From 0e8514ef999df77e5538208f48ca6ce7d58dd428 Mon Sep 17 00:00:00 2001 From: John Wickerson <> Date: Mon, 1 Nov 2021 06:34:44 +0000 Subject: [PATCH] finalised cw for 2021 --- README.md | 5 + dafny/2019/dafny_exercises_2019.pdf | Bin 0 -> 85849 bytes dafny/2019/solutions_2019.dfy | 175 ++++++++ dafny/2019/tasks_2019.dfy | 110 +++++ dafny/2020/dafny_exercises_2020.pdf | Bin 0 -> 73159 bytes dafny/2020/solutions_2020.dfy | 200 +++++++++ dafny/2020/tasks_2020.dfy | 122 ++++++ dafny/2021/dafny_exercises_2021.pdf | Bin 0 -> 85995 bytes dafny/2021/tasks_2021.dfy | 162 ++++++++ dafny/dafny.pdf | Bin 0 -> 110020 bytes dafny/dafny_worksheet.dfy | 71 ++++ isabelle/2019/HSV_tasks_2019.thy | 21 + isabelle/2019/HSV_tasks_2019_solutions.thy | 392 ++++++++++++++++++ .../2019/HSV_tasks_2019_solutions_task11.thy | 204 +++++++++ isabelle/2019/isabelle_exercises_2019.pdf | Bin 0 -> 132477 bytes isabelle/2020/HSV_tasks_2020.thy | 92 ++++ isabelle/2020/isabelle_exercises_2020.pdf | Bin 0 -> 142088 bytes isabelle/HSV_chapter3.thy | 59 +++ isabelle/HSV_chapter4.thy | 83 ++++ isabelle/HSV_chapter5.thy | 136 ++++++ isabelle/isabelle.pdf | Bin 0 -> 284046 bytes lectures/lecture10_solving.pdf | Bin 0 -> 3755743 bytes lectures/lecture1_intro_john.pdf | Bin 0 -> 4510929 bytes lectures/lecture3_dafny.pdf | Bin 0 -> 585987 bytes lectures/lecture4_isabelle.pdf | Bin 0 -> 952162 bytes lectures/lecture5_isabelle2.pdf | Bin 0 -> 337133 bytes 26 files changed, 1832 insertions(+) create mode 100644 README.md create mode 100644 dafny/2019/dafny_exercises_2019.pdf create mode 100644 dafny/2019/solutions_2019.dfy create mode 100644 dafny/2019/tasks_2019.dfy create mode 100644 dafny/2020/dafny_exercises_2020.pdf create mode 100644 dafny/2020/solutions_2020.dfy create mode 100644 dafny/2020/tasks_2020.dfy create mode 100644 dafny/2021/dafny_exercises_2021.pdf create mode 100644 dafny/2021/tasks_2021.dfy create mode 100644 dafny/dafny.pdf create mode 100644 dafny/dafny_worksheet.dfy create mode 100644 isabelle/2019/HSV_tasks_2019.thy create mode 100644 isabelle/2019/HSV_tasks_2019_solutions.thy create mode 100644 isabelle/2019/HSV_tasks_2019_solutions_task11.thy create mode 100644 isabelle/2019/isabelle_exercises_2019.pdf create mode 100644 isabelle/2020/HSV_tasks_2020.thy create mode 100644 isabelle/2020/isabelle_exercises_2020.pdf create mode 100644 isabelle/HSV_chapter3.thy create mode 100644 isabelle/HSV_chapter4.thy create mode 100644 isabelle/HSV_chapter5.thy create mode 100644 isabelle/isabelle.pdf create mode 100644 lectures/lecture10_solving.pdf create mode 100644 lectures/lecture1_intro_john.pdf create mode 100644 lectures/lecture3_dafny.pdf create mode 100644 lectures/lecture4_isabelle.pdf create mode 100644 lectures/lecture5_isabelle2.pdf diff --git a/README.md b/README.md new file mode 100644 index 0000000..04e42dc --- /dev/null +++ b/README.md @@ -0,0 +1,5 @@ +# Hardware & Software Verification + +This is the homepage of the above-named module, which is offered to MSc and 4th-year MEng students at Imperial College London. + +The module is run by Dr John Wickerson (software) and Professor Pete Harrod (hardware). diff --git a/dafny/2019/dafny_exercises_2019.pdf b/dafny/2019/dafny_exercises_2019.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aad82c5156a362c1c7096a91774daf59f800be6a GIT binary patch literal 85849 zcma&NV~j39yRO-`u^OxSwr$&7ZQHhO+qSV9t8Lr1ZToG^H)rN#PO^70yM8^D)UT>k zuKTGgMW!GsPRB&g21hoxG`t4KPQ*xLZ)63>%L~UKYieiiVnM{j#lc1NzXlwGxTTGY zsS^={xQ(HUsi>*3y@@FtKR=wai<7CLEu6>3wWf^yaSP@SP`!d@+lYR;wL&P-D2mj! zL2^DWu)v|AfNVqQFAbO15L?vuE5U=pt299@ANr^umgZE#YXUECeEcc#;t_e_xc=-J z%b`B~%uMeJ?S6{Iw5pS1E~AJPJ$el%Y7iNgsZ*w(W!)ye^7)|MEasnRy9?;nL+b8b zrYyHFx`o``NBf`r%V;l$!JE-&`&mjuSTeltV=3fSdcW^SQYn;RDL~1IUJSwuRq|CR zK-?PRaI@CxzUE@9v*Nmxw8eRYHdms!2X` zjJQOdN}{biM>*`t_j>@d?OpPmz z4IZAQ>*)i99KA!0mDQai0fmX^N*qErtx}U#EG};!SK~UJ#lBR^4xK1^74eLjel(#o z!iu7NgU-jgP)$eTc0-nU0g~Wk5_PPgpA-$9QFcV?7*aUiUT7FeoUd4N6-|hhT$&LB 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index <= j < A.Length ==> A[i] <= A[j] +} + +// Task 2 +method bubble_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant partitioned(A, A.Length - i) + invariant sorted_between(A, A.Length - i, A.Length) + decreases A.Length - i + { + var j := 1; + while j < A.Length - i + invariant 0 <= i <= A.Length + invariant 0 < j <= A.Length - i + invariant partitioned(A, A.Length - i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant forall m :: 0 <= m < j-1 ==> A[m] <= A[j-1] + decreases A.Length - j + { + if A[j-1] > A[j] { + A[j-1], A[j] := A[j], A[j-1]; + } + j := j+1; + } + i := i+1; + } +} + +// Task 3 +method selection_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant sorted_between(A, 0, i) + invariant partitioned(A, i) + decreases A.Length - i + { + var k := i; + var j := i+1; + while j < A.Length + invariant 0 <= i <= k < j <= A.Length + invariant sorted_between(A, 0, i) + invariant partitioned(A, i) + invariant forall n :: i <= n < j ==> A[k] <= A[n] + decreases A.Length - j + { + if A[k] > A[j] { + k := j; + } + j := j+1; + } + A[k], A[i] := A[i], A[k]; + i := i + 1; + } +} + +// Task 4 +method insertion_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant sorted_between(A, 0, i) + decreases A.Length - i + { + var j := i; + var tmp := A[j]; + while 1 <= j && tmp < A[j-1] + invariant 0 <= j <= i <= A.Length + invariant forall l :: j <= l < i ==> tmp < A[l] + invariant sorted_between(A, 0, i) + invariant j < i ==> sorted_between(A, 0, i+1) + decreases j + { + A[j] := A[j-1]; + j := j-1; + } + A[j] := tmp; + i := i+1; + } +} + +// Task 5 +method shellsort(A:array) + modifies A + ensures sorted(A) +{ + var stride := A.Length / 2; + while 0 < stride + invariant 0 <= stride + invariant stride == 0 ==> sorted(A) + decreases stride + { + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant stride==1 ==> sorted_between(A, 0, i) + decreases A.Length - i + { + var j := i; + var tmp := A[j]; + while stride <= j && tmp < A[j-stride] + invariant 0 <= j <= i <= A.Length + invariant stride==1 ==> forall l :: j <= l < i ==> tmp < A[l] + invariant stride==1 ==> sorted_between(A, 0, i) + invariant stride==1 ==> j < i ==> sorted_between(A, 0, i+1) + decreases j + { + A[j] := A[j-stride]; + j := j-stride; + } + A[j] := tmp; + i := i+1; + } + stride := stride / 2; + } +} + +// Task 6 +method john_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant forall j :: 0 <= j < i ==> A[j] == 42 + decreases A.Length - i + { + A[i] := 42; + i := i + 1; + } +} + +method Main() { + var A:array := new int[7] [4,0,1,9,7,1,2]; + print "Before: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; + bubble_sort(A); + print "After: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; +} \ No newline at end of file diff --git a/dafny/2019/tasks_2019.dfy b/dafny/2019/tasks_2019.dfy new file mode 100644 index 0000000..a925541 --- /dev/null +++ b/dafny/2019/tasks_2019.dfy @@ -0,0 +1,110 @@ +// Dafny coursework tasks +// Autumn term, 2019 +// +// Authors: John Wickerson and Matt Windsor + +// Task 1 +predicate sorted(A:array) + reads A +{ + forall m,n :: 0 <= m < n < A.Length ==> A[m] <= A[n] +} + +// Task 2 +method bubble_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length { + var j := 1; + while j < A.Length - i { + if A[j-1] > A[j] { + A[j-1], A[j] := A[j], A[j-1]; + } + j := j+1; + } + i := i+1; + } +} + +// Task 3 +method selection_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length { + var k := i; + var j := i+1; + while j < A.Length { + if A[k] > A[j] { + k := j; + } + j := j+1; + } + A[k], A[i] := A[i], A[k]; + i := i + 1; + } +} + +// Task 4 +method insertion_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length { + var j := i; + var tmp := A[j]; + while 1 <= j && tmp < A[j-1] { + A[j] := A[j-1]; + j := j-1; + } + A[j] := tmp; + i := i+1; + } +} + +// Task 5 +method shellsort(A:array) + modifies A + ensures sorted(A) +{ + var stride := A.Length / 2; + while 0 < stride { + var i := 0; + while i < A.Length { + var j := i; + var tmp := A[j]; + while stride <= j && tmp < A[j-stride] { + A[j] := A[j-stride]; + j := j-stride; + } + A[j] := tmp; + i := i+1; + } + stride := stride / 2; + } +} + +// Task 6 +method john_sort(A:array) + modifies A + ensures sorted(A) +{ + var i := 0; + while i < A.Length { + A[i] := 42; + i := i + 1; + } +} + +method Main() { + var A:array := new int[7] [4,0,1,9,7,1,2]; + print "Before: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; + bubble_sort(A); + print "After: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; +} \ No newline at end of file diff --git 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Difficulty: * +method create_multiples_of_two(A:array) +ensures forall n :: 0 <= n < A.Length ==> A[n] == 2*n +modifies A +{ + // ... +} + +// Task 2. Difficulty: ** +method exchange_sort (A:array) +ensures sorted(A) +modifies A +{ + var i := 0; + while i < A.Length - 1 { + var j := i + 1; + while j < A.Length { + if A[i] > A[j] { + A[i], A[j] := A[j], A[i]; + } + j := j + 1; + } + i := i + 1; + } +} + +// Task 3. Difficulty: ** +method fung_sort (A:array) +ensures sorted(A) +modifies A +{ + var i := 0; + while i < A.Length { + var j := 0; + while j < A.Length { + if A[i] < A[j] { + A[i], A[j] := A[j], A[i]; + } + j := j+1; + } + i := i+1; + } +} + +// Task 4. Difficulty: *** +predicate odd_sorted(A:array, hi:int) + reads A +{ + forall i :: 0 <= i ==> 2*i+2 < hi <= A.Length ==> A[2*i + 1] <= A[2*i+2] +} + +method odd_even_sort(A:array) + ensures sorted(A) + modifies A + decreases * +{ + var is_sorted := false; + while !is_sorted { + is_sorted := true; + var i := 0; + while 2*i+2 < A.Length { + if A[2*i+2] < A[2*i+1] { + A[2*i+1], A[2*i+2] := A[2*i+2], A[2*i+1]; + is_sorted := false; + } + i := i+1; + } + i := 0; + while 2*i+1 < A.Length { + if A[2*i+1] < A[2*i] { + A[2*i], A[2*i+1] := A[2*i+1], A[2*i]; + is_sorted := false; + } + i := i+1; + } + } +} + + +// Task 5. Difficulty: *** +method bubble_sort3(A:array) + ensures sorted(A) + modifies A + decreases * +{ + var stable := false; + while !stable { + stable := true; + var j := 0; + while 2*j+2 <= A.Length { + if 2*j+2 == A.Length { + if A[2*j+1] < A[2*j] { + A[2*j+1], A[2*j] := A[2*j], A[2*j+1]; + stable := false; + } + } else { + if A[2*j+1] < A[2*j] { + A[2*j+1], A[2*j] := A[2*j], A[2*j+1]; + stable := false; + } + if A[2*j+2] < A[2*j+1] { + A[2*j+2], A[2*j+1] := A[2*j+1], A[2*j+2]; + stable := false; + } + if A[2*j+1] < A[2*j] { + A[2*j+1], A[2*j] := A[2*j], A[2*j+1]; + stable := false; + } + } + j := j + 1; + } + } +} + +// Task 6. Difficulty: * +function method get_min(a:int, b:int) : int +{ + if a <= b then a else b +} + +method sort3(a:int, b:int, c:int) returns (min:int, mid:int, max:int) +ensures min in {a,b,c} +ensures mid in {a,b,c} +ensures max in {a,b,c} +ensures a in {min, mid, max} +ensures b in {min, mid, max} +ensures c in {min, mid, max} +ensures min <= mid <= max +{ + min := get_min(a, get_min(b,c)); + max := -get_min(-a, get_min(-b,-c)); + mid := a + b + c - max - min; +} + +method print_array(A:array) +{ + var i := 0; + while i < A.Length { + print A[i]; + i := i + 1; + } +} + +method Main() +decreases * +{ + var A:array := new int[7] [4,0,1,9,7,1,2]; + print "Before: "; print_array(A); print "\n"; + exchange_sort(A); + print "After: "; print_array(A); print "\n"; +} \ No newline at end of file diff --git a/dafny/dafny.pdf b/dafny/dafny.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7103bb048262fef73533fd1eec7a62fa5b289246 GIT binary patch literal 110020 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i + { + i := i+1; + t := t+i; + } + return t; +} + +method max (a:int, b:int) returns (r:int) + ensures r >= a + ensures r >= b + ensures r == a || r == b +{ + if a>b { + return a; + } else { + return b; + } +} + +predicate contains_upto(A:array, hi:int, v:int) + reads A + requires hi <= A.Length +{ + exists j :: 0 <= j < hi && v == A[j] +} + +predicate contains(A:array, v:int) + reads A +{ + contains_upto(A, A.Length, v) +} + +method maxarray (A:array) returns (r:int) + requires A.Length > 0 + ensures contains(A, r) + ensures forall j :: 0 <= j < A.Length ==> r >= A[j] +{ + r := A[0]; + var i := 1; + while i < A.Length + invariant 1 <= i <= A.Length + invariant contains_upto(A, i, r) + invariant forall j :: 0 <= j < i ==> r >= A[j] + decreases A.Length - i + { + if r < A[i] { + r := A[i]; + } + i := i+1; + } + assert i == A.Length; +} + +method Main() { + var m:int := max(3,5); + print m, "\n"; +} \ No newline at end of file diff --git a/isabelle/2019/HSV_tasks_2019.thy b/isabelle/2019/HSV_tasks_2019.thy new file mode 100644 index 0000000..786fe80 --- /dev/null +++ b/isabelle/2019/HSV_tasks_2019.thy @@ -0,0 +1,21 @@ +theory HSV_tasks_2019 imports Main begin + +datatype "circuit" = + NOT "circuit" +| AND "circuit" "circuit" +| OR "circuit" "circuit" +| TRUE +| FALSE +| INPUT "int" + +section \Starting point for Task 8\ + +text \The following function calculates the area of a circuit (i.e. number of gates).\ + +fun area :: "circuit \ nat" where + "area (NOT c) = 1 + area c" +| "area (AND c1 c2) = 1 + area c1 + area c2" +| "area (OR c1 c2) = 1 + area c1 + area c2" +| "area _ = 0" + +end \ No newline at end of file diff --git a/isabelle/2019/HSV_tasks_2019_solutions.thy b/isabelle/2019/HSV_tasks_2019_solutions.thy new file mode 100644 index 0000000..1ae9ae2 --- /dev/null +++ b/isabelle/2019/HSV_tasks_2019_solutions.thy @@ -0,0 +1,392 @@ +theory HSV_tasks_2019_solutions imports Complex_Main begin + +section \Task 1: proving that 2 * sqrt 2 is irrational.