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uploading 2021 solutions
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dafny/2021/solutions_2021.dfy
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227
dafny/2021/solutions_2021.dfy
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// Dafny coursework solutions
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// Autumn term, 2021
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//
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// Authors: John Wickerson
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predicate sorted_between(A:array<int>, lo:int, hi:int)
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reads A
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{
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forall m,n :: 0 <= lo <= m < n < hi <= A.Length ==> A[m] <= A[n]
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}
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predicate sorted(A:array<int>)
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reads A
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{
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sorted_between(A,0,A.Length)
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}
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// Task 1. Difficulty: *
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method create_multiples_of_two(A:array<int>)
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ensures forall n :: 0 <= n < A.Length ==> A[n] == 2*n
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modifies A
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{
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var i := 0;
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while i < A.Length
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invariant 0 <= i <= A.Length
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invariant forall n :: 0 <= n < i ==> A[n] == 2*n
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{
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A[i] := 2 * i;
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i := i + 1;
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}
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}
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// Task 2. Difficulty: **
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predicate partitioned(A:array<int>, index:int)
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reads A
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requires 0 <= index <= A.Length
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{
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forall i,j :: 0 <= i < index <= j < A.Length ==> A[i] <= A[j]
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}
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method exchange_sort (A:array<int>)
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ensures sorted(A)
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modifies A
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{
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var i := 0;
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while i < A.Length - 1
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invariant 0 < A.Length ==> 0 <= i < A.Length
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invariant 0 < A.Length ==> partitioned(A, i)
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invariant sorted_between(A,0,i+1);
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{
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var j := i + 1;
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while j < A.Length
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invariant i < j <= A.Length
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invariant partitioned(A, i)
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invariant sorted_between(A,0,i+1);
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invariant forall k :: i <= k < j ==> A[i] <= A[k]
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{
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if A[i] > A[j] {
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A[i], A[j] := A[j], A[i];
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}
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j := j + 1;
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}
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i := i + 1;
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}
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}
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// Task 3. Difficulty: **
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method fung_sort (A:array<int>)
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ensures sorted(A)
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modifies A
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{
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var i := 0;
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while i < A.Length
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invariant i <= A.Length
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invariant sorted_between (A, 0, i)
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{
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var j := 0;
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while j < A.Length
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invariant j <= A.Length
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invariant sorted_between (A, 0, i)
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invariant 0 < j ==> A[j-1] <= A[i]
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invariant i <= j ==> sorted_between (A, 0, i+1)
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{
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if A[i] < A[j] {
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A[i], A[j] := A[j], A[i];
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}
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j := j+1;
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}
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i := i+1;
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}
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}
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// Task 4. Difficulty: ***
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predicate odd_sorted(A:array<int>, hi:int)
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reads A
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{
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forall i :: 0 <= i ==> 2*i+2 < hi <= A.Length ==> A[2*i + 1] <= A[2*i+2]
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}
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method odd_even_sort(A:array<int>)
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ensures sorted(A)
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modifies A
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decreases *
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{
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var is_sorted := false;
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while !is_sorted
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invariant is_sorted ==> sorted(A)
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decreases *
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{
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is_sorted := true;
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var i := 0;
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while 2*i+2 < A.Length
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invariant 0 < A.Length ==> 0 <= 2*i < A.Length
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invariant odd_sorted (A, 2*i+1)
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{
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if A[2*i+2] < A[2*i+1] {
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A[2*i+1], A[2*i+2] := A[2*i+2], A[2*i+1];
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is_sorted := false;
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}
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i := i+1;
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}
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i := 0;
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while 2*i+1 < A.Length
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invariant 0 <= 2*i <= A.Length
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invariant is_sorted ==> odd_sorted (A, A.Length)
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invariant is_sorted ==> 2*i+1 < A.Length ==> sorted_between(A, 0, 2*i+1);
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invariant is_sorted ==> 2*i+1 >= A.Length ==> sorted_between(A, 0, A.Length);
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{
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if A[2*i+1] < A[2*i] {
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A[2*i], A[2*i+1] := A[2*i+1], A[2*i];
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is_sorted := false;
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}
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i := i+1;
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}
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}
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}
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// Task 5. Difficulty: ***
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method bubble_sort3(A:array<int>)
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ensures sorted(A)
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modifies A
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decreases *
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{
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var stable := false;
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while !stable
