mirror of
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New Verilog and Coq sample files added
New Verilog examples and Coq examples for additional training have been added since linguist is currently failing Coq/Verilog recognition tasks (see #201). In case it wasn't obvious, linguist will not currently pass these new, added test cases.
This commit is contained in:
Schuyler Eldridge
591
samples/coq/Lists.v
Executable file
591
samples/coq/Lists.v
Executable file
@@ -0,0 +1,591 @@
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Require Export Basics.
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Module NatList.
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Import Playground1.
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Inductive natprod : Type :=
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pair : nat -> nat -> natprod.
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Definition fst (p : natprod) : nat :=
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match p with
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| pair x y => x
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end.
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Definition snd (p : natprod) : nat :=
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match p with
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| pair x y => y
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end.
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Notation "( x , y )" := (pair x y).
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Definition swap_pair (p : natprod) : natprod :=
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match p with
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| (x, y) => (y, x)
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end.
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Theorem surjective_pairing' : forall (n m : nat),
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(n, m) = (fst (n, m), snd (n, m)).
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Proof.
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reflexivity. Qed.
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Theorem surjective_pairing : forall (p : natprod),
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p = (fst p, snd p).
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Proof.
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intros p.
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destruct p as (n, m).
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simpl.
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reflexivity.
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Qed.
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Theorem snd_fst_is_swap : forall (p : natprod),
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(snd p, fst p) = swap_pair p.
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Proof.
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intros p.
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destruct p.
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reflexivity.
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Qed.
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Theorem fst_swap_is_snd : forall (p : natprod),
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fst (swap_pair p) = snd p.
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Proof.
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intros p.
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destruct p.
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reflexivity.
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Qed.
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Inductive natlist : Type :=
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| nil : natlist
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| cons : nat -> natlist -> natlist.
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Definition l_123 := cons (S O) (cons (S (S O)) (cons (S (S (S O))) nil)).
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Notation "x :: l" := (cons x l) (at level 60, right associativity).
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Notation "[ ]" := nil.
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Notation "[]" := nil.
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Notation "[ x , .. , y ]" := (cons x .. (cons y nil) ..).
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Fixpoint repeat (n count : nat) : natlist :=
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match count with
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| O => nil
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| S count' => n :: (repeat n count')
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end.
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Fixpoint length (l:natlist) : nat :=
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match l with
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| nil => O
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| h :: t => S (length t)
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end.
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Fixpoint app (l1 l2 : natlist) : natlist :=
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match l1 with
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| nil => l2
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| h :: t => h :: (app t l2)
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end.
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Notation "x ++ y" := (app x y) (right associativity, at level 60).
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(*
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Example test_app1: [1,2,3] ++ [4,5] = [1,2,3,4,5].
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Proof. reflexivity. Qed.
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Example test_app2: nil ++ [4,5] = [4,5].
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Proof. reflexivity. Qed.
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Example test_app3: [1,2,3] ++ [] = [1,2,3].
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Proof. reflexivity. Qed.
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*)
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Definition head (l : natlist) : nat :=
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match l with
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| nil => O
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| h :: t => h
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end.
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Definition tl (l : natlist) : natlist :=
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match l with
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| nil => nil
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| h :: t => t
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end.
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(*
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Example test_tl: tl [1,2,3] = [2,3].
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Proof. reflexivity. Qed.
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*)
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Fixpoint nonzeros (l:natlist) : natlist :=
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match l with
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| nil => nil
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| O :: r => nonzeros r
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| n :: r => n :: (nonzeros r)
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end.
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Example test_nonzeros: nonzeros [O,S O,O,S (S O), S (S (S O)),O,O] = [S O,S (S O), S (S (S O))].
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Proof. reflexivity. Qed.
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Fixpoint oddmembers (l:natlist) : natlist :=
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match l with
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| nil => nil
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| n :: r => match (oddb n) with
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| true => n :: (oddmembers r)
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| false => oddmembers r
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end
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end.