\ + +(* The following theorem is copied from Chapter 3 of the worksheet *) +theorem sqrt2_irrational: "sqrt 2 \ \" +proof auto + assume "sqrt 2 \ \" + then obtain m n where + "n \ 0" and "\sqrt 2\ = real m / real n" and "coprime m n" + by (rule Rats_abs_nat_div_natE) + hence "\sqrt 2\^2 = (real m / real n)^2" by auto + hence "2 = (real m / real n)^2" by simp + hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto + hence "2 * (real n)^2 = (real m)^2" + by (simp add: nonzero_eq_divide_eq `n \ 0`) + hence "real (2 * n^2) = (real m)^2" by auto + hence *: "2 * n^2 = m^2" + using of_nat_power_eq_of_nat_cancel_iff by blast + hence "even (m^2)" by presburger + hence "even m" by simp + then obtain m' where "m = 2 * m'" by auto + with * have "2 * n^2 = (2 * m')^2" by auto + hence "2 * n^2 = 4 * m'^2" by simp + hence "n^2 = 2 * m'^2" by simp + hence "even (n^2)" by presburger + hence "even n" by simp + with `even m` and `coprime m n` show False by auto +qed + +theorem "2 * sqrt 2 \ \" +proof auto + assume *: "2 * sqrt 2 \ \" + have "2 \ \" by simp + with * have "2 * sqrt 2 / 2 \ \" using Rats_divide by blast + hence "sqrt 2 \ \" by simp + with sqrt2_irrational show False by simp +qed + +section \Task 2: L-numbers.\ + +fun L :: "nat \ nat" where + "L n = (if n < 2 then n else 2 + L (n - 1))" + +value "L 0" (* should be 0 *) +value "L 1" (* should be 1 *) +value "L 2" (* should be 3 *) +value "L 3" (* should be 5 *) + +lemma L_helper: "L (Suc n) = 2 * (Suc n) - 1" + apply (induct n) + apply simp+ + done + +theorem "L n = (if n = 0 then 0 else 2 * n - 1)" + apply (cases n) + apply simp + using L_helper apply auto + done + +section \Task 3: Pyramidal numbers.\ + +fun py :: "nat \ nat" where + "py n = (if n = 0 then 0 else n^2 + py (n-1))" + +value "py 0" (* 0 *) +value "py 1" (* 1 *) +value "py 2" (* 5 *) +value "py 3" (* 14 *) +value "py 4" (* 30 *) +value "py 5" (* 55 *) +value "py 6" (* 91 *) + + +theorem "py n = ((2 * n + 1) * (n + 1) * n) div 6" +proof (induct n) + case 0 + thus ?case by simp +next + case (Suc n) + assume IH: "py n = ((2 * n + 1) * (n + 1) * n) div 6" (* induction hypothesis *) + + have "py (Suc n) = (Suc n)^2 + py n" by simp + also have "... = (Suc n)^2 + ((2 * n + 1) * (n + 1) * n) div 6" using IH by simp + also have "... = (6 * (n + 1)^2 + (2 * n + 1) * (n + 1) * n) div 6" by simp + also have "... = (6 * (n^2 + 2*n + 1) + (2 * n + 1) * (n + 1) * n) div 6" + by (metis Suc_eq_plus1 add_Suc mult_2 one_add_one plus_1_eq_Suc power2_sum power_one) + also have "... = (2 * n * n + 4 * n + 3 * n + 6) * (n + 1) div 6" + by (simp add: add_mult_distrib power2_eq_square distrib_right) + also have "... = (2 * n * (n + 2) + 3 * (n + 2)) * (n + 1) div 6" + by (smt add.assoc distrib_right mult.commute mult_2_right numeral_Bit0) + also have "... = (2 * n + 3) * (n + 2) * (n + 1) div 6" + by (simp add: distrib_right) + also have "... = (2 * Suc n + 1) * (Suc n + 1) * Suc n div 6" + by (metis One_nat_def Suc_1 Suc_eq_plus1 add_Suc_shift distrib_left mult_2 numeral_3_eq_3) + finally show "py (Suc n) = (2 * Suc n + 1) * (Suc n + 1) * Suc n div 6" by assumption +qed + +section \Task 4: the opt_NOT optimisation leaves no consecutive NOTs in the circuit.\ + +datatype "circuit" = + NOT "circuit" +| AND "circuit" "circuit" +| OR "circuit" "circuit" +| TRUE +| FALSE +| INPUT "int" + +text \Simulates a circuit given a valuation for each input wire\ + +fun simulate where + "simulate (AND c1 c2) \ = ((simulate c1 \) \ (simulate c2 \))" +| "simulate (OR c1 c2) \ = ((simulate c1 \) \ (simulate c2 \))" +| "simulate (NOT c) \ = (\ (simulate c \))" +| "simulate TRUE \ = True" +| "simulate FALSE \ = False" +| "simulate (INPUT i) \ = \ i" + +text \A function that optimises a circuit by removing pairs of consecutive NOT gates\ + +fun opt_NOT where + "opt_NOT (NOT (NOT c)) = opt_NOT c" +| "opt_NOT (NOT c) = NOT (opt_NOT c)" +| "opt_NOT (AND c1 c2) = AND (opt_NOT c1) (opt_NOT c2)" +| "opt_NOT (OR c1 c2) = OR (opt_NOT c1) (opt_NOT c2)" +| "opt_NOT TRUE = TRUE" +| "opt_NOT FALSE = FALSE" +| "opt_NOT (INPUT i) = INPUT i" + +text \We can prove that the optimiser does its job: that the optimised +circuit has no consecutive NOT gates.\ + +fun has_double_NOT where + "has_double_NOT (NOT (NOT c)) = True" +| "has_double_NOT (NOT c) = has_double_NOT c" +| "has_double_NOT (AND c1 c2) = (has_double_NOT c1 \ has_double_NOT c2)" +| "has_double_NOT (OR c1 c2) = (has_double_NOT c1 \ has_double_NOT c2)" +| "has_double_NOT _ = False" + +theorem opt_NOT_is_complete: "\ has_double_NOT (opt_NOT c)" + by (induct c rule: opt_NOT.induct, auto) + +section \Task 5: Removing NOT-NOT is idempotent.\ + +text \We can prove that the optimiser is 'idempotent': applying it twice is the +same as applying it just once.\ + +theorem opt_NOT_is_idempotent: "opt_NOT (opt_NOT c) = opt_NOT c" + by (induct c rule: opt_NOT.induct, auto) + +section \Task 6: Applying De Morgan's laws is sound.\ + +text \Optimising based on De Morgan's laws\ + +fun opt_DM where + "opt_DM (NOT c) = NOT (opt_DM c)" +| "opt_DM (AND (NOT c1) (NOT c2)) = NOT (OR (opt_DM c1) (opt_DM c2))" +| "opt_DM (AND c1 c2) = AND (opt_DM c1) (opt_DM c2)" +| "opt_DM (OR (NOT c1) (NOT c2)) = NOT (AND (opt_DM c1) (opt_DM c2))" +| "opt_DM (OR c1 c2) = OR (opt_DM c1) (opt_DM c2)" +| "opt_DM TRUE = TRUE" +| "opt_DM FALSE = FALSE" +| "opt_DM (INPUT i) = INPUT i" + +theorem opt_DM_is_sound: "simulate (opt_DM c) \ = simulate c \" + by (induct rule: opt_DM.induct, auto) + +text \The following non-theorem is easily contradicted, e.g. + using c = AND (NOT (NOT TRUE)) (NOT (NOT TRUE)). Therefore, + opt_DM isn't idempotent.\ + +theorem opt_DM_is_idempotent: "opt_DM (opt_DM c) = opt_DM c" + oops + +section \Task 7: Applying both optimisations successively is sound.\ + +(* The following theorem was provided in the worksheet *) +theorem opt_NOT_is_sound: "simulate (opt_NOT c) \ = simulate c \" + by (induct rule: opt_NOT.induct, auto) + +theorem both_sound: "simulate (opt_DM (opt_NOT c)) \ = simulate c \" + apply (subst opt_DM_is_sound) + apply (subst opt_NOT_is_sound) + apply auto + done + +section \Task 8: Neither optimisation increases the circuit's area.\ + +text \The following function calculates the area of a circuit (i.e. number of gates).\ + +fun area :: "circuit \ nat" where + "area (NOT c) = 1 + area c" +| "area (AND c1 c2) = 1 + area c1 + area c2" +| "area (OR c1 c2) = 1 + area c1 + area c2" +| "area _ = 0" + +theorem opt_NOT_doesnt_increase_area: "area (opt_NOT c) \ area c" + by (induct rule: opt_NOT.induct, auto) + +theorem opt_DM_doesnt_increase_area: "area (opt_DM c) \ area c" + by (induct rule: opt_DM.induct, auto) + +section \Task 9: Neither optimisation increases the circuit's delay.