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invariant stable ==> sorted(A)
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decreases *
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//decreases entropy(A)
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{
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stable := true;
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var j := 0;
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while 2*j+2 <= A.Length
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invariant stable ==> sorted_between(A, 0, 2*j+1)
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invariant stable ==> 2*j+2 == A.Length + 2 ==> sorted_between(A, 0, A.Length)
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invariant 2*j+2 <= A.Length + 2
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decreases A.Length - j
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{
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if 2*j+2 == A.Length {
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if A[2*j+1] < A[2*j] {
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A[2*j+1], A[2*j] := A[2*j], A[2*j+1];
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stable := false;
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}
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} else {
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if A[2*j+1] < A[2*j] {
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A[2*j+1], A[2*j] := A[2*j], A[2*j+1];
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stable := false;
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}
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if A[2*j+2] < A[2*j+1] {
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A[2*j+2], A[2*j+1] := A[2*j+1], A[2*j+2];
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stable := false;
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}
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if A[2*j+1] < A[2*j] {
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A[2*j+1], A[2*j] := A[2*j], A[2*j+1];
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stable := false;
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}
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}
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j := j + 1;
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}
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}
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}
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// Task 5, bonus
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function method entropy_elem(A:array<int>, i:int, j:int) : int
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reads A
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requires 0 <= i < j <= A.Length
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ensures entropy_elem(A, i, j) >= 0
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decreases A.Length - j
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{
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if j == A.Length then 0 else
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entropy_elem(A, i, j+1) + (if A[i] > A[j] then 1 else 0)
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}
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function method entropy_from(A:array<int>, i:int) : int
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reads A
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requires 0 <= i <= A.Length
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ensures entropy_from(A, i) >= 0
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decreases A.Length - i
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{
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if i == A.Length then 0 else
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entropy_elem(A, i, i+1) + entropy_from(A, i + 1)
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}
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function method entropy(A:array<int>) : int
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reads A
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ensures entropy(A) >= 0
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{
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entropy_from(A, 0)
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}
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method print_array(A:array<int>)
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{
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var i := 0;
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while i < A.Length {
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print A[i];
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i := i + 1;
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}
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}
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method Main()
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decreases *
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{
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var A:array<int> := new int[7] [4,0,1,9,7,1,2];
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print "Before: "; print_array(A); print "\n";
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exchange_sort(A);
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print "After: "; print_array(A); print "\n";
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}
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689
isabelle/2021/HSV_tasks_2021_solutions.thy
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689
isabelle/2021/HSV_tasks_2021_solutions.thy
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@ -0,0 +1,689 @@
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theory HSV_tasks_2021_solutions imports Complex_Main begin
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section \<open>Task 1: Factorising circuits.\<close>
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(* Datatype for representing simple circuits. *)
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datatype "circuit" =
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NOT "circuit"
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| AND "circuit" "circuit"
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| OR "circuit" "circuit"
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| TRUE
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| FALSE
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| INPUT "int"
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(* Simulates a circuit given a valuation for each input wire. *)
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fun simulate where
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"simulate (AND c1 c2) \<rho> = ((simulate c1 \<rho>) \<and> (simulate c2 \<rho>))"
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| "simulate (OR c1 c2) \<rho> = ((simulate c1 \<rho>) \<or> (simulate c2 \<rho>))"
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| "simulate (NOT c) \<rho> = (\<not> (simulate c \<rho>))"
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| "simulate TRUE \<rho> = True"
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| "simulate FALSE \<rho> = False"
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| "simulate (INPUT i) \<rho> = \<rho> i"
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(* Equivalence between circuits. *)
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fun circuits_equiv (infix "\<sim>" 50) (* the "50" indicates the operator precedence *) where
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"c1 \<sim> c2 = (\<forall>\<rho>. simulate c1 \<rho> = simulate c2 \<rho>)"
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(* An optimisation that exploits the following Boolean identities:
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`(a | b) & (a | c) = a | (b & c)`
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`(a | b) & (c | a) = a | (b & c)`
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`(a | b) & (b | c) = b | (a & c)`
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`(a | b) & (c | b) = b | (a & c)`
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*)
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fun factorise where
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"factorise (NOT c) = NOT (factorise c)"
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| "factorise (AND (OR c1 c2) (OR c3 c4)) = (
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let c1' = factorise c1; c2' = factorise c2; c3' = factorise c3; c4' = factorise c4 in
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if c1' = c3' then OR c1' (AND c2' c4')
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else if c1' = c4' then OR c1' (AND c2' c3')
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else if c2' = c3' then OR c2' (AND c1' c4')
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else if c2' = c4' then OR c2' (AND c1' c3')
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else AND (OR c1' c2') (OR c3' c4'))"
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| "factorise (AND c1 c2) = AND (factorise c1) (factorise c2)"
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| "factorise (OR c1 c2) = OR (factorise c1) (factorise c2)"
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| "factorise TRUE = TRUE"
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| "factorise FALSE = FALSE"
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| "factorise (INPUT i) = INPUT i"
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lemma (* test case *)
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"factorise (AND TRUE TRUE) = AND TRUE TRUE"
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by eval
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lemma (* test case *)
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"factorise (AND (OR (INPUT 1) FALSE) (OR TRUE (INPUT 1))) =
