From d2e9a960ce94dd2e4384f361828f08e73b50ad6b Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 19:26:50 +1200
Subject: [PATCH 1/8] Update README.md

---
 ch02/README.md | 19 ++++++++++++++-----
 1 file changed, 14 insertions(+), 5 deletions(-)

diff --git a/ch02/README.md b/ch02/README.md
index a7d7396..507ba47 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -315,11 +315,20 @@ I.M.T -> LI holds.
  * Time complexity : theta(n)
  * pseudocode
 ```python
-Naive-Polynomial-Evaluation(arr_of_terms, x)
-  y = 0;
-  for term in arr_of_terms
+Naive-Polynomial-Evaluation(arr, x)
+  y = 0
+  for i = 0 to arr.length - 1
     X = 1
-    for i = 1 to term.k
+    for k = i downto 1
       X *= x
-    y += term.ak * X
+    y += X * arr[i]
+  return y
+```
+
+ * LI and proof
+```cpp
+(To avoid Greek letter Sigma, here is using sum(head, tail)() function instead)
+LI:
+  At the start of each iteration of the for loop of lines 2–3,
+  y = sum(k = 0, k = n - (i + 1)) ()
 ```

From 6a4d35dc698f87b91603477cfe86a4d0c748afad Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 20:58:14 +1200
Subject: [PATCH 2/8] add p2.3.c

---
 ch02/README.md | 37 +++++++++++++++++++++++++++++++++++--
 1 file changed, 35 insertions(+), 2 deletions(-)

diff --git a/ch02/README.md b/ch02/README.md
index 507ba47..36b71fc 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -328,7 +328,40 @@ Naive-Polynomial-Evaluation(arr, x)
  * LI and proof
 ```cpp
 (To avoid Greek letter Sigma, here is using sum(head, tail)() function instead)
-LI:
+L.I.:
   At the start of each iteration of the for loop of lines 2–3,
-  y = sum(k = 0, k = n - (i + 1)) ()
+  y = sum(k = 0, k = n - (i + 1)) (a(k + i + 1) * x ^ k)
+  
+I.:
+  Prior to the first iteration, 
+    y = sum(k = 0, k = -1) (a(k + i + 1) * x ^ k)
+    upper limit is lower than lower limit, this equation is meaningless, y = 0
+  Hence, I holds.
+
+M.:
+  Suppose L.I. holds before i = i
+    y = sum(k = 0, k = n - (i + 1)) (a(k + i + 1) * x ^ k)
+    
+    by line 2 - 3,            LHS becomes:
+      y = a(i) + x * y
+        = a(i) + x * ( a(i + 1) + a(i + 2) * x + ... + a(n) * x^(n - i - 1)
+        = a(i) + a(i + 1) * x + a(i + 2) * x^2 + ... + a(n) * x^(n - i)
+        
+    when i = i - 1            RHS becomes
+      sum(k = 0, k = n - i) (a(k + i) * x ^ k)  
+        = a(i) + a(i + 1) * x + a(i + 2) * x^2 + ... + a(n) * x^(n - i)
+      
+  LHS = RHS, hence M holds for next iteration.
+  
+T.:
+  The termination condition is i = -1
+  y = sum(k = 0, k = n - (-1 + 1)) (a(k - 1 + 1) * x ^ k)
+    = sum(k = 0, k = n) (a(k) * x ^ k)
+  Hence, T holds.
+  
+L.I. holds.
+    
+  
+      
+
 ```

From 15261f5ef985368e69f22df2712a6ef9b7940cc6 Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:05:22 +1200
Subject: [PATCH 3/8] Update README.md

---
 ch02/README.md | 6 ++----
 1 file changed, 2 insertions(+), 4 deletions(-)

diff --git a/ch02/README.md b/ch02/README.md
index 36b71fc..9081d04 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -360,8 +360,6 @@ T.:
   Hence, T holds.
   