\ + +text \Delay (assuming all gates have a delay of 1, and constants and inputs are instantaneous)\ + +fun delay :: "circuit \ nat" where + "delay (NOT c) = 1 + delay c" +| "delay (AND c1 c2) = 1 + max (delay c1) (delay c2)" +| "delay (OR c1 c2) = 1 + max (delay c1) (delay c2)" +| "delay _ = 0" + +text \We can prove that our optimisations never increase the maximum delay of a circuit.\ + +theorem opt_NOT_doesnt_increase_delay: "delay (opt_NOT c) \ delay c" + by (induct rule: opt_NOT.induct, auto) + +theorem opt_DM_doesnt_increase_delay: "delay (opt_DM c) \ delay c" + by (induct rule: opt_DM.induct, auto) + +section \Task 10: Constant folding.\ + +text \Here is an optimiser that performs 'constant folding': wherever it sees a gate with +constants (TRUE and FALSE) as inputs, it tries to replace the gate with TRUE, FALSE, or the other +input depending on the logical behaviour of the gate. + +For instance, it replaces NOT FALSE with TRUE, AND FALSE c with FALSE for all inputs c, and +OR c FALSE with c. + +This optimiser is somewhat more complicated than the other optimisers: it has a lot of rules, and +needs to do analysis on the results of recursively applying itself. + +First, we define how it translates NOT gates, AND gates, and OR gates:\ + +fun opt_CF_NOT where + "opt_CF_NOT TRUE = FALSE" +| "opt_CF_NOT FALSE = TRUE" +| "opt_CF_NOT c = NOT c" + +fun opt_CF_AND where + "opt_CF_AND TRUE c2 = c2" +| "opt_CF_AND c1 TRUE = c1" +| "opt_CF_AND FALSE _ = FALSE" +| "opt_CF_AND _ FALSE = FALSE" +| "opt_CF_AND c1 c2 = AND c1 c2" + +fun opt_CF_OR where + "opt_CF_OR FALSE c2 = c2" +| "opt_CF_OR c1 FALSE = c1" +| "opt_CF_OR TRUE _ = TRUE" +| "opt_CF_OR _ TRUE = TRUE" +| "opt_CF_OR c1 c2 = OR c1 c2" + +text \The actual optimiser just runs the above rewrites on each NOT, AND, and OR gate.\ + +fun opt_CF where + "opt_CF (NOT c) = opt_CF_NOT (opt_CF c)" +| "opt_CF (AND c1 c2) = opt_CF_AND (opt_CF c1) (opt_CF c2)" +| "opt_CF (OR c1 c2) = opt_CF_OR (opt_CF c1) (opt_CF c2)" +| "opt_CF x = x" + +text \To prove the constant folder sound, first consider the soundness of the sub-cases.\ + +lemma opt_CF_NOT_is_sound: "simulate (opt_CF_NOT c) \ = simulate (NOT c) \" + by (cases c, auto) + +lemma opt_CF_AND_is_sound: "simulate (opt_CF_AND c1 c2) \ = simulate (AND c1 c2) \" + by (induct c1 c2 rule: opt_CF_AND.induct, auto) + +lemma opt_CF_OR_is_sound: "simulate (opt_CF_OR c1 c2) \ = simulate (OR c1 c2) \" + by (induct c1 c2 rule: opt_CF_OR.induct, auto) + +theorem opt_CF_is_sound: "simulate (opt_CF c) \ = simulate c \" + by (induct c rule: opt_CF.induct, + auto simp add: opt_CF_NOT_is_sound opt_CF_AND_is_sound opt_CF_OR_is_sound) + +text \To get an idea of whether the constant folder is complete, we can see whether it reduces a +circuit with only constant inputs into a single TRUE or FALSE constant. First, define what it +means for a circuit to have no inputs, and to be only a constant.\ + +fun no_inputs where + "no_inputs (NOT c) = no_inputs c" +| "no_inputs (AND c1 c2) = (no_inputs c1 \ no_inputs c2)" +| "no_inputs (OR c1 c2) = (no_inputs c1 \ no_inputs c2)" +| "no_inputs (INPUT _) = False" +| "no_inputs _ = True" + +fun is_constant where + "is_constant c = (c = TRUE \ c = FALSE)" + +lemma opt_CF_NOT_reduce: + assumes "is_constant c" + shows "is_constant (opt_CF_NOT c)" + using assms + by (cases c, auto) + +lemma opt_CF_AND_reduce: + assumes "is_constant c1" and "is_constant c2" + shows "is_constant (opt_CF_AND c1 c2)" + using assms + by (induct c1 c2 rule: opt_CF_AND.induct, auto) + +lemma opt_CF_OR_reduce: + assumes "is_constant c1" and "is_constant c2" + shows "is_constant (opt_CF_OR c1 c2)" + using assms + by (induct c1 c2 rule: opt_CF_OR.induct, auto) + +theorem opt_CF_reduce: + assumes "no_inputs c" + shows "is_constant (opt_CF c)" + using assms + by (induct c rule: opt_CF.induct, + auto simp add: opt_CF_NOT_reduce opt_CF_AND_reduce opt_CF_OR_reduce) + +lemma opt_CF_NOT_doesnt_increase_area: "area (opt_CF_NOT c) \ area (NOT c)" + by (cases c, auto) + +lemma opt_CF_AND_doesnt_increase_area: "area (opt_CF_AND c1 c2) \ area (AND c1 c2)" + by (induct c1 c2 rule: opt_CF_AND.induct, auto) + +lemma opt_CF_OR_doesnt_increase_area: "area (opt_CF_OR c1 c2) \ area (OR c1 c2)" + by (induct c1 c2 rule: opt_CF_OR.induct, auto) + +theorem opt_CF_doesnt_increase_area: "area (opt_CF c) \ area c" +proof (induct rule: opt_CF.induct) + case (1 c) + have "area (opt_CF (NOT c)) + = area (opt_CF_NOT (opt_CF c))" by simp + also have "... \ area (NOT (opt_CF c))" + using opt_CF_NOT_doesnt_increase_area by assumption + also have "... \ area (NOT c)" using 1 by simp + finally show ?case by assumption +next + case (2 c1 c2) + have "area (opt_CF (AND c1 c2)) + = area (opt_CF_AND (opt_CF c1) (opt_CF c2))" by simp + also have "... \ area (AND (opt_CF c1) (opt_CF c2))" + using opt_CF_AND_doesnt_increase_area by assumption + also have "... \ area (AND c1 c2)" using 2 by simp + finally show ?case by assumption +next + case (3 c1 c2) + have "area (opt_CF (OR c1 c2)) + = area (opt_CF_OR (opt_CF c1) (opt_CF c2))" by simp + also have "... \ area (OR (opt_CF c1) (opt_CF c2))" + using opt_CF_OR_doesnt_increase_area by assumption + also have "... \ area (OR c1 c2)" using 3 by simp + finally show ?case by assumption +qed simp+ + +lemma opt_CF_NOT_doesnt_increase_delay: "delay (opt_CF_NOT c) \ delay (NOT c)" + by (cases c, auto) + +lemma opt_CF_AND_doesnt_increase_delay: "delay (opt_CF_AND c1 c2) \ delay (AND c1 c2)" + by (induct c1 c2 rule: opt_CF_AND.induct, auto) + +lemma opt_CF_OR_doesnt_increase_delay: "delay (opt_CF_OR c1 c2) \ delay (OR c1 c2)" + by (induct c1 c2 rule: opt_CF_OR.induct, auto) + +theorem opt_CF_doesnt_increase_delay: "delay (opt_CF c) \ delay c" +proof (induct rule: opt_CF.induct) + case (1 c) + have "delay (opt_CF (NOT c)) + = delay (opt_CF_NOT (opt_CF c))" by simp + also have "... \ delay (NOT (opt_CF c))" + using opt_CF_NOT_doesnt_increase_delay by assumption + also have "... \ delay (NOT c)" using 1 by simp + finally show ?case by assumption +next + case (2 c1 c2) + have "delay (opt_CF (AND c1 c2)) + = delay (opt_CF_AND (opt_CF c1) (opt_CF c2))" by simp + also have "... \ delay (AND (opt_CF c1) (opt_CF c2))" + using opt_CF_AND_doesnt_increase_delay by assumption + also have "... \ delay (AND c1 c2)" using 2 by auto + finally show ?case by assumption +next + case (3 c1 c2) + have "delay (opt_CF (OR c1 c2)) + = delay (opt_CF_OR (opt_CF c1) (opt_CF c2))" by simp + also have "... \ delay (OR (opt_CF c1) (opt_CF c2))" + using opt_CF_OR_doesnt_increase_delay by assumption + also have "... \ delay (OR c1 c2)" using 3 by auto + finally show ?case by assumption +qed simp+ + +section \Task 11: Adding fan-out.