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OR (INPUT 1) (AND FALSE TRUE)"
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by eval
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lemma (* test case *)
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"factorise (NOT (AND (OR FALSE (INPUT 2)) (OR TRUE (INPUT 2)))) =
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NOT (OR (INPUT 2) (AND FALSE TRUE))"
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by eval
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theorem factorise_is_sound: "factorise c \<sim> c"
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proof (induct rule: factorise.induct)
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case (2 c1 c2 c3 c4)
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let ?c1' = "factorise c1"
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let ?c2' = "factorise c2"
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let ?c3' = "factorise c3"
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let ?c4' = "factorise c4"
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from 2 have IH: "?c1' \<sim> c1" "?c2' \<sim> c2" "?c3' \<sim> c3" "?c4' \<sim> c4" by auto
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have "factorise (AND (OR c1 c2) (OR c3 c4)) = (
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if ?c1' = ?c3' then OR ?c1' (AND ?c2' ?c4')
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else if ?c1' = ?c4' then OR ?c1' (AND ?c2' ?c3')
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else if ?c2' = ?c3' then OR ?c2' (AND ?c1' ?c4')
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else if ?c2' = ?c4' then OR ?c2' (AND ?c1' ?c3')
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else AND (OR ?c1' ?c2') (OR ?c3' ?c4'))"
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by auto
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also have "... \<sim> AND (OR c1 c2) (OR c3 c4)" using IH by auto
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finally show ?case by auto
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qed(simp_all)
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fun factorise2 where
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"factorise2 (NOT c) = NOT (factorise2 c)"
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| "factorise2 (AND (OR c1 c2) (OR c3 c4)) = (
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let c1' = factorise2 c1; c2' = factorise2 c2; c3' = factorise2 c3; c4' = factorise2 c4 in
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if c1' = c3' then OR c1' (AND c2' c4')
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else if c1' = c4' then OR c1' (AND c2' c3')
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else if c2' = c3' then OR c2' (AND c1' c4')
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else if c2' = c4' then OR c2' (AND c1' c3')
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else AND (OR c1' c2') (OR c3' c4'))"
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| "factorise2 (OR (AND c1 c2) (AND c3 c4)) = (
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let c1' = factorise2 c1; c2' = factorise2 c2; c3' = factorise2 c3; c4' = factorise2 c4 in
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if c1' = c3' then AND c1' (OR c2' c4')
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else if c1' = c4' then AND c1' (OR c2' c3')
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else if c2' = c3' then AND c2' (OR c1' c4')
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else if c2' = c4' then AND c2' (OR c1' c3')
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else OR (AND c1' c2') (AND c3' c4'))"
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| "factorise2 (AND c1 c2) = AND (factorise2 c1) (factorise2 c2)"
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| "factorise2 (OR c1 c2) = OR (factorise2 c1) (factorise2 c2)"
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| "factorise2 TRUE = TRUE"
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| "factorise2 FALSE = FALSE"
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| "factorise2 (INPUT i) = INPUT i"
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lemma (* test case *)
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"factorise2 (OR (AND (INPUT 1) (INPUT 2)) (AND TRUE (INPUT 1))) =
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AND (INPUT 1) (OR (INPUT 2) TRUE)"
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by eval
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theorem factorise2_is_sound: "factorise2 c \<sim> c"
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proof (induct rule: factorise2.induct)
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case (2 c1 c2 c3 c4)
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let ?c1' = "factorise2 c1"
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let ?c2' = "factorise2 c2"
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let ?c3' = "factorise2 c3"
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let ?c4' = "factorise2 c4"
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from 2 have IH: "?c1' \<sim> c1" "?c2' \<sim> c2" "?c3' \<sim> c3" "?c4' \<sim> c4" by auto
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have "factorise2 (AND (OR c1 c2) (OR c3 c4)) = (
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if ?c1' = ?c3' then OR ?c1' (AND ?c2' ?c4')
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else if ?c1' = ?c4' then OR ?c1' (AND ?c2' ?c3')
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else if ?c2' = ?c3' then OR ?c2' (AND ?c1' ?c4')
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else if ?c2' = ?c4' then OR ?c2' (AND ?c1' ?c3')
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else AND (OR ?c1' ?c2') (OR ?c3' ?c4'))"
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by auto
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also have "... \<sim> AND (OR c1 c2) (OR c3 c4)" using IH by auto
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finally show ?case by auto
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next
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case (3 c1 c2 c3 c4)
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let ?c1' = "factorise2 c1"
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let ?c2' = "factorise2 c2"
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let ?c3' = "factorise2 c3"
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let ?c4' = "factorise2 c4"
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from 3 have IH: "?c1' \<sim> c1" "?c2' \<sim> c2" "?c3' \<sim> c3" "?c4' \<sim> c4" by auto
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have "factorise2 (OR (AND c1 c2) (AND c3 c4)) = (
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if ?c1' = ?c3' then AND ?c1' (OR ?c2' ?c4')
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else if ?c1' = ?c4' then AND ?c1' (OR ?c2' ?c3')
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else if ?c2' = ?c3' then AND ?c2' (OR ?c1' ?c4')
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else if ?c2' = ?c4' then AND ?c2' (OR ?c1' ?c3')
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else OR (AND ?c1' ?c2') (AND ?c3' ?c4'))"
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by auto
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also have "... \<sim> OR (AND c1 c2) (AND c3 c4)" using IH by auto
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finally show ?case by auto
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qed(simp_all)
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section \<open>Task 2: A theorem about divisibility.\<close>
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theorem plus_dvd_odd_power:
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"(a::int) + b dvd a ^ (2 * n + 1) + b ^ (2 * n + 1)"
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proof (induct n)
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case 0
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thus ?case by auto
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next
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case (Suc n)
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then obtain k::int where "a ^ (2 * n + 1) + b ^ (2 * n + 1) = (a + b) * k"
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unfolding dvd_class.dvd_def by auto
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hence IH: "a ^ (2 * n + 1) = (a + b) * k - b ^ (2 * n + 1)" by auto
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have "a ^ (2 * Suc n + 1) + b ^ (2 * Suc n + 1) = a ^ (2 * n + 2 + 1) + b ^ (2 * n + 2 + 1)"
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by simp
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also have "... = a\<^sup>2 * a ^ (2 * n + 1) + b\<^sup>2 * b ^ (2 * n + 1)"
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by (metis (no_types, lifting) add.commute add_Suc_right plus_1_eq_Suc power_add)
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also have "... = a\<^sup>2 * ((a + b) * k - b ^ (2 * n + 1)) + b\<^sup>2 * b ^ (2 * n + 1)"
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unfolding IH by auto
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also have "... = a\<^sup>2 * (a + b) * k - a\<^sup>2 * b ^ (2 * n + 1) + b\<^sup>2 * b ^ (2 * n + 1)"
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by algebra
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also have "... = a\<^sup>2 * (a + b) * k - ((a\<^sup>2 - b\<^sup>2) * b ^ (2 * n + 1))"
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by algebra
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also have "... = a\<^sup>2 * (a + b) * k - ((a + b) * (a - b) * b ^ (2 * n + 1))"
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by algebra
|
||||
also have "... = (a + b) * (a\<^sup>2 * k - ((a - b) * b ^ (2 * n + 1)))"
|
||||
by algebra
|
||||
finally have "a ^ (2 * Suc n + 1) + b ^ (2 * Suc n + 1) =
|
||||
(a + b) * (a ^ 2 * k - ((a - b) * b ^ (2 * n + 1)))" .