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Example test_oddmembers: oddmembers [O, S O, O, S (S O), S (S (S O)), O, O] = [S O, S (S (S O))].
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Proof. reflexivity. Qed.
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Fixpoint countoddmembers (l:natlist) : nat :=
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length (oddmembers l).
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Example test_countoddmembers2: countoddmembers [O, S (S O), S (S (S (S O)))] = O.
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Proof. reflexivity. Qed.
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Example test_countoddmembers3: countoddmembers [] = O.
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Proof. reflexivity. Qed.
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Fixpoint alternate (l1 l2 : natlist) : natlist :=
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match l1 with
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| nil => l2
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| a :: r1 => match l2 with
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| nil => l1
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| b :: r2 => a :: b :: (alternate r1 r2)
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end
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end.
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Example test_alternative1: alternate [S O, S (S O), S (S (S O))] [S (S (S (S O))), S (S (S (S (S O)))), S (S (S (S (S (S O)))))] =
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[S O, S (S (S (S O))), S (S O), S (S (S (S (S O)))), S (S (S O)), S (S (S (S (S (S O)))))].
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Proof. reflexivity. Qed.
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Definition bag := natlist.
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Fixpoint count (v : nat) (s: bag) : nat :=
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match s with
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| nil => O
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| v' :: r => match (beq_nat v' v) with
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| true => S (count v r)
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| false => count v r
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end
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end.
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Example test_count1: count (S O) [S O, S (S O), S (S (S O)), S O, S (S (S (S O))), S O] = S (S (S O)).
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Proof. reflexivity. Qed.
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Definition sum : bag -> bag -> bag := app.
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Example test_sum1: count (S O) (sum [S O, S (S O), S (S (S O))] [S O, S (S (S (S O))), S O]) = S (S (S O)).
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Proof. reflexivity. Qed.
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Definition add (v:nat) (s:bag) : bag := v :: s.
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Example test_add1: count (S O) (add (S O) [S O, S (S (S (S O))), S O]) = S (S (S O)).
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Proof. reflexivity. Qed.
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Definition member (v:nat) (s:bag) : bool :=
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ble_nat (S O) (count v s).
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Example test_member1: member (S O) [S O, S (S (S (S O))), S O] = true.
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Proof. reflexivity. Qed.
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Example test_member2: member (S (S O)) [S O, S (S (S (S O))), S O] = false.
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Proof. reflexivity. Qed.
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Fixpoint remove_one (v:nat) (s:bag) : bag :=
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match s with
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| nil => nil
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| v' :: r => match (beq_nat v v') with
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| true => r
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| false => v' :: (remove_one v r)
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end
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end.
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Example test_remove_one1: count (S (S (S (S (S O)))))
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(remove_one (S (S (S (S (S O)))))
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[S (S O), S O, S (S (S (S (S O)))), S (S (S (S O))), S O]) = O.
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Proof. reflexivity. Qed.
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Fixpoint remove_all (v:nat) (s:bag) : bag :=
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match s with
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| nil => nil
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| v' :: r => match (beq_nat v v') with
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| true => remove_all v r
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| false => v' :: (remove_all v r)
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end
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end.
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Example test_remove_all1: count (S (S (S (S (S O)))))
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(remove_all (S (S (S (S (S O)))))
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[S (S O), S O, S (S (S (S (S O)))), S (S (S (S O))), S O]) = O.
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Proof. reflexivity. Qed.
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Fixpoint subset (s1:bag) (s2:bag) : bool :=
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match s1 with
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| nil => true
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| v :: r => andb (member v s2)
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(subset r (remove_one v s2))
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end.
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Definition test_subset1: subset [S O, S (S O)] [S (S O), S O, S (S (S (S O))), S O] = true.
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Proof. reflexivity. Qed.
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Definition test_subset2: subset [S O, S (S O), S (S O)] [S (S O), S O, S (S (S (S O))), S O] = false.