\ + +text \Please see separate file.\ + +end \ No newline at end of file diff --git a/isabelle/2019/HSV_tasks_2019_solutions_task11.thy b/isabelle/2019/HSV_tasks_2019_solutions_task11.thy new file mode 100644 index 0000000..d50fc48 --- /dev/null +++ b/isabelle/2019/HSV_tasks_2019_solutions_task11.thy @@ -0,0 +1,204 @@ +theory HSV_tasks_2019_solutions_task11 imports Main begin + +section \Representation of circuits.\ + +text \A wire is represented as an integer.\ + +type_synonym wire = int + +datatype gate = + NOT "wire" "wire" +| AND "wire" "wire" "wire" +| OR "wire" "wire" "wire" +| TRUE "wire" +| FALSE "wire" + +text \A circuit is represented as a list gates together with a list of output wires.\ + +type_synonym circuit = "gate list \ wire list" + +text \Here are some examples of circuits.\ + +definition "circuit1 == ([NOT 1 2], [2])" +definition "circuit2 == ([NOT 1 3, NOT 2 4, AND 3 4 5, NOT 5 6], [6])" +definition "circuit3 == ([NOT 1 3, NOT 2 4, NOT 1 7, AND 3 4 5, NOT 5 6], [6])" +definition "circuit4 == ([NOT 1 3, NOT 2 4, NOT 1 7, NOT 7 8, AND 3 4 5, NOT 5 6], [6])" + +text \Return the input wire(s) of a gate.\ + +fun inputs_of where + "inputs_of (NOT wi _) = {wi}" +| "inputs_of (AND wi1 wi2 _) = {wi1, wi2}" +| "inputs_of (OR wi1 wi2 _) = {wi1, wi2}" +| "inputs_of (TRUE _) = {}" +| "inputs_of (FALSE _) = {}" + +text \Return the output wire of a gate.\ + +fun output_of where + "output_of (NOT _ wo) = wo" +| "output_of (AND _ _ wo) = wo" +| "output_of (OR _ _ wo) = wo" +| "output_of (TRUE wo) = wo" +| "output_of (FALSE wo) = wo" + +section \Evaluating circuits.\ + +text \A valuation associates every wire with a truth-value.\ + +type_synonym valuation = "wire \ bool" + +text \A few examples of valuations.\ + +definition "\0 == \_. True" +definition "\1 == \0(1 := True, 2 := False, 3 := True)" +definition "\2 == \0(1 := True, 2 := True, 3 := True)" + +text \Calculate the output of a single gate, given a valuation.\ + +fun sim_gate :: "valuation \ gate \ bool" where + "sim_gate \ (NOT wi wo) = (\ \ wi)" +| "sim_gate \ (AND wi1 wi2 wo) = (\ wi1 \ \ wi2)" +| "sim_gate \ (OR wi1 wi2 wo) = (\ wi1 \ \ wi2)" +| "sim_gate \ (TRUE wo) = True" +| "sim_gate \ (FALSE wo) = False" + +text \Simulates a list of gates, given an initial valuation. Produces a new valuation.\ + +fun sim_gates :: "valuation \ gate list \ valuation" where + "sim_gates \ [] = \" +| "sim_gates \ (g # gs) = sim_gates (\ (output_of g := sim_gate \ g)) gs" + +text \Simulates a circuit, given an initial valuation. Produces a list of + truth-values, one truth-value per output.\ + +fun sim :: "valuation \ circuit \ bool list" where + "sim \ (gs, wos) = map (sim_gates \ gs) wos" + +text \Testing the simulator.\ + +value "sim \1 circuit1" +value "sim \2 circuit1" +value "sim \1 circuit2" +value "sim \2 circuit2" +value "sim \1 circuit3" +value "sim \2 circuit3" + +section \Optimising circuits by removing dead gates.\ + +text \Return the set of wires that lead to an output.\ + +fun live_wires where + "live_wires ([], wos) = set wos" +| "live_wires (g # gs, wos) = (let ws = live_wires (gs, wos) in + (if output_of g \ ws then inputs_of g else {}) \ ws)" + +value "live_wires ([NOT 1 3, NOT 2 4, NOT 1 7, AND 3 4 5, NOT 5 6], [6])" +value "live_wires ([NOT 1 3, NOT 2 4, NOT 1 7, NOT 7 8, AND 3 4 5, NOT 5 6], [6])" +value "live_wires ([NOT 1 3, NOT 2 4, NOT 1 7, NOT 7 8, AND 3 4 5, NOT 5 6], [6, 8])" +value "live_wires ([NOT 1 3, NOT 2 4, NOT 7 8, AND 3 4 5, NOT 5 6], [6])" + +text \This is a helper function for the next function .\ + +fun remove_dead_inner where + "remove_dead_inner [] wos = []" +| "remove_dead_inner (g # gs) wos = + (let gs' = remove_dead_inner gs wos in + (if output_of g \ live_wires (gs', wos) then [g] else []) @ gs')" + +text \This function strips out gates that are not needed.\ + +fun remove_dead where + "remove_dead (gs, wos) = (remove_dead_inner gs wos, wos)" + +value "remove_dead circuit2" +value "remove_dead circuit3" +value "remove_dead circuit4" + +section \Proving that removing dead gates does not change a circuit's behaviour.\ + +text \This lemma is obviously false -- it wrongly claims that remove_dead + has no effect on a circuit.\ + +lemma "remove_dead c = c" oops + +text \We shall say that two functions are 'congruent on X' if they coincide on all inputs in X.\ + +fun cong_on where + "cong_on X f g = (\x \ X. f x = g x)" + +text \Congruency is transitive.\ + +lemma cong_on_trans: + assumes "cong_on X g h" + assumes "cong_on X f g" + shows "cong_on X f h" + using assms by simp + +text \This is a rather technical lemma. It says that if two valuations \ and \' are + congruent on all wires that are live in circuit (g # gs, wos), then the + respective valuations obtained after simulating g remain congruent on all wires + that are live in circuit (gs, wos). \ + +lemma cong_on_live_wires: + assumes "cong_on (live_wires (g # gs, wos)) \ \'" + shows "cong_on (live_wires (gs, wos)) (\(output_of g := sim_gate \ g)) (\'(output_of g := sim_gate \' g))" + using assms + apply (cases "output_of g \ live_wires (gs, wos)", auto) + apply (cases g, auto)+ + done + +text \If valuations \ and \' are congruent on all wires that are live in circuit (gs, wos), then + the respective valuations obtained after simulating gs are congruent on all wires in wos.\ + +lemma cong_sim_gates: + assumes "cong_on (live_wires (gs, wos)) \ \'" + shows "cong_on (set wos) (sim_gates \ gs) (sim_gates \' gs)" + using assms +proof (induct gs arbitrary: \ \') + case Nil + thus ?case by auto +next + case (Cons g gs) + show ?case + apply (clarsimp simp del: cong_on.simps) + apply (rule Cons.hyps[OF cong_on_live_wires[OF Cons.prems]]) + done +qed + +text \This is a slightly unwrapped version of the main theorem below.\ + +lemma sim_remove_dead: + "cong_on (set wos) (sim_gates \ (remove_dead_inner gs wos)) (sim_gates \ gs)" +proof (induct gs arbitrary: \) + case Nil + thus ?case by simp +next + case (Cons g gs \) + show ?case + proof (cases "output_of g \ live_wires (remove_dead (gs, wos))") + case True + thus ?thesis using Cons.hyps by auto + next + case False + thus ?thesis + apply (simp del: cong_on.simps) + apply (rule cong_on_trans) + apply (rule Cons.hyps[of "\(output_of g := sim_gate \ g)"]) + apply (rule cong_sim_gates) + apply clarsimp + done + qed +qed + +text \We define a convenient shorthand for expressing that two circuits have the same behaviour.\ + +fun sim_equiv (infix "\" 50) where + "c1 \ c2 = (\\. sim \ c1 = sim \ c2)" + +text \Main theorem: removing the dead gates from a circuit does not change its behaviour.