|
||||
thus ?case by simp
|
||||
qed
|
||||
|
||||
theorem plus_dvd_power:
|
||||
"(a::int) + b dvd a ^ (2 * n + 2) + b ^ (2 * n + 2)"
|
||||
oops
|
||||
|
||||
section \<open>Task 3: Proving that the shift-and-add-3 algorithm is correct.\<close>
|
||||
|
||||
subsection \<open>Binary and its conversion to nat\<close>
|
||||
|
||||
type_synonym bit = "bool"
|
||||
|
||||
abbreviation B0 where "B0 == False"
|
||||
abbreviation B1 where "B1 == True"
|
||||
|
||||
(* The following lemma says that if I want to prove a property of
|
||||
all 5-bit binary numbers, it suffices to just consider all 32 bit patterns. *)
|
||||
lemma cases_b5:
|
||||
fixes v w x y z :: "bit"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B0, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B0, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B0, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B0, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B1, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B1, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B1, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B0, B1, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B0, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B0, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B0, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B0, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B1, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B1, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B1, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B0, B1, B1, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B0, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B0, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B0, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B0, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B1, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B1, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B1, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B0, B1, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B0, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B0, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B0, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B0, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B1, B0, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B1, B0, B1) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B1, B1, B0) \<Longrightarrow> P v w x y z"
|
||||
assumes "(v, w, x, y, z) = (B1, B1, B1, B1, B1) \<Longrightarrow> P v w x y z"
|
||||
shows "P v w x y z"
|
||||
using assms
|
||||
apply (cases v)
|
||||
apply (cases w)
|
||||
apply (cases x)
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases x)
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases w)
|
||||
apply (cases x)
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases x)
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases y)
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
apply (cases z)
|
||||
apply auto[2]
|
||||
done
|
||||
|
||||
fun binary_to_nat :: "bit list \<Rightarrow> nat"
|
||||
where
|
||||
"binary_to_nat [] = 0"
|
||||
| "binary_to_nat (b # bs) = (if b then 2 ^ length bs else 0) + binary_to_nat bs"
|
||||
|
||||
|
||||
lemma (* test case *) "binary_to_nat [B0, B1, B0, B1] = 5" by eval
|
||||
lemma (* test case *) "binary_to_nat [B0, B0, B1, B0, B1] = 5" by eval
|
||||
lemma (* test case *) "binary_to_nat [B1] = 1" by eval
|
||||
lemma (* test case *) "binary_to_nat [B0] = 0" by eval
|
||||
|
||||
subsection \<open>BCD and its conversion to nat\<close>
|
||||
|
||||
type_synonym nibble = "bit * bit * bit * bit"
|
||||
|
||||
fun nibble_to_nat :: "nibble \<Rightarrow> nat"
|
||||
where
|
||||
"nibble_to_nat (B0,B0,B0,B0) = 0"
|
||||
| "nibble_to_nat (B0,B0,B0,B1) = 1"
|
||||
| "nibble_to_nat (B0,B0,B1,B0) = 2"
|
||||
| "nibble_to_nat (B0,B0,B1,B1) = 3"
|
||||
| "nibble_to_nat (B0,B1,B0,B0) = 4"
|
||||
| "nibble_to_nat (B0,B1,B0,B1) = 5"
|
||||
| "nibble_to_nat (B0,B1,B1,B0) = 6"
|
||||
| "nibble_to_nat (B0,B1,B1,B1) = 7"
