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### Definitions File

### Template File

### Check File

theory Defs imports Main begin inductive subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr "\<sqsubseteq>" 50) where [intro]: "[] \<sqsubseteq> _" | [intro]: "xs \<sqsubseteq> ys \<Longrightarrow> x # xs \<sqsubseteq> x # ys" | [intro]: "xs \<sqsubseteq> ys \<Longrightarrow> xs \<sqsubseteq> y # ys" definition proper_subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr "\<sqsubset>" 50) where "xs \<sqsubset> ys \<longleftrightarrow> xs \<noteq> ys \<and> xs \<sqsubseteq> ys" definition all_proper_subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where "all_proper_subseq xs ys \<longleftrightarrow> (\<forall>xs'. xs' \<sqsubset> xs \<longrightarrow> xs' \<sqsubseteq> ys)" fun aux where "aux [] _ = True" | "aux _ [] = False" | "aux (x#xs) (y#ys) = (if x = y then aux xs ys else aux (x # xs) ys)" fun aux2 where "aux2 ys acc [] = True" | "aux2 ys acc (x # xs) \<longleftrightarrow> aux (acc @ xs) ys \<and> aux2 ys (acc @ [x]) xs" definition "judge1 xs ys \<longleftrightarrow> aux2 ys [] xs" end

theory Submission imports Defs begin theorem judge1_correct: "judge1 xs ys \<longleftrightarrow> all_proper_subseq xs ys" sorry definition "judge2 xs ys \<longleftrightarrow> judge1 xs ys" theorem judge2_correct: "judge2 xs ys \<longleftrightarrow> all_proper_subseq xs ys" unfolding judge2_def by (rule judge1_correct) theorem judge2_is_executable: "judge2 ''ab'' ''ab'' \<longleftrightarrow> True" "judge2 ''ba'' ''ab'' \<longleftrightarrow> True" "judge2 ''abcd'' ''cdabc'' \<longleftrightarrow> False" by eval+ end

theory Check imports Submission begin theorem judge1_correct: "judge1 xs ys \<longleftrightarrow> all_proper_subseq xs ys" by (rule Submission.judge1_correct) theorem judge2_correct: "judge2 xs ys \<longleftrightarrow> all_proper_subseq xs ys" by (rule Submission.judge2_correct) theorem judge2_is_executable: "judge2 ''ab'' ''ab'' \<longleftrightarrow> True" "judge2 ''ba'' ''ab'' \<longleftrightarrow> True" "judge2 ''abcd'' ''cdabc'' \<longleftrightarrow> False" by eval+ end

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### Definitions File

### Template File

### Check File

import data.list.basic variable {α : Type*} -- `<+` is notation for `is_sublist` def is_strict_sublist (xs ys : list α) : Prop := xs <+ ys ∧ xs ≠ ys infix ` <<+ `:50 := is_strict_sublist def all_proper_sublist (xs ys : list α) : Prop := ∀ (xs' <<+ xs), xs' <+ ys variable [decidable_eq α] def judge_aux : list α → list α → Prop | [] _ := true | _ [] := false | (x::xs) (y::ys) := if x = y then judge_aux xs ys else judge_aux (x :: xs) ys def judge_aux2 (ys : list α) : list α → list α → Prop | acc [] := true | acc (x :: xs) := judge_aux (acc ++ xs) ys ∧ judge_aux2 (acc ++ [x]) xs def judge1 (xs ys : list α) : Prop := xs ≠ ys ∧ xs.length = ys.length ∧ judge_aux2 ys [] xs ----------------------just some definitions to prevent cheating------------------------ definition judge1_correct_prop : Prop := ∀ {α : Type*} [eq_inst : decidable_eq α] (xs ys : list α), @judge1 α eq_inst xs ys ↔ all_proper_sublist xs ys notation `judge1_correct_prop` := judge1_correct_prop

import .defs variables {α : Type*} [decidable_eq α] /- Task -/ theorem judge1_correct : ∀ (xs ys : list α), judge1 xs ys ↔ all_proper_sublist xs ys := sorry def judge2 (xs ys : list α) : Prop := judge1 xs ys theorem judge2_correct : ∀ (xs ys : list α), judge2 xs ys ↔ all_proper_sublist xs ys := sorry

import .defs import .submission theorem check_judge1_correct : judge1_correct_prop := @judge1_correct definition judge2_correct_prop : Prop := ∀ {α : Type*} [eq_inst : decidable_eq α] (xs ys : list α), @judge2 α eq_inst xs ys ↔ all_proper_sublist xs ys theorem check_judge2_correct : judge2_correct_prop := @judge2_correct