\ + +theorem "remove_dead c \ c" + using sim_remove_dead by (cases c, auto) + +end \ No newline at end of file diff --git a/isabelle/2019/isabelle_exercises_2019.pdf b/isabelle/2019/isabelle_exercises_2019.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d9b69e862f3f0633d96e30894cf747b5ce6d3e4 GIT binary patch literal 132477 zcma&NLzE^!vuImi**3qjZQHipWp~-OZQDkdZQJUy(Ph2=y?4*xu5$(_M-h{p7A?JPejYO_G`$#QcVDHG9i^Z ztO@9E@GUT}AZ}zWNjtjI`4Ldxk#BliPn012AnT`nj#k%KH}q9suCsE4w!C{m@I853 z>zZlbqsDNQNjYTc@P^hxP3A%gOv-^C2pJ?TzgyIj9rhwCg=)`D`*{ ztI3Xc`;} zd~1risFW=|8v8VanLw-&oKhRbeJdqQ`=fD3f+Wp&^~lX}I*VR*NYu}#3C^r_j>!2{ z*|h^X97*1{N!PQD_NwA0s#}=0FYwQzvBtH<@8+_JHo6+bx88J{e0g61^a6MWL{9mL z6h~!8JA7Wjw3WJv=C&F{qq0Nfe*UwjAt;|HM?(Q4AOFIX;JBT)b$+{4co=IbAJzNP zVu-QbKQbS6hXq>X$l%rQSTyHwCQsMyBu}N#MXIY)ugw@tpcX~QcO;?ZakUUkfoBun z3HMZ+#`hl?w>&P?buO*0V?Dh0=X!4T>u&NIgi#DlekVZyqfFj1iNushS5XmbVY3%db)x zd$+yO0Z*P`tn8pjOGzLC=4K^`30o@}29fk3(Jh{TzBU$V$Vw{cA2(f6D)WM9dZB3o 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z(<$Y*GnTg|M>Ht6v#Cl3d}jG!{f^296asykZx-cs9WD}b&jA)%_|njK;iWB2k5jAfP3K4C7;QR^kujkMO|s<74`W#$u`=B~0u zxN8lPnGtOvEeKS7$!pm^;-V}0)p&zh3X;+q7cw2*3g>dH+T)zXRmj%sn-s3HD!6Nw zDJjiBsZ`~SKPP}^wO%usn(MSiY^CbE-wbz^#l_vTEFmpJ(Uni73O=@RJK}1EB}M7=rt6sY>0Z-q;w=zaBqX824zYYBR%nYTL9aC zCKdlNgaDe{3=ASn93reN!h%f9OsvBE48m*-0`&ZX0_^lGf&y&(ynz4zB0sH28QYjT znE@CX*#FlU{Qn|+9(R(9ZU7i1ZIy5+mFmf z&T0AkC1om$zK*U)jV8QxdC8@9MWrE+YfTask 1: proving that "3 / sqrt 2" is irrational.\ + +(* In case it is helpful, the following theorem is copied from Chapter 3 of the worksheet. *) +theorem sqrt2_irrational: "sqrt 2 \ \" +proof auto + assume "sqrt 2 \ \" + then obtain m n where + "n \ 0" and "\sqrt 2\ = real m / real n" and "coprime m n" + by (rule Rats_abs_nat_div_natE) + hence "\sqrt 2\^2 = (real m / real n)^2" by auto + hence "2 = (real m / real n)^2" by simp + hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto + hence "2 * (real n)^2 = (real m)^2" + by (simp add: nonzero_eq_divide_eq `n \ 0`) + hence "real (2 * n^2) = (real m)^2" by auto + hence *: "2 * n^2 = m^2" + using of_nat_power_eq_of_nat_cancel_iff by blast + hence "even (m^2)" by presburger + hence "even m" by simp + then obtain m' where "m = 2 * m'" by auto + with * have "2 * n^2 = (2 * m')^2" by auto + hence "2 * n^2 = 4 * m'^2" by simp + hence "n^2 = 2 * m'^2" by simp + hence "even (n^2)" by presburger + hence "even n" by simp + with `even m` and `coprime m n` show False by auto +qed + +theorem "3 / sqrt 2 \ \" + sorry (* TODO: Complete this proof. *) + +section \Task 2: Centred pentagonal numbers.\ + +fun pent :: "nat \ nat" where + "pent n = (if n = 0 then 1 else 5 * n + pent (n - 1))" + +value "pent 0" (* should be 1 *) +value "pent 1" (* should be 6 *) +value "pent 2" (* should be 16 *) +value "pent 3" (* should be 31 *) + +theorem "pent n = (5 * n^2 + 5 * n + 2) div 2" + sorry (* TODO: Complete this proof. *) + + +section \Task 3: Lucas numbers.\ + +fun fib :: "nat \ nat" where + "fib n = (if n = 0 then 0 else if n = 1 then 1 else fib (n - 1) + fib (n - 2))" + +value "fib 0" (* should be 0 *) +value "fib 1" (* should be 1 *) +value "fib 2" (* should be 1 *) +value "fib 3" (* should be 2 *) + +thm fib.induct (* rule induction theorem for fib *) + +(* TODO: Complete this task. *) + + +section \Task 4: Balancing circuits.\ + +(* Here is a datatype for representing circuits, copied from the worksheet *) + +datatype "circuit" = + NOT "circuit" +| AND "circuit" "circuit" +| OR "circuit" "circuit" +| TRUE +| FALSE +| INPUT "int" + +text \Delay (assuming all gates have a delay of 1)\ + +(* The following "delay" function also appeared in the 2019 coursework exercises. *) + +fun delay :: "circuit \ nat" where + "delay (NOT c) = 1 + delay c" +| "delay (AND c1 c2) = 1 + max (delay c1) (delay c2)" +| "delay (OR c1 c2) = 1 + max (delay c1) (delay c2)" +| "delay _ = 0" + +(* TODO: Complete this task. *) + + +section \Task 5: Extending with NAND gates.\ + +(* TODO: Complete this task. *) + +end \ No newline at 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zjSiyEGN(&lQcH|{=Au^W>61(Rn%&F%hDz7xMaNT#fBp9bbA_0>Ks;UEm*`!H@o)lB LXlbQYWKjMMXHXb0 literal 0 HcmV?d00001 diff --git a/isabelle/HSV_chapter3.thy b/isabelle/HSV_chapter3.thy new file mode 100644 index 0000000..37ff570 --- /dev/null +++ b/isabelle/HSV_chapter3.thy @@ -0,0 +1,59 @@ +theory HSV_chapter3 imports Complex_Main begin + +(* We use the following command to search Isabelle's library + for theorems that contain a particular pattern. *) +find_theorems "_ \ \" +thm Rats_abs_nat_div_natE + +find_theorems "(_ / _)^_" +thm power_divide + +(* A proof that the square root of 2 is irrational. *) +theorem sqrt2_irrational: "sqrt 2 \ \" +proof auto + assume "sqrt 2 \ \" + then obtain m n where + "n \ 0" and "\sqrt 2\ = real m / real n" and "coprime m n" + by (rule Rats_abs_nat_div_natE) + hence "\sqrt 2\^2 = (real m / real n)^2" by auto + hence "2 = (real m / real n)^2" by simp + hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto + hence "2 * (real n)^2 = (real m)^2" + by (simp add: nonzero_eq_divide_eq `n \ 0`) + hence "real (2 * n^2) = (real m)^2" by auto + hence *: "2 * n^2 = m^2" + using of_nat_power_eq_of_nat_cancel_iff by blast + hence "even (m^2)" by presburger + hence "even m" by simp + then obtain m' where "m = 2 * m'" by auto + with * have "2 * n^2 = (2 * m')^2" by auto + hence "2 * n^2 = 4 * m'^2" by simp + hence "n^2 = 2 * m'^2" by simp + hence "even (n^2)" by presburger + hence "even n" by simp + with `even m` and `coprime m n` show False by auto +qed + +(* A proof that there is no greatest even number. *) +theorem "\n::int. even n \ (\m. even m \ m > n)" +proof clarify + fix n::int + assume "even n" + hence "even (n+2)" by simp + moreover + have "n < (n+2)" by simp + ultimately + show "\m. even m \ n < m" by blast +qed + +(* The same proof in the procedural style. *) +theorem "\n::int. even n \ (\m. even m \ m > n)" + apply clarify + apply (rule_tac x="n+2" in exI) + apply (rule conjI) + apply simp + apply (thin_tac "even n") + apply simp + done + +end \ No newline at end of file diff --git a/isabelle/HSV_chapter4.thy b/isabelle/HSV_chapter4.thy new file mode 100644 index 0000000..960f626 --- /dev/null +++ b/isabelle/HSV_chapter4.thy @@ -0,0 +1,83 @@ +theory HSV_chapter4 imports Complex_Main begin + +(* Triangle numbers *) +fun triangle :: "nat \ nat" where + "triangle n = (if n = 0 then 0 else n + triangle (n-1))" + +value "triangle 1" +value "triangle 2" +value "triangle 3" + +theorem triangle_closed_form: "triangle n = (n+1) * n div 2" + apply (induct n) + apply simp+ + done + +(* Tetrahedral numbers *) +fun tet :: "nat \ nat" where + "tet n = (if n = 0 then 0 