|
||||
| "nibble_to_nat (B1,B0,B0,B0) = 8"
|
||||
| "nibble_to_nat (B1,B0,B0,B1) = 9"
|
||||
| "nibble_to_nat (B1,B0,B1,B0) = 10"
|
||||
| "nibble_to_nat (B1,B0,B1,B1) = 11"
|
||||
| "nibble_to_nat (B1,B1,B0,B0) = 12"
|
||||
| "nibble_to_nat (B1,B1,B0,B1) = 13"
|
||||
| "nibble_to_nat (B1,B1,B1,B0) = 14"
|
||||
| "nibble_to_nat (B1,B1,B1,B1) = 15"
|
||||
|
||||
fun bcd_to_nat :: "nibble list \<Rightarrow> nat"
|
||||
where
|
||||
"bcd_to_nat [] = 0"
|
||||
| "bcd_to_nat (n # ns) = bcd_to_nat ns + nibble_to_nat n * 10 ^ length ns"
|
||||
|
||||
lemma (* test case *) "bcd_to_nat [(B0,B1,B1,B0)] = 6" by eval
|
||||
lemma (* test case *) "bcd_to_nat [(B0,B1,B1,B0),(B1,B0,B0,B1)] = 69" by eval
|
||||
lemma (* test case *) "bcd_to_nat [(B0,B0,B0,B0),(B1,B0,B0,B1)] = 9" by eval
|
||||
lemma (* test case *) "bcd_to_nat [(B0,B0,B1,B1),(B0,B0,B0,B0)] = 30" by eval
|
||||
|
||||
|
||||
subsection \<open>Left-shifting BCD numbers\<close>
|
||||
|
||||
(*
|
||||
"Add-three-and-shift"
|
||||
bcd_part bin_part combined
|
||||
([ ], [1,0,1,0,1,0,1]) 0 85 0 + 85 = 85
|
||||
\<longrightarrow> ([ (0,0,0,1)], [0,1,0,1,0,1]) 1 21 1*64 + 21 = 85
|
||||
\<longrightarrow> ([ (0,0,1,0)], [1,0,1,0,1]) 2 21 2*32 + 21 = 85
|
||||
\<longrightarrow> ([ (0,1,0,1)], [0,1,0,1]) 5 5 5*16 + 5 = 85
|
||||
\<longrightarrow> ([(0,0,0,1),(0,0,0,0)], [1,0,1]) 10 5 10*8 + 5 = 85
|
||||
\<longrightarrow> ([(0,0,1,0),(0,0,0,1)], [0,1]) 21 1 21*4 + 1 = 85
|
||||
\<longrightarrow> ([(0,1,0,0),(0,0,1,0)], [1]) 42 1 42*2 + 1 = 85
|
||||
\<longrightarrow> ([(1,0,0,0),(0,1,0,1)], []) 85 0 85 + 0 = 85
|
||||
\<longrightarrow> done
|
||||
*)
|
||||
|
||||
fun shift_helper :: "nibble list \<Rightarrow> bit \<Rightarrow> (bit * nibble list)"
|
||||
where
|
||||
"shift_helper [] b = (b, [])"
|
||||
| "shift_helper ((b1, b2, b3, b4) # ns) b = (
|
||||
let (c, ns') = shift_helper ns b in
|
||||
(b1, (b2,b3,b4,c) # ns'))"
|
||||
|
||||
fun shift :: "nibble list \<Rightarrow> bit \<Rightarrow> nibble list"
|
||||
where
|
||||
"shift ns b = (let (c,ns') = shift_helper ns b in
|
||||
if c = B1 then (B0,B0,B0,c) # ns' else ns')"
|
||||
|
||||
lemma (* test case *)
|
||||
"shift [(B0, B1, B0, B1), (B1, B0, B1, B0)] B0
|
||||
= [(B1, B0, B1, B1), (B0, B1, B0, B0)]"
|
||||
by eval
|
||||
|
||||
lemma (* test case *)
|
||||
"shift [(B1, B1, B0, B1), (B1, B0, B1, B0)] B0
|
||||
= [(B0, B0, B0, B1), (B1, B0, B1, B1), (B0, B1, B0, B0)]"
|
||||
by eval
|
||||
|
||||
lemma (* test case *)
|
||||
"shift [] B1
|
||||
= [(B0, B0, B0, B1)]"
|
||||
by eval
|
||||
|
||||
subsection \<open>Functions for adding 3 to BCD digits\<close>
|
||||
|
||||
(* Even though this function is only used on nibbles 0 to 9, I've
|
||||
defined it as a total, bijective function so that it is invertible;
|
||||
this seems to make the proof easier later on. *)
|
||||
fun maybe_add3 :: "nibble \<Rightarrow> nibble"
|
||||
where
|
||||
"maybe_add3 (B0,B0,B0,B0) = (B0,B0,B0,B0)" (* 0 \<rightarrow> 0 *)
|
||||
| "maybe_add3 (B0,B0,B0,B1) = (B0,B0,B0,B1)" (* 1 \<rightarrow> 1 *)
|
||||
| "maybe_add3 (B0,B0,B1,B0) = (B0,B0,B1,B0)" (* 2 \<rightarrow> 2 *)
|
||||
| "maybe_add3 (B0,B0,B1,B1) = (B0,B0,B1,B1)" (* 3 \<rightarrow> 3 *)
|
||||
| "maybe_add3 (B0,B1,B0,B0) = (B0,B1,B0,B0)" (* 4 \<rightarrow> 4 *)
|
||||
| "maybe_add3 (B0,B1,B0,B1) = (B1,B0,B0,B0)" (* 5 \<rightarrow> 8 *)
|
||||
| "maybe_add3 (B0,B1,B1,B0) = (B1,B0,B0,B1)" (* 6 \<rightarrow> 9 *)
|
||||
| "maybe_add3 (B0,B1,B1,B1) = (B1,B0,B1,B0)" (* 7 \<rightarrow> 10 *)
|
||||
| "maybe_add3 (B1,B0,B0,B0) = (B1,B0,B1,B1)" (* 8 \<rightarrow> 11 *)
|
||||
| "maybe_add3 (B1,B0,B0,B1) = (B1,B1,B0,B0)" (* 9 \<rightarrow> 12 *)
|
||||
| "maybe_add3 (B1,B0,B1,B0) = (B1,B1,B0,B1)" (* 10 \<rightarrow> 13 *)
|
||||
| "maybe_add3 (B1,B0,B1,B1) = (B1,B1,B1,B0)" (* 11 \<rightarrow> 14 *)
|
||||
| "maybe_add3 (B1,B1,B0,B0) = (B1,B1,B1,B1)" (* 12 \<rightarrow> 15 *)
|
||||
| "maybe_add3 (B1,B1,B0,B1) = (B0,B1,B0,B1)" (* 13 \<rightarrow> 5 *)
|
||||
| "maybe_add3 (B1,B1,B1,B0) = (B0,B1,B1,B0)" (* 14 \<rightarrow> 6 *)
|
||||
