From d158b4ee3ac6ed900c96c73f6d1146ec88a9a80f Mon Sep 17 00:00:00 2001 From: Hykilpikonna Date: Tue, 9 Nov 2021 20:54:30 -0500 Subject: [PATCH] [F] Use lists, not sets --- assignments/a4/a4.tex | 32 ++++++++++++++++---------------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/assignments/a4/a4.tex b/assignments/a4/a4.tex index 947c97c..f5214b8 100644 --- a/assignments/a4/a4.tex +++ b/assignments/a4/a4.tex @@ -138,7 +138,7 @@ That is $\forall k_2 \in N, gcd(k_2, 2) = 1 \land gcd(k_2, 3) = 1$ \\ Assumption 2: The statement proven in Part 1.1: \\ $\forall a,k,n \in \Z, gcd(a,n) = 1 \implies gcd(a + kn, n) = 1$ \\ \\ -We need to prove: $\forall k \in N \cup \{ N[-2] + 6 \}, gcd(k, 2) = 1 \land gcd(k, 3) = 1$ \\ +We need to prove: $\forall k \in N + [N[-2] + 6], gcd(k, 2) = 1 \land gcd(k, 3) = 1$ \\ Which is equivalent to: $\forall k \in N, gcd(k, 2) = 1 \land gcd(k, 3) = 1$ and \\ $gcd(N[-2] + 6, 2) = 1 \land gcd(N[-2] + 6, 3) = 1$ \\ \\ @@ -168,10 +168,10 @@ By assumption 1, we know that $gcd(N[-2], 2) = 1$ and $gcd(N[-2], 3) = 1$ Variables: In this proof, $N$ is the abbreviation for the list \texttt{nums\_so\_far}, and $|N|$ represents the size of $N$. \\ \\ Assumption 1: The loop invariant 2 is true for the previous iteration. \\ -That is $\forall i_2 \in \{ 0, \dots, |N| - 3\}, N[i_2] + 6 = N[i_2 + 2]$ \\ +That is $\forall i_2 \in [0, \dots, |N| - 3], N[i_2] + 6 = N[i_2 + 2]$ \\ \\ -Let $M = N \cup \{ N[-2] + 6 \}$ be the list of the current iteration. \\ -We need to prove: $\forall i \in \{0, \dots, |M| - 3\}, M[i] + 6 = M[i + 2] $ +Let $M = N + [N[-2] + 6]$ be the list of the current iteration. \\ +We need to prove: $\forall i \in [0, \dots, |M| - 3], M[i] + 6 = M[i + 2] $ \begin{enumerate} \item[1.] Let $i < |M| - 3$ \\ @@ -183,11 +183,11 @@ We need to prove: $\forall i \in \{0, \dots, |M| - 3\}, M[i] + 6 = M[i + 2] $ \\ Let's start with a true statement: \\ $N[-2] = N[-2]$ \\ - Since $N[-2] = (N \cup \{ N[-2] + 6 \})[-3] = M[-3]$, \\ + Since $N[-2] = (N + [N[-2] + 6])[-3] = M[-3]$, \\ $M[-3] = N[-2]$ \\ Add 6 to both sides: \\ $M[-3] + 6 = N[-2] + 6$ \\ - Since $M[-1] = (N \cup \{ N[-2] + 6 \})[-1] = N[-2] + 6$ is it's last entry, \\ + Since $M[-1] = (N + [N[-2] + 6])[-1] = N[-2] + 6$ is it's last entry, \\ $M[-3] + 6 = M[-1]$ \\ Which is what we want to show. @@ -199,20 +199,20 @@ We need to prove: $\forall i \in \{0, \dots, |M| - 3\}, M[i] + 6 = M[i + 2] $ Variables: In this proof, $N$ is the abbreviation for the list \texttt{nums\_so\_far}, and $|N|$ represents the size of $N$. \\ \\ Assumption 1: Loop invariant 3 is true for the previous iteration. \\ -That is $\forall i_2 \in \{0, \dots, |N| - 2\}, N[i_2] < N[i_2 + 1]$ \\ +That is $\forall i_2 \in [0, \dots, |N| - 2], N[i_2] < N[i_2 + 1]$ \\ \\ Assumption 2: Loop invariant 2 is true for the previous iteration. \\ -That is $\forall i_3 \in \{0, \dots, |N| - 3\}, N[i_3] + 6 = N[i_3 + 2]$ \\ +That is $\forall i_3 \in [0, \dots, |N| - 3], N[i_3] + 6 = N[i_3 + 2]$ \\ \\ -Let $M = N \cup \{ N[-2] + 6 \}$ be the list of the current iteration. \\ -We need to prove: $\forall i \in \{0, \dots, |M| - 2\}, M[i] < M[i + 1] $ \\ +Let $M = N + [N[-2] + 6]$ be the list of the current iteration. \\ +We need to prove: $\forall i \in [0, \dots, |M| - 2], M[i] < M[i + 1] $ \\ \\ Assumption 3: Loop invariant 2 is true for the current iteration. \\ -That is $\forall i_4 \in \{0, \dots, |M| - 3\}, M[i_4] + 6 = M[i_4 + 2]$ \\ +That is $\forall i_4 \in [0, \dots, |M| - 3], M[i_4] + 6 = M[i_4 + 2]$ \\ \\ Let's first prove an intermediate statment, statment 4: \\ -$\forall i_5 \in \{0, \dots, |N| - 2\}, 0 < N[i_5 + 1] - N[i_5] < 6$ \\ -Let $i_5 \in \{0, \dots, |N| - 2\}$, \\ +$\forall i_5 \in [0, \dots, |N| - 2], 0 < N[i_5 + 1] - N[i_5] < 6$ \\ +Let $i_5 \in [0, \dots, |N| - 2]$, \\ We want to show $0 < N[i_5 + 1] - N[i_5] < 6$ \\ \\ Pick $i_2 = i_5$, \\ @@ -233,7 +233,7 @@ $N[(i_5 + 1) + 1] < N[i_5 + 1] + 6$ \\ Which is the end of our induction. \\ \\ We have proven the intermediate statement 4. \\ -We now need to prove: $\forall i \in \{0, \dots, |M| - 2\}, M[i] < M[i + 1] $ +We now need to prove: $\forall i \in [0, \dots, |M| - 2], M[i] < M[i + 1] $ \begin{enumerate} \item[1.] Let $i < |M| - 2$ \\ @@ -245,8 +245,8 @@ We now need to prove: $\forall i \in \{0, \dots, |M| - 2\}, M[i] < M[i + 1] $ \\ Pick $i_5 = |N| - 2$ \\ We know that $0 < N[-1] - N[-2] < 6$ by statement 4 \\ - Since $N[-1] = (N \cup \{ N[-2] + 6 \})[-2] = M[-2]$, \\ - Since $N[-2] = (N \cup \{ N[-2] + 6 \})[-3] = M[-3]$, \\ + Since $N[-1] = (N + [N[-2] + 6])[-2] = M[-2]$, \\ + Since $N[-2] = (N + [N[-2] + 6])[-3] = M[-3]$, \\ $0 < M[-2] - M[-3] < 6$ \\ $M[-2] < M[-3] + 6$ \\ Pick $i_4 = |M| - 3$ \\