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Proof. reflexivity. Qed.
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Theorem bag_count_add : forall n t: nat, forall s : bag,
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count n s = t -> count n (add n s) = S t.
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Proof.
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intros n t s.
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intros H.
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induction s.
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simpl.
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rewrite <- beq_nat_refl.
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rewrite <- H.
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reflexivity.
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rewrite <- H.
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simpl.
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rewrite <- beq_nat_refl.
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reflexivity.
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Qed.
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Theorem nil_app : forall l:natlist,
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[] ++ l = l.
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Proof.
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reflexivity. Qed.
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Theorem tl_length_pred : forall l:natlist,
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pred (length l) = length (tl l).
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Proof.
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intros l. destruct l as [| n l'].
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Case "l = nil".
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reflexivity.
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Case "l = cons n l'".
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reflexivity. Qed.
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Theorem app_ass:forall l1 l2 l3 : natlist,
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(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3).
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Proof.
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intros l1 l2 l3. induction l1 as [| n l1'].
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Case "l1 = nil".
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reflexivity.
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Case "l1 = cons n l1'".
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simpl. rewrite -> IHl1'. reflexivity. Qed.
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Theorem app_length: forall l1 l2 : natlist,
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length (l1 ++ l2) = (length l1) + (length l2).
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Proof.
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intros l1 l2. induction l1 as [| n l1'].
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Case "l1 = nil".
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reflexivity.
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Case "l1 = cons".
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simpl. rewrite -> IHl1'. reflexivity. Qed.
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Fixpoint snoc (l:natlist) (v:nat) : natlist :=
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match l with
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| nil => [v]
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| h :: t => h :: (snoc t v)
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end.
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Fixpoint rev (l:natlist) : natlist :=
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match l with
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| nil => nil
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| h :: t => snoc (rev t) h
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end.
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Example test_rev1: rev [S O, S (S O), S (S (S O))] = [S (S (S O)), S (S O), S O].
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Proof. reflexivity. Qed.
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Theorem length_snoc : forall n : nat, forall l : natlist,
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length (snoc l n) = S (length l).
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Proof.
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intros n l. induction l as [| n' l'].
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Case "l = nil".
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reflexivity.
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Case "l = cons n' l'".
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simpl. rewrite -> IHl'. reflexivity. Qed.
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Theorem rev_length : forall l : natlist,
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length (rev l) = length l.
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Proof.
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intros l. induction l as [| n l'].
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Case "l = nil".
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reflexivity.
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Case "l = cons".
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simpl. rewrite -> length_snoc.
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rewrite -> IHl'. reflexivity. Qed.
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Theorem app_nil_end : forall l :natlist,
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l ++ [] = l.
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Proof.
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intros l.
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induction l.
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Case "l = nil".
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reflexivity.
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Case "l = cons".
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simpl. rewrite -> IHl. reflexivity. Qed.
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Theorem rev_snoc : forall l: natlist, forall n : nat,
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rev (snoc l n) = n :: (rev l).
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Proof.
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intros l n.
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induction l.
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Case "l = nil".
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reflexivity.
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Case "l = cons".
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simpl.
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rewrite -> IHl.
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reflexivity.
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Qed.
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Theorem rev_involutive : forall l : natlist,
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rev (rev l) = l.
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Proof.
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intros l.
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induction l.
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Case "l = nil".
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reflexivity.
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Case "l = cons".
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simpl.
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rewrite -> rev_snoc.
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rewrite -> IHl.
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reflexivity.
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Qed.
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Theorem app_ass4 : forall l1 l2 l3 l4 : natlist,
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l1 ++ (l2 ++ (l3 ++ l4)) = ((l1 ++ l2) ++ l3) ++ l4.
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Proof.
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intros l1 l2 l3 l4.
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rewrite -> app_ass.
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rewrite -> app_ass.
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reflexivity.
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Qed.