| "maybe_add3 (B1,B1,B1,B1) = (B0,B1,B1,B1)" (* 15 \<rightarrow> 7 *)
|
||||
|
||||
fun maybe_add3_inv :: "nibble \<Rightarrow> nibble"
|
||||
(* It's sometimes handy to be able to run the `maybe_add3` function backwards *)
|
||||
where
|
||||
"maybe_add3_inv (B0,B0,B0,B0) = (B0,B0,B0,B0)"
|
||||
| "maybe_add3_inv (B0,B0,B0,B1) = (B0,B0,B0,B1)"
|
||||
| "maybe_add3_inv (B0,B0,B1,B0) = (B0,B0,B1,B0)"
|
||||
| "maybe_add3_inv (B0,B0,B1,B1) = (B0,B0,B1,B1)"
|
||||
| "maybe_add3_inv (B0,B1,B0,B0) = (B0,B1,B0,B0)"
|
||||
| "maybe_add3_inv (B1,B0,B0,B0) = (B0,B1,B0,B1)"
|
||||
| "maybe_add3_inv (B1,B0,B0,B1) = (B0,B1,B1,B0)"
|
||||
| "maybe_add3_inv (B1,B0,B1,B0) = (B0,B1,B1,B1)"
|
||||
| "maybe_add3_inv (B1,B0,B1,B1) = (B1,B0,B0,B0)"
|
||||
| "maybe_add3_inv (B1,B1,B0,B0) = (B1,B0,B0,B1)"
|
||||
| "maybe_add3_inv (B1,B1,B0,B1) = (B1,B0,B1,B0)"
|
||||
| "maybe_add3_inv (B1,B1,B1,B0) = (B1,B0,B1,B1)"
|
||||
| "maybe_add3_inv (B1,B1,B1,B1) = (B1,B1,B0,B0)"
|
||||
| "maybe_add3_inv (B0,B1,B0,B1) = (B1,B1,B0,B1)"
|
||||
| "maybe_add3_inv (B0,B1,B1,B0) = (B1,B1,B1,B0)"
|
||||
| "maybe_add3_inv (B0,B1,B1,B1) = (B1,B1,B1,B1)"
|
||||
|
||||
lemma maybe_add3_inv1:
|
||||
"maybe_add3_inv (maybe_add3 n) = n"
|
||||
by (cases n, metis (full_types) maybe_add3.simps maybe_add3_inv.simps)
|
||||
|
||||
lemma maybe_add3_inv2:
|
||||
"maybe_add3 (maybe_add3_inv n) = n"
|
||||
by (cases n, metis (full_types) maybe_add3.simps maybe_add3_inv.simps)
|
||||
|
||||
|
||||
subsection \<open>Converting binary to BCD\<close>
|
||||
|
||||
fun binary_to_bcd_helper :: "nibble list \<Rightarrow> bit list \<Rightarrow> nibble list"
|
||||
where
|
||||
"binary_to_bcd_helper ns [] = ns"
|
||||
| "binary_to_bcd_helper ns (b # bs) = binary_to_bcd_helper (shift (map maybe_add3 ns) b) bs"
|
||||
|
||||
fun binary_to_bcd :: "bit list \<Rightarrow> nibble list"
|
||||
where
|
||||
"binary_to_bcd bs = binary_to_bcd_helper [] bs"
|
||||
|
||||
lemma (* test case *)
|
||||
"binary_to_bcd [B1,B0,B1,B0,B1,B0,B1] = [(B1,B0,B0,B0), (B0,B1,B0,B1)]"
|
||||
by eval
|
||||
|
||||
(* more test cases *)
|
||||
lemma "binary_to_bcd [B1,B1,B1] = [(B0,B1,B1,B1)]" by eval
|
||||
lemma "binary_to_bcd [B1,B0,B0,B0,B0,B0,B0,B0] = [(B0,B0,B0,B1),(B0,B0,B1,B0),(B1,B0,B0,B0)]" by eval
|
||||
lemma "binary_to_bcd [B0,B0,B0,B0,B0,B1] = [(B0,B0,B0,B1)]" by eval
|
||||
lemma "binary_to_bcd [B1,B1,B0,B0,B0,B0,B0] = [(B1,B0,B0,B1),(B0,B1,B1,B0)]" by eval
|
||||
|
||||
subsection \<open>Checking that nibbles correspond to valid BCD digits\<close>
|
||||
|
||||
fun valid_nibble :: "nibble \<Rightarrow> bool"
|
||||
where
|
||||
"valid_nibble (B0,B0,B0,B0) = True"
|
||||
| "valid_nibble (B0,B0,B0,B1) = True"
|
||||
| "valid_nibble (B0,B0,B1,B0) = True"
|
||||
| "valid_nibble (B0,B0,B1,B1) = True"
|
||||
| "valid_nibble (B0,B1,B0,B0) = True"
|
||||
| "valid_nibble (B0,B1,B0,B1) = True"
|
||||
| "valid_nibble (B0,B1,B1,B0) = True"
|
||||
| "valid_nibble (B0,B1,B1,B1) = True"
|
||||
| "valid_nibble (B1,B0,B0,B0) = True"
|
||||
| "valid_nibble (B1,B0,B0,B1) = True"
|
||||
| "valid_nibble (B1,B0,B1,B0) = False"
|
||||
| "valid_nibble (B1,B0,B1,B1) = False"
|
||||
| "valid_nibble (B1,B1,B0,B0) = False"
|
||||
| "valid_nibble (B1,B1,B0,B1) = False"
|
||||
| "valid_nibble (B1,B1,B1,B0) = False"
|
||||
| "valid_nibble (B1,B1,B1,B1) = False"
|
||||
|
||||
lemma shift_helper_valid:
|
||||
"list_all valid_nibble ns \<Longrightarrow>
|
||||
list_all valid_nibble (snd (shift_helper (map maybe_add3 ns) b))"
|
||||
proof (induct ns)
|
||||
case Nil
|
||||
thus ?case by simp
|
||||
next
|
||||
case (Cons n ns)
|
||||
thus ?case
|
||||
apply auto
|
||||
apply (cases "maybe_add3 n")
|
||||
apply auto
|
||||
apply (split prod.split)
|
||||
apply auto
|
||||
apply (cases n)
|
||||
apply (smt (z3) maybe_add3.simps valid_nibble.simps prod.inject)
|
||||
done
|
||||
qed
|
||||
|
||||
lemma binary_to_bcd_helper_step_valid:
|
||||
assumes "list_all valid_nibble ns"
|
||||
shows "list_all valid_nibble (shift (map maybe_add3 ns) b)"
|
||||
using assms
|
||||
apply auto
|
||||
apply (split prod.split)