else triangle n + tet (n-1))" + +value "tet 1" (* 1 *) +value "tet 2" (* 4 *) +value "tet 3" (* 10 *) +value "tet 4" (* 20 *) +value "tet 5" (* 35 *) +value "tet 6" (* 56 *) + +find_theorems "_ div ?x + _ div ?x" +thm div_add + +(* Proving that closed form is equivalent to recursive definition *) +theorem "tet n = ((n + 2) * (n + 1) * n) div 6" +proof (induct n) + case 0 + show ?case by simp +next + case (Suc k) + + (* induction hypothesis *) + assume IH: "tet k = (k + 2) * (k + 1) * k div 6" + + (* establish a useful fact and label it "*" *) + have "2 dvd (k + 2) * (k + 1)" by simp + hence *: "6 dvd (k + 2) * (k + 1) * 3" by presburger + + (* establish another useful fact and label it "**" *) + have "2 dvd (k + 2) * (k + 1) * k" by simp + moreover have "3 dvd (k + 2) * (k + 1) * k" + proof - + { + assume "k mod 3 = 0" + hence "3 dvd k" by presburger + hence "3 dvd (k + 2) * (k + 1) * k" by fastforce + } moreover { + assume "k mod 3 = 1" + hence "3 dvd (k + 2)" by presburger + hence "3 dvd (k + 2) * (k + 1) * k" by fastforce + } moreover { + assume "k mod 3 = 2" + hence "3 dvd (k + 1)" by presburger + hence "3 dvd (k + 2) * (k + 1) * k" by fastforce + } ultimately + show "3 dvd (k + 2) * (k + 1) * k" by linarith + qed + ultimately have **: "6 dvd (k + 2) * (k + 1) * k" by presburger + + (* the actual proof *) + have "tet (Suc k) = triangle (Suc k) + tet k" + by simp + also have "... = (k + 2) * (k + 1) div 2 + tet k" + using triangle_closed_form by simp + also have "... = (k + 2) * (k + 1) div 2 + (k + 2) * (k + 1) * k div 6" + using IH by simp + also have "... = ((k + 2) * (k + 1) * 3 + (k + 2) * (k + 1) * k) div 6" + using div_add[OF * **] by simp + also have "... = (k + 2) * (k + 1) * (k + 3) div 6" + by (simp add: distrib_left) + also have "... = (Suc k + 2) * (Suc k + 1) * Suc k div 6" + by (metis One_nat_def Suc_1 add.commute add_Suc_shift mult.assoc + mult.commute numeral_3_eq_3 plus_1_eq_Suc) + finally show ?case by assumption +qed + +end \ No newline at end of file diff --git a/isabelle/HSV_chapter5.thy b/isabelle/HSV_chapter5.thy new file mode 100644 index 0000000..a9cb48f --- /dev/null +++ b/isabelle/HSV_chapter5.thy @@ -0,0 +1,136 @@ +theory HSV_chapter5 imports Main begin + +section \Representing circuits (cf. worksheet Section 5.1)\ + +text \Defining a data structure to represent fan-out-free circuits with numbered inputs\ + +datatype "circuit" = + NOT "circuit" +| AND "circuit" "circuit" +| OR "circuit" "circuit" +| TRUE +| FALSE +| INPUT "int" + +text \A few example circuits\ + +definition "circuit1 == AND (INPUT 1) (INPUT 2)" +definition "circuit2 == OR (NOT circuit1) FALSE" +definition "circuit3 == NOT (NOT circuit2)" +definition "circuit4 == AND circuit3 (INPUT 3)" + +section \Simulating circuits (cf. worksheet Section 5.2)\ + +text \Simulates a circuit given a valuation for each input wire\ + +fun simulate where + "simulate (AND c1 c2) \ = ((simulate c1 \) \ (simulate c2 \))" +| "simulate (OR c1 c2) \ = ((simulate c1 \) \ (simulate c2 \))" +| "simulate (NOT c) \ = (\ (simulate c \))" +| "simulate TRUE \ = True" +| "simulate FALSE \ = False" +| "simulate (INPUT i) \ = \ i" + +text \A few example valuations\ + +definition "\0 == \_. True" +definition "\1 == \0(1 := True, 2 := False, 3 := True)" +definition "\2 == \0(1 := True, 2 := True, 3 := True)" + +text \Trying out the simulator\ + +value "simulate circuit1 \1" +value "simulate circuit2 \1" +value "simulate circuit3 \1" +value "simulate circuit4 \1" +value "simulate circuit1 \2" +value "simulate circuit2 \2" +value "simulate circuit3 \2" +value "simulate circuit4 \2" + +section \Structural induction on circuits (cf. worksheet Section 5.3)\ + +text \A function that switches each pair of wires entering an OR or AND gate\ + +fun mirror where + "mirror (NOT c) = NOT (mirror c)" +| "mirror (AND c1 c2) = AND (mirror c2) (mirror c1)" +| "mirror (OR c1 c2) = OR (mirror c2) (mirror c1)" +| "mirror TRUE = TRUE" +| "mirror FALSE = FALSE" +| "mirror (INPUT i) = INPUT i" + +value "circuit1" +value "mirror circuit1" +value "circuit2" +value "mirror circuit2" + +text \The following non-theorem is easily contradicted.\ + +theorem "mirror c = c" + oops + +text \Proving that mirroring doesn't affect simulation behaviour.\ + +theorem mirror_is_sound: "simulate (mirror c) \ = simulate c \" + by (induct c, auto) + +section \A simple circuit optimiser (cf. worksheet Section 5.4)\ + +text \A function that optimises a circuit by removing pairs of consecutive NOT gates\ + +fun opt_NOT where + "opt_NOT (NOT (NOT c)) = opt_NOT c" +| "opt_NOT (NOT c) = NOT (opt_NOT c)" +| "opt_NOT (AND c1 c2) = AND (opt_NOT c1) (opt_NOT c2)" +| "opt_NOT (OR c1 c2) = OR (opt_NOT c1) (opt_NOT c2)" +| "opt_NOT TRUE = TRUE" +| "opt_NOT FALSE = FALSE" +| "opt_NOT (INPUT i) = INPUT i" + +text \Trying out the optimiser\ + +value "circuit1" +value "opt_NOT circuit1" +value "circuit2" +value "opt_NOT circuit2" +value "circuit3" +value "opt_NOT circuit3" +value "circuit4" +value "opt_NOT circuit4" + +section \Rule induction (cf. worksheet Section 5.5)\ + +text \A Fibonacci function that demonstrates complex recursion schemes\ + +fun f :: "nat \ nat" where + "f (Suc (Suc n)) = f n + f (Suc n)" +| "f (Suc 0) = 1" +| "f 0 = 1" + +thm f.induct (* rule induction theorem for f *) + +text \We need to prove a stronger version of the theorem below + first, in order to make the inductive step work. Just like how + it often goes with loop invariants in Dafny!\ +lemma helper: "f n \ n \ f n \ 1" + by (rule f.induct[of "\n. f n \ n \ f n \ 1"], auto) + +text \The nth Fibonacci number is greater than or equal to n\ +theorem "f n \ n" + using helper by simp + +section \Verifying our optimiser (cf. worksheet Section 5.6)\ + +text \The following non-theorem is easily contradicted.\ + +theorem "opt_NOT c = c" + oops + +text \The following theorem says that the optimiser is sound.\ + +theorem opt_NOT_is_sound: "simulate (opt_NOT c) \ = simulate c \" + by (induct rule:opt_NOT.induct, auto) + + +end diff --git a/isabelle/isabelle.pdf b/isabelle/isabelle.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e5311adf047ab136ef88c43affcf3a6660717a03 GIT binary patch literal 284046 zcma%iQ*65wQbwBjot3rwr$(CZQHi7d*1)x>ixzzU#@bI5zMS4bFQ^Q zrXV6l2cTz#A)8woUV~vHWF)jVvV!5^fnks_wKI3IAY@`@CHy}HhC$5I#>Lc$kU`AG z(8W~5)Y#s{6o!uv#@WTm)X)~@_eOM_to$G&T*&P=Iv*Pf>r80p2}hCtnm=g75U}G~ 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ze5>0tu+ig|oY8-yt+T~m9`4c@uqK@w9S4MMsw=H5ek!V_I~uW)JGh1 z>)-4Hlt=lCcz3AJ?So@3pRRLzm#w#PQ2o|UQoAX;pLA^B;F=I}%}IfH?>t3IN@o>7!6s07U#=UjJY0NEF~c zvi_S;1enVxDnpbMB}9}&B_*KBU@=h$7^Ex$7FUFdi7SFcpnp zLB#$Ywf_fi|F0 z*KQAfFh5Y!IZ*;WeFk{eNEK=lX_0Ju`PYM=2?$(Ed+jtqtp&9Bv=&_kTMb ZgN389Ug&!