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Theorem snoc_append : forall (l : natlist) (n : nat),
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snoc l n = l ++ [n].
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Proof.
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intros l n.
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induction l.
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Case "l = nil".
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reflexivity.
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Case "l = cons".
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simpl.
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rewrite -> IHl.
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reflexivity.
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Qed.
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Theorem nonzeros_length : forall l1 l2 : natlist,
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nonzeros (l1 ++ l2) = (nonzeros l1) ++ (nonzeros l2).
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Proof.
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intros l1 l2.
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induction l1.
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Case "l1 = nil".
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reflexivity.
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Case "l1 = cons".
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simpl.
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rewrite -> IHl1.
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destruct n.
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reflexivity.
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reflexivity.
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Qed.
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Theorem distr_rev : forall l1 l2 : natlist,
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rev (l1 ++ l2) = (rev l2) ++ (rev l1).
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Proof.
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intros l1 l2.
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induction l1.
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Case "l1 = nil".
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simpl.
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rewrite -> app_nil_end.
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reflexivity.
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Case "l1 = cons".
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simpl.
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rewrite -> IHl1.
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simpl.
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rewrite -> snoc_append.
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rewrite -> snoc_append.
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rewrite -> app_ass.
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reflexivity.
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Qed.
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Theorem count_number_nonzero : forall (s : bag),
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ble_nat O (count (S O) (S O :: s)) = true.
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Proof.
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intros s.
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induction s.
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reflexivity.
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reflexivity.
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Qed.
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||||
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Theorem ble_n_Sn : forall n,
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ble_nat n (S n) = true.
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Proof.
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||||
intros n. induction n as [| n'].
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Case "0".
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||||
simpl. reflexivity.
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||||
Case "S n'".
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simpl. rewrite -> IHn'. reflexivity. Qed.
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||||
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Theorem remove_decreases_count: forall (s : bag),
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ble_nat (count O (remove_one O s)) (count O s) = true.
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Proof.
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intros s.
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induction s.
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Case "s = nil".
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reflexivity.
|
||||
Case "s = cons".
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simpl.
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induction n.
|
||||
SCase "n = O".
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simpl. rewrite -> ble_n_Sn.
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||||
reflexivity.
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SCase "n = S n'".
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simpl.
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rewrite -> IHs.
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reflexivity.
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Qed.
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||||
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Inductive natoption : Type :=
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||||
| Some : nat -> natoption
|
||||
| None : natoption.
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Fixpoint index (n:nat) (l:natlist) : natoption :=
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||||
match l with
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| nil => None
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||||
| a :: l' => if beq_nat n O then Some a else index (pred n) l'
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||||
end.
|
||||
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||||
Definition option_elim (o : natoption) (d : nat) : nat :=
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||||
match o with
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||||
| Some n' => n'
|
||||
| None => d
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||||
end.
|
||||
|
||||
Definition hd_opt (l : natlist) : natoption :=
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||||
match l with
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||||
| nil => None
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||||
| v :: r => Some v
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||||
end.
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||||
|
||||
Example test_hd_opt1 : hd_opt [] = None.
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||||
Proof. reflexivity. Qed.
|
||||
Example test_hd_opt2 : hd_opt [S O] = Some (S O).
|
||||
Proof. reflexivity. Qed.
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||||
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||||
Theorem option_elim_hd : forall l:natlist,
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||||
head l = option_elim (hd_opt l) O.
|
||||
Proof.
|
||||
intros l.
|
||||
destruct l.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
Qed.
|
||||
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Fixpoint beq_natlist (l1 l2 : natlist) : bool :=
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||||
match l1 with
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||||
| nil => match l2 with
|
||||
| nil => true
|
||||
| _ => false
|
||||
end
|
||||
| v1 :: r1 => match l2 with
|
||||
| nil => false
|
||||
| v2 :: r2 => if beq_nat v1 v2 then beq_natlist r1 r2
|
||||
else false
|
||||
end
|
||||
end.
|
||||
|
||||
Example test_beq_natlist1 : (beq_natlist nil nil = true).
|
||||
Proof. reflexivity. Qed.
|
||||
Example test_beq_natlist2 : (beq_natlist [S O, S (S O), S (S (S O))]
|
||||
[S O, S (S O), S (S (S O))] = true).