|
||||
apply (smt (z3) list_all_simps(1) shift_helper_valid snd_conv valid_nibble.simps(2))
|
||||
done
|
||||
|
||||
lemma binary_to_bcd_helper_valid:
|
||||
"list_all valid_nibble ns \<Longrightarrow>
|
||||
list_all valid_nibble (binary_to_bcd_helper ns bs)"
|
||||
proof (induct bs arbitrary: ns)
|
||||
case Nil
|
||||
thus ?case by auto
|
||||
next
|
||||
case (Cons b bs)
|
||||
thus ?case using binary_to_bcd_helper_step_valid by auto
|
||||
qed
|
||||
|
||||
theorem binary_to_bcd_valid:
|
||||
"list_all valid_nibble (binary_to_bcd bs)"
|
||||
using binary_to_bcd_helper_valid by auto
|
||||
|
||||
subsection \<open>Proof that the binary_to_bcd translation is correct.\<close>
|
||||
|
||||
(* `shift_helper` doesn't change the length of its list *)
|
||||
lemma length_shift_helper:
|
||||
"length ns = length (snd (shift_helper ns b))"
|
||||
proof (induct ns)
|
||||
case Nil
|
||||
thus ?case by auto
|
||||
next
|
||||
case (Cons a ns)
|
||||
show ?case
|
||||
apply auto
|
||||
apply (cases a)
|
||||
apply auto
|
||||
apply (cases "shift_helper ns b")
|
||||
apply (auto simp add: Cons)
|
||||
done
|
||||
qed
|
||||
|
||||
lemma shift_helper:
|
||||
"list_all valid_nibble ns \<Longrightarrow>
|
||||
shift_helper (map maybe_add3 ns) b = (c, ns') \<Longrightarrow>
|
||||
bcd_to_nat ((B0,B0,B0,c) # ns') = bcd_to_nat ns * 2 + (if b then 1 else 0)"
|
||||
proof (induct ns arbitrary: c ns')
|
||||
case Nil
|
||||
thus ?case by simp
|
||||
next
|
||||
case (Cons n ns)
|
||||
|
||||
obtain c' ns'' where *: "shift_helper (map maybe_add3 ns) b = (c', ns'')" by fastforce
|
||||
with Cons have IH:
|
||||
"bcd_to_nat ((B0, B0, B0, c') # ns'') = bcd_to_nat ns * 2 + (if b then 1 else 0)" by simp
|
||||
|
||||
obtain b1 b2 b3 b4 where maybe_add3_n: "maybe_add3 n = (b1, b2, b3, b4)"
|
||||
by (cases rule: prod_cases4)
|
||||
hence "maybe_add3_inv (maybe_add3 n) = maybe_add3_inv (b1, b2, b3, b4)" by auto
|
||||
with maybe_add3_inv1 have n_def: "n = maybe_add3_inv (b1, b2, b3, b4)" by auto
|
||||
|
||||
have "c = b1" and ns'_def: "ns' = (b2, b3, b4, c') # ns''" using maybe_add3_n * Cons by auto
|
||||
|
||||
have "length ns'' = length ns"
|
||||
by (metis * length_map length_shift_helper snd_conv)
|
||||
|
||||
have "nibble_to_nat (b2, b3, b4, c') * 10 ^ length ns +
|
||||
nibble_to_nat (B0, B0, B0, b1) * 10 * 10 ^ length ns =
|
||||
nibble_to_nat (B0, B0, B0, c') * 10 ^ length ns +
|
||||
nibble_to_nat n * 10 ^ length ns * 2"
|
||||
apply (rule cases_b5[of b1 b2 b3 b4 c'])
|
||||
using Cons apply (auto simp add: n_def)
|
||||
done
|
||||
thus ?case using ns'_def IH `length ns'' = length ns` `c = b1` by auto
|
||||
qed
|
||||
|
||||
(* Determines the "current state" of a translation-in-progress. See examples below. *)
|
||||
fun bcd_binary_to_nat :: "nibble list \<Rightarrow> bit list \<Rightarrow> nat"
|
||||
where
|
||||
"bcd_binary_to_nat ns bs = bcd_to_nat ns * 2 ^ length bs + binary_to_nat bs"
|
||||
|
||||
lemma "bcd_binary_to_nat [] [B1,B0,B1,B0,B1,B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B0,B0,B1)] [B0,B1,B0,B1,B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B0,B1,B0)] [B1,B0,B1,B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B1,B0,B1)] [B0,B1,B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B0,B0,B1), (B0,B0,B0,B0)] [B1,B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B0,B1,B0), (B0,B0,B0,B1)] [B0,B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B0,B1,B0,B0), (B0,B0,B1,B0)] [B1] = 85" by eval
|
||||
lemma "bcd_binary_to_nat [(B1,B0,B0,B0), (B0,B1,B0,B1)] [] = 85" by eval
|
||||
|
||||
lemma binary_to_bcd_helper_step_correct:
|
||||
assumes "list_all valid_nibble ns"
|
||||
shows "bcd_binary_to_nat ns (b # bs) =
|
||||
bcd_binary_to_nat (shift (map maybe_add3 ns) b) bs"
|
||||
proof -
|
||||
have "bcd_to_nat (shift (map maybe_add3 ns) B0) = bcd_to_nat ns * 2"
|
||||
apply (auto simp del: bcd_to_nat.simps)
|
||||
apply (split prod.split)
|
||||
apply (auto simp del: bcd_to_nat.simps)
|
||||
using assms shift_helper apply presburger
|
||||