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### Definitions File

### Template File

From Coq Require Export Lists.List. Export List.ListNotations. Require Export PeanoNat. Inductive subseq : list nat -> list nat -> Prop := subseq_nil : subseq [] [] | subseq_take xs ys x: subseq xs ys -> subseq (x::xs) (x::ys) | subseq_drop xs ys x: subseq xs ys -> subseq xs (x::ys). Definition proper_subseq (xs ys : list nat) := xs <> ys /\ subseq xs ys. Definition all_proper_subseq (xs ys : list nat) := forall xs', proper_subseq xs' xs -> subseq xs' ys. Fixpoint aux (xs ys : list nat) :bool := match xs,ys with [],_ => true | _,[] => false | x::xs,y::ys => if Nat.eq_dec x y then aux xs ys else aux (x::xs) ys end. Fixpoint aux2 (ys acc xs : list nat) : bool := match xs with [] => true | x::xs => aux (acc++xs) ys && aux2 ys (acc++[x]) xs end. Definition judge1 xs ys := aux2 ys [] xs.

Require Import Defs. (** * Task 1: completion of this gives full points*) Lemma judge1_correct xs ys: (all_proper_subseq xs ys) <-> judge1 xs ys = true. Admitted. (** * Alternative Task *) (** Show any implementation correct for half the points *) Definition judge2:= judge1. Lemma judge2_correct xs ys: (all_proper_subseq xs ys) <-> judge2 xs ys = true. Proof. apply judge1_correct. Qed.

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### Definitions File

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theory Defs imports Main begin inductive subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr "\<sqsubseteq>" 50) where [intro]: "[] \<sqsubseteq> _" | [intro]: "xs \<sqsubseteq> ys \<Longrightarrow> x # xs \<sqsubseteq> x # ys" | [intro]: "xs \<sqsubseteq> ys \<Longrightarrow> xs \<sqsubseteq> y # ys" definition proper_subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr "\<sqsubset>" 50) where "xs \<sqsubset> ys \<longleftrightarrow> xs \<noteq> ys \<and> xs \<sqsubseteq> ys" definition all_proper_subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where "all_proper_subseq xs ys \<longleftrightarrow> (\<forall>xs'. xs' \<sqsubset> xs \<longrightarrow> xs' \<sqsubseteq> ys)" fun aux where "aux [] _ = True" | "aux _ [] = False" | "aux (x#xs) (y#ys) = (if x = y then aux xs ys else aux (x # xs) ys)" fun aux2 where "aux2 ys acc [] = True" | "aux2 ys acc (x # xs) \<longleftrightarrow> aux (acc @ xs) ys \<and> aux2 ys (acc @ [x]) xs" definition "judge1 xs ys \<longleftrightarrow> aux2 ys [] xs" end

theory Submission imports Defs begin theorem judge1_correct: "judge1 xs ys \<longleftrightarrow> all_proper_subseq xs ys" sorry definition "judge2 xs ys \<longleftrightarrow> judge1 xs ys" theorem judge2_correct: "judge2 xs ys \<longleftrightarrow> all_proper_subseq xs ys" unfolding judge2_def by (rule judge1_correct) theorem judge2_is_executable: "judge2 ''ab'' ''ab'' \<longleftrightarrow> True" "judge2 ''ba'' ''ab'' \<longleftrightarrow> True" "judge2 ''abcd'' ''cdabc'' \<longleftrightarrow> False" by eval+ end

theory Check imports Submission begin theorem judge1_correct: "judge1 xs ys \<longleftrightarrow> all_proper_subseq xs ys" by (rule Submission.judge1_correct) theorem judge2_correct: "judge2 xs ys \<longleftrightarrow> all_proper_subseq xs ys" by (rule Submission.judge2_correct) theorem judge2_is_executable: "judge2 ''ab'' ''ab'' \<longleftrightarrow> True" "judge2 ''ba'' ''ab'' \<longleftrightarrow> True" "judge2 ''abcd'' ''cdabc'' \<longleftrightarrow> False" by eval+ end

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