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Theorem beq_natlist_refl : forall l:natlist,
|
||||
beq_natlist l l = true.
|
||||
Proof.
|
||||
intros l.
|
||||
induction l.
|
||||
Case "l = nil".
|
||||
reflexivity.
|
||||
Case "l = cons".
|
||||
simpl.
|
||||
rewrite <- beq_nat_refl.
|
||||
rewrite -> IHl.
|
||||
reflexivity.
|
||||
Qed.
|
||||
|
||||
Theorem silly1 : forall (n m o p : nat),
|
||||
n = m -> [n, o] = [n, p] -> [n, o] = [m, p].
|
||||
Proof.
|
||||
intros n m o p eq1 eq2.
|
||||
rewrite <- eq1.
|
||||
apply eq2. Qed.
|
||||
|
||||
Theorem silly2a : forall (n m : nat),
|
||||
(n,n) = (m,m) ->
|
||||
(forall (q r : nat), (q, q) = (r, r) -> [q] = [r]) ->
|
||||
[n] = [m].
|
||||
Proof.
|
||||
intros n m eq1 eq2.
|
||||
apply eq2.
|
||||
apply eq1.
|
||||
Qed.
|
||||
|
||||
Theorem silly_ex :
|
||||
(forall n, evenb n = true -> oddb (S n) = true) ->
|
||||
evenb (S (S (S O))) = true ->
|
||||
oddb (S (S (S (S O)))) = true.
|
||||
Proof.
|
||||
intros eq1 eq2.
|
||||
apply eq1.
|
||||
apply eq2.
|
||||
Qed.
|
||||
|
||||
Theorem silly3 : forall (n : nat),
|
||||
true = beq_nat n (S (S (S (S (S O))))) ->
|
||||
beq_nat (S (S n)) (S (S (S (S (S (S (S O))))))) = true.
|
||||
Proof.
|
||||
intros n H.
|
||||
symmetry.
|
||||
apply H.
|
||||
Qed.
|
||||
|
||||
Theorem rev_exercise : forall (l l' : natlist),
|
||||
l = rev l' -> l' = rev l.
|
||||
Proof.
|
||||
intros l l' H.
|
||||
rewrite -> H.
|
||||
rewrite -> rev_involutive.
|
||||
reflexivity.
|
||||
Qed.
|
||||
|
||||
Theorem beq_nat_sym : forall (n m:nat), forall (b: bool),
|
||||
beq_nat n m = b -> beq_nat m n = b.
|
||||
Proof.
|
||||
intros n.
|
||||
induction n as [| n'].
|
||||
Case "n = O".
|
||||
intros m b eq1.
|
||||
induction m.
|
||||
SCase "m = 0".
|
||||
apply eq1.
|
||||
SCase "m = S m'".
|
||||
apply eq1.
|
||||
Case "n = S n'".
|
||||
induction m.
|
||||
SCase "m = 0".
|
||||
intros b eq1.
|
||||
apply eq1.
|
||||
SCase "m = S m'".
|
||||
intros b eq1.
|
||||
apply IHn'.
|
||||
apply eq1.
|
||||
Qed.
|
||||
|
||||
Theorem app_ass' : forall l1 l2 l3 : natlist,
|
||||
(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3).
|
||||
Proof.
|
||||
intros l1. induction l1 as [ | n l1'].
|
||||
reflexivity.
|
||||
simpl.
|
||||
intros l2 l3.
|
||||
rewrite -> IHl1'.
|
||||
reflexivity.
|
||||
Qed.
|
||||
|
||||
End NatList.
|
||||
Reference in New Issue
Block a user