
 
@@ -125,7 +125,7 @@ Proof.
 Qed.
 
 
-Theorem subsequence2_cons_eq : forall (l1 l2: list Type) (a: Type),
+Theorem subsequence2_cons_eq {X: Type}: forall (l1 l2: list X) (a: X),
   subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2.
 Proof.
   intros l s a. split. intro H. unfold subsequence2 in H.
@@ -167,9 +167,9 @@ Qed.
  *)
 
 
-Theorem subsequence2_dec :
-  (forall x y : Type, {x = y} + {x <> y})
-  -> forall (l s : list Type), { subsequence2 l s } + { ~ subsequence2 l s }.
+Theorem subsequence2_dec {X: Type}:
+  (forall x y : X, {x = y} + {x <> y})
+  -> forall (l s : list X), { subsequence2 l s } + { ~ subsequence2 l s }.
 Proof.
   intro H.
   intros l. induction l. intro s. destruct s. left. apply subsequence2_nil_r.
@@ -182,25 +182,25 @@ Proof.
   apply subsequence2_cons_r in s0. left. assumption.
 
   destruct s. left. apply subsequence2_nil_r.
-  assert ({T=a}+{T<>a}). apply H. destruct H0. rewrite e.
+  assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
   destruct IHl with (s := s).
   left. rewrite subsequence2_cons_eq. assumption.
   right. rewrite subsequence2_cons_eq. assumption.
 
   right. unfold not in n. unfold not. intro I.
-  destruct I. destruct H0. destruct x.
+  destruct I. destruct H0. destruct x0.
   symmetry in H1. apply nil_cons in H1. contradiction H1.
   destruct b.
   inversion H1. rewrite H3 in n0. contradiction n0. reflexivity.
 
-  assert (subsequence2 l (T::s)). exists x. split.
+  assert (subsequence2 l (x::s)). exists x0. split.
   inversion H0. reflexivity. assumption. apply n in H2. contradiction H2.
 Qed.
 
 
-Theorem subsequence_eq_def :
-  (forall x y : Type, {x = y} + {x <> y})
-  -> (forall l s : list Type, subsequence l s <-> subsequence2 l s).
+Theorem subsequence_eq_def {X: Type} :
+  (forall (x y : X), {x = y} + {x <> y})
+  -> (forall l s : list X, subsequence l s <-> subsequence2 l s).
 Proof.
   intro I. intro l. induction l.
   (* first part of the induction *)
@@ -218,7 +218,7 @@ Proof.
   symmetry in H2. apply nil_cons in H2. contradiction H2.
   exists nil. exists nil. destruct s. simpl. easy.
   destruct H. destruct H.
-  assert (x = nil). destruct x. reflexivity. simpl in H.
+  assert (x0 = nil). destruct x0. reflexivity. simpl in H.
   apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
   rewrite H1 in H0. simpl in H0.
   symmetry in H0. apply nil_cons in H0. contradiction H0.
@@ -227,38 +227,38 @@ Proof.
   exists (repeat false (S (length l))). rewrite repeat_length.
   split. easy. simpl.
   assert (forall u,
-    (nil: list Type)
+    (nil: list X)
        = map snd (filter fst (combine (repeat false (length u)) u))).
   intro u. induction u. reflexivity. simpl. assumption. apply H0.
   exists (a::l). exists (nil). simpl. split; try rewrite app_nil_r; reflexivity.
   (* deux cas : a = n ou non *)
-  assert ({a=T} + {a<>T}). apply I. destruct H.
+  assert ({a=x} + {a<>x}). apply I. destruct H.
   rewrite e. rewrite subsequence2_cons_eq. rewrite <- IHl.
   rewrite subsequence_cons_eq. split; intro; assumption.
 
   split; intro H. apply subsequence2_cons_l. apply IHl.
   destruct H. destruct H. destruct H.
-  destruct x. destruct x0.
+  destruct x0. destruct x1.
   apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
   simpl in H0. inversion H0. rewrite H2 in n. contradiction n.
   reflexivity. inversion H0.
-  exists x. exists x0. split. assumption.
+  exists x2. exists x1. split. assumption.
   reflexivity.
 
   apply subsequence_cons_l. apply IHl.
   destruct H. destruct H.
-  destruct x. simpl in H0.
+  destruct x0. simpl in H0.
   symmetry in H0. apply nil_cons in H0. contradiction H0.
   destruct b. simpl in H0. inversion H0. rewrite H2 in n.
   contradiction n. reflexivity.
-  exists x. inversion H. inversion H0.
+  exists x0. inversion H. inversion H0.
   split; reflexivity.
 Qed.
 
 
-Theorem subsequence_dec :
-  (forall x y : Type, {x = y} + {x <> y})
-  -> forall (l s : list Type), { subsequence l s } + { ~ subsequence l s }.
+Theorem subsequence_dec {X: Type}:
+  (forall x y : X, {x = y} + {x <> y})
+  -> forall (l s : list X), { subsequence l s } + { ~ subsequence l s }.
 Proof.
   intro H. intros l s.
   assert ({ subsequence2 l s } + { ~ subsequence2 l s }).
@@ -285,3 +285,7 @@ Proof.
   exists [1].
   exists [[3];[5]]. simpl. easy.
 Qed.
+
+Example test3: subsequence [1;2;3;4;5] [1;3;5].
+Proof.
+  rewrite subsequence_eq_def.