_L%~pS2njd0l8!RT{{o0{-gW>0 literal 0 HcmV?d00001 diff --git a/dafny/2020/solutions_2020.dfy b/dafny/2020/solutions_2020.dfy new file mode 100644 index 0000000..2ed8285 --- /dev/null +++ b/dafny/2020/solutions_2020.dfy @@ -0,0 +1,200 @@ +// Dafny coursework solutions +// Autumn term, 2020 +// +// Authors: John Wickerson and Matt Windsor + +// Task 1 +predicate zeroes(A:array) + reads A +{ + forall i :: 0 <= i < A.Length ==> A[i] == 0 +} + +method resetArray(A:array) + ensures zeroes(A) + modifies A +{ + var i := 0; + while i < A.Length + invariant 0 <= i <= A.Length + invariant forall j :: 0 <= j < i ==> A[j] == 0 + decreases A.Length - i + { + A[i] := 0; + i := i + 1; + } +} + +predicate sorted_between(A:array, lo:int, hi:int) + reads A +{ + forall m,n :: 0 <= lo <= m < n < hi <= A.Length ==> A[m] <= A[n] +} +predicate sorted(A:array) + reads A +{ + sorted_between(A,0,A.Length) +} + +predicate partitioned(A:array, index:int) + reads A +{ + forall i,j :: 0 <= i < index <= j < A.Length ==> A[i] <= A[j] +} + +// Task 2 +method backwards_selection_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := A.Length; + while 0 < i + invariant 0 <= i <= A.Length + invariant sorted_between(A, i, A.Length) + invariant partitioned(A, i) + decreases i + { + var max := 0; + var j := 1; + while j < i + invariant 0 <= max < j <= i + invariant forall n :: 0 <= n < j ==> A[n] <= A[max] + decreases i - j + { + if A[max] < A[j] { + max := j; + } + j := j+1; + } + + A[max], A[i-1] := A[i-1], A[max]; + i := i - 1; + } +} + +// Task 3 +method recursive_selection_sort_inner(A:array, i:int) + requires 0 <= i <= A.Length + requires sorted_between(A, 0, i) + requires partitioned(A, i) + ensures sorted(A) + modifies A + decreases A.Length - i +{ + if i == A.Length { + return; + } + var k := i; + var j := i + 1; + while j < A.Length + invariant 0 <= i <= k < j <= A.Length + invariant forall n :: i <= n < j ==> A[k] <= A[n] + decreases A.Length - j + { + if A[k] > A[j] { + k := j; + } + j := j+1; + } + A[k], A[i] := A[i], A[k]; + recursive_selection_sort_inner(A, i + 1); +} +method recursive_selection_sort(A:array) + ensures sorted(A) + modifies A +{ + recursive_selection_sort_inner(A, 0); +} + +// Task 4 +method bubble_sort_with_early_stop(A:array) + ensures sorted(A) + modifies A +{ + var stable := false; + var i := 0; + while !stable + invariant 0 <= i <= A.Length + 1 + invariant partitioned(A, A.Length - i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant if stable then sorted(A) else i < A.Length + 1 + decreases A.Length - i + { + var j := 0; + stable := true; + while j + 1 < A.Length - i + invariant 0 <= i <= A.Length + invariant 0 < A.Length ==> 0 <= j < A.Length + invariant partitioned(A, A.Length - i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant if stable then sorted_between(A, 0, j + 1) else i < A.Length + invariant forall m :: 0 <= m < j < A.Length ==> A[m] <= A[j] + invariant A.Length - i <= j ==> partitioned(A, A.Length - i - 1) + decreases A.Length - j + { + if A[j+1] < A[j] { + A[j], A[j+1] := A[j+1], A[j]; + stable := false; + } + j := j+1; + } + i := i+1; + } +} + +// Task 5 +method cocktail_shaker_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length / 2 + invariant i <= A.Length + invariant partitioned(A, A.Length - i) + invariant partitioned(A, i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant sorted_between(A, 0, i) + decreases A.Length - i + { + var j := i; + while j < A.Length - i - 1 + invariant j < A.Length - i + invariant partitioned(A, A.Length - i) + invariant partitioned(A, i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant sorted_between(A, 0, i) + invariant forall m :: 0 <= m < j ==> A[m] <= A[j] + decreases A.Length - j + { + if A[j+1] < A[j] { + A[j], A[j+1] := A[j+1], A[j]; + } + j := j+1; + } + while i < j + invariant i <= j < A.Length - i + invariant partitioned(A, A.Length - i) + invariant partitioned(A, A.Length - i - 1) + invariant partitioned(A, i) + invariant sorted_between(A, A.Length - i, A.Length) + invariant sorted_between(A, 0, i) + invariant forall m :: j < m < A.Length ==> A[j] <= A[m] + decreases j + { + if A[j-1] > A[j] { + A[j-1], A[j] := A[j], A[j-1]; + } + j := j-1; + } + i := i+1; + } +} + +method Main() { + var A:array := new int[7] [4,0,1,9,7,1,2]; + print "Before: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; + recursive_selection_sort(A); + print "After: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; +} \ No newline at end of file diff --git a/dafny/2020/tasks_2020.dfy b/dafny/2020/tasks_2020.dfy new file mode 100644 index 0000000..a5171b1 --- /dev/null +++ b/dafny/2020/tasks_2020.dfy @@ -0,0 +1,122 @@ +// Dafny coursework tasks +// Autumn term, 2020 +// +// Authors: John Wickerson and Matt Windsor + +predicate sorted(A:array) + reads A +{ + forall m,n :: 0 <= m < n < A.Length ==> A[m] <= A[n] +} + +// Task 1 +method resetArray(A:array) + ensures zeroes(A) + modifies A +{ + var i := 0; + while i < A.Length { + A[i] := 0; + i := i + 1; + } +} + +// Task 2 +method backwards_selection_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := A.Length; + while 0 < i { + var max := 0; + var j := 1; + while j < i { + if A[max] < A[j] { + max := j; + } + j := j+1; + } + A[max], A[i-1] := A[i-1], A[max]; + i := i - 1; + } +} + +// Task 3 +method recursive_selection_sort_inner(A:array, i:int) + modifies A +{ + if i == A.Length { + return; + } + var k := i; + var j := i + 1; + while j < A.Length { + if A[k] > A[j] { + k := j; + } + j := j+1; + } + A[k], A[i] := A[i], A[k]; + recursive_selection_sort_inner(A, i + 1); +} + +method recursive_selection_sort(A:array) + ensures sorted(A) + modifies A +{ + recursive_selection_sort_inner(A, 0); +} + +// Task 4 +method bubble_sort_with_early_stop(A:array) + ensures sorted(A) + modifies A +{ + var stable := false; + var i := 0; + while !stable { + var j := 0; + stable := true; + while j + 1 < A.Length - i { + if A[j+1] < A[j] { + A[j], A[j+1] := A[j+1], A[j]; + stable := false; + } + j := j+1; + } + i := i+1; + } +} + +// Task 5 +method cocktail_shaker_sort(A:array) + ensures sorted(A) + modifies A +{ + var i := 0; + while i < A.Length / 2 { + var j := i; + while j < A.Length - i - 1 { + if A[j+1] < A[j] { + A[j], A[j+1] := A[j+1], A[j]; + } + j := j+1; + } + while i < j { + if A[j-1] > A[j] { + A[j-1], A[j] := A[j], A[j-1]; + } + j := j-1; + } + i := i+1; + } +} + +method Main() { + var A:array := new int[7] [4,0,1,9,7,1,2]; + print "Before: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; + recursive_selection_sort(A); + print "After: ", A[0], A[1], A[2], A[3], + A[4], A[5], A[6], "\n"; +} \ No newline at 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