apply (smt (z3) assms shift_helper add.right_neutral bcd_to_nat.simps(2)
|
||||
mult_zero_left nibble_to_nat.simps(1))
|
||||
done
|
||||
moreover
|
||||
have "bcd_to_nat (shift (map maybe_add3 ns) B1) = bcd_to_nat ns * 2 + 1"
|
||||
apply (auto simp del: bcd_to_nat.simps)
|
||||
apply (split prod.split)
|
||||
apply (auto simp del: bcd_to_nat.simps)
|
||||
using assms shift_helper apply presburger
|
||||
apply (smt (z3) One_nat_def add_Suc_shift add_right_imp_eq assms bcd_to_nat.simps(2)
|
||||
mult_zero_left nibble_to_nat.simps(1) shift_helper)
|
||||
done
|
||||
ultimately show ?thesis using assms by auto
|
||||
qed
|
||||
|
||||
lemma binary_to_bcd_helper_correct:
|
||||
"list_all valid_nibble ns \<Longrightarrow>
|
||||
bcd_to_nat (binary_to_bcd_helper ns bs) = bcd_binary_to_nat ns bs"
|
||||
proof (induct bs arbitrary: ns)
|
||||
case Nil
|
||||
thus ?case by simp
|
||||
next
|
||||
case Cons
|
||||
thus ?case
|
||||
using binary_to_bcd_helper_step_correct and binary_to_bcd_helper_step_valid
|
||||
by auto
|
||||
qed
|
||||
|
||||
theorem binary_to_bcd_correct:
|
||||
"bcd_to_nat (binary_to_bcd bs) = binary_to_nat bs"
|
||||
using binary_to_bcd_helper_correct by auto
|
||||
|
||||
section \<open> Alternative implementation of binary_to_bcd that converts to nat and then to bcd \<close>
|
||||
|
||||
lemma mod10_induct [case_names 0 1 2 3 4 5 6 7 8 9]:
|
||||
fixes n :: nat
|
||||
assumes "P 0" "P 1" "P 2" "P 3" "P 4" "P 5" "P 6" "P 7" "P 8" "P 9"
|
||||
shows "P (n mod 10)"
|
||||
proof -
|
||||
have "n mod 10 \<in> {..<10}"
|
||||
by simp
|
||||
also have "{..<10} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 :: nat}"
|
||||
by (simp add: lessThan_nat_numeral lessThan_Suc insert_commute)
|
||||
finally show ?thesis using assms
|
||||
by fastforce
|
||||
qed
|
||||
|
||||
fun nat_to_nibble :: "nat \<Rightarrow> nibble"
|
||||
where
|
||||
"nat_to_nibble n = (
|
||||
if n = 0 then (B0,B0,B0,B0) else
|
||||
if n = 1 then (B0,B0,B0,B1) else
|
||||
if n = 2 then (B0,B0,B1,B0) else
|
||||
if n = 3 then (B0,B0,B1,B1) else
|
||||
if n = 4 then (B0,B1,B0,B0) else
|
||||
if n = 5 then (B0,B1,B0,B1) else
|
||||
if n = 6 then (B0,B1,B1,B0) else
|
||||
if n = 7 then (B0,B1,B1,B1) else
|
||||
if n = 8 then (B1,B0,B0,B0) else
|
||||
if n = 9 then (B1,B0,B0,B1) else
|
||||
(B1,B1,B1,B1))" (* unreachable *)
|
||||
|
||||
fun nat_to_bcd :: "nat \<Rightarrow> nibble list"
|
||||
where
|
||||
"nat_to_bcd n = (if n = 0 then [] else
|
||||
nat_to_bcd (n div 10) @ [nat_to_nibble (n mod 10)])"
|
||||
|
||||
value "nat_to_bcd 420"
|
||||
value "nat_to_bcd 42"
|
||||
value "nat_to_bcd 4"
|
||||
value "nat_to_bcd 0"
|
||||
|
||||
fun binary_to_bcd2 :: "bit list \<Rightarrow> nibble list"
|
||||
where
|
||||
"binary_to_bcd2 bs = nat_to_bcd (binary_to_nat bs)"
|
||||
|
||||
lemma nat_to_nibble_valid:
|
||||
"valid_nibble (nat_to_nibble (n mod 10))"
|
||||
by (induct rule: mod10_induct, auto)
|
||||
|
||||
lemma nat_to_bcd_valid:
|
||||
"list_all valid_nibble (nat_to_bcd n)"
|
||||
using nat_to_nibble_valid
|
||||
by (induct rule: nat_to_bcd.induct, simp)
|
||||
|
||||
theorem binary_to_bcd2_valid:
|
||||
"list_all valid_nibble (binary_to_bcd2 bs)"
|
||||
using nat_to_bcd_valid by auto
|
||||
|
||||
lemma bcd_to_nat_snoc:
|
||||
"bcd_to_nat (ns @ [n]) = bcd_to_nat ns * 10 + bcd_to_nat [n]"
|
||||
by (induct ns, auto)
|
||||
|
||||
lemma bcd_to_nat_nibble:
|
||||
"bcd_to_nat [nat_to_nibble (n mod 10)] = n mod 10"
|
||||
by (induct rule: mod10_induct, auto)
|
||||
|
||||
lemma bcd_to_nat_inv:
|
||||
"bcd_to_nat (nat_to_bcd n) = n"
|
||||
apply (induct rule: nat_to_bcd.induct)
|
||||
apply (subst nat_to_bcd.simps)
|
||||
apply (case_tac "n=0")
|
||||
apply simp
|
||||
apply (simp del: nat_to_bcd.simps nat_to_nibble.simps)
|
||||
apply (subst bcd_to_nat_snoc)
|
||||
apply (subst bcd_to_nat_nibble)
|
||||
apply simp
|
||||
done
|
||||
|
||||
theorem binary_to_bcd2_correct:
|
||||
"bcd_to_nat (binary_to_bcd2 bs) = binary_to_nat bs"
|
||||
using bcd_to_nat_inv by auto
|
||||
|
||||
end
|
||||
Loading…
Reference in a new issue