 L.I. holds.
-    
-  
-      
-
 ```
+
+ * As shown above, this code fragment correctly evaluates a polynomial.

From a3727a773a99631d31252c03db018a44a405e4bb Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:11:57 +1200
Subject: [PATCH 4/8] Update README.md

---
 ch02/README.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/ch02/README.md b/ch02/README.md
index 9081d04..94944fb 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -363,3 +363,9 @@ L.I. holds.
 ```
 
  * As shown above, this code fragment correctly evaluates a polynomial.
+
+##Problem 2-3 Inversions
+ *  Five inversions:
+```cpp
+{2,1}, {3,1}, {8,1}, {6,1}, {8,6}
+```

From 1f9e1499c5ecc504494569946f2ad969755d6ced Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:33:09 +1200
Subject: [PATCH 5/8] Update README.md

---
 ch02/README.md | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/ch02/README.md b/ch02/README.md
index 94944fb..e47052a 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -369,3 +369,13 @@ L.I. holds.
 ```cpp
 {2,1}, {3,1}, {8,1}, {6,1}, {8,6}
 ```
+ * set {n, n-1, n-2, ...,2, 1}, i.e. numbers in descending order has most inversions.
+ * As shown below, the expression `A[i] > key` is in essence checking for an inversion. So a function can be made to describe the relationship between the running time and number of inversions:
+```cpp
+  T(n) = theta(n * number_of_inversions + n)
+```
+
+```cpp
+//from Insertion-Sort
+5     while i > 0 and A[i] > key
+```

From 0731bac3dd27c3c9e6966000b1da574344eb7ed8 Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:34:43 +1200
Subject: [PATCH 6/8] Update README.md

---
 ch02/README.md | 7 +------
 1 file changed, 1 insertion(+), 6 deletions(-)

diff --git a/ch02/README.md b/ch02/README.md
index e47052a..87cecee 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -370,12 +370,7 @@ L.I. holds.
 {2,1}, {3,1}, {8,1}, {6,1}, {8,6}
 ```
  * set {n, n-1, n-2, ...,2, 1}, i.e. numbers in descending order has most inversions.
- * As shown below, the expression `A[i] > key` is in essence checking for an inversion. So a function can be made to describe the relationship between the running time and number of inversions:
+ * As shown below, the expression `A[i] > key` in line 5 Insertion-Sort is in essence checking for an inversion. So a function can be made to describe the relationship between the running time and number of inversions:
 ```cpp
   T(n) = theta(n * number_of_inversions + n)
 ```
-
-```cpp
-//from Insertion-Sort
-5     while i > 0 and A[i] > key
-```

From f3fc2c2d154f1133a6ccad62e22dee0eda4a85a3 Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:56:22 +1200
Subject: [PATCH 7/8] Update README.md

---
 ch02/README.md | 7 +++++--
 1 file changed, 5 insertions(+), 2 deletions(-)

diff --git a/ch02/README.md b/ch02/README.md
index 87cecee..55df9f3 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -364,13 +364,16 @@ L.I. holds.
 
  * As shown above, this code fragment correctly evaluates a polynomial.
 
-##Problem 2-3 Inversions
+##Problem 2-4 Inversions
  *  Five inversions:
 ```cpp
 {2,1}, {3,1}, {8,1}, {6,1}, {8,6}
 ```
  * set {n, n-1, n-2, ...,2, 1}, i.e. numbers in descending order has most inversions.
+```
+number of inversions = n(n - 1)/2
+```
  * As shown below, the expression `A[i] > key` in line 5 Insertion-Sort is in essence checking for an inversion. So a function can be made to describe the relationship between the running time and number of inversions:
 ```cpp
-  T(n) = theta(n * number_of_inversions + n)
+  T(n) = theta(number_of_inversions + n)
 ```

From 111e598e62b497d7234f7e7bcc321e0d53458272 Mon Sep 17 00:00:00 2001
From: Yue Wang <Moophyalan@gmail.com>
Date: Mon, 6 Apr 2015 21:56:58 +1200
Subject: [PATCH 8/8] Update README.md

---
 ch02/README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ch02/README.md b/ch02/README.md
index 55df9f3..e66b946 100644
--- a/ch02/README.md
+++ b/ch02/README.md
@@ -375,5 +375,5 @@ number of inversions = n(n - 1)/2
 ```
  * As shown below, the expression `A[i] > key` in line 5 Insertion-Sort is in essence checking for an inversion. So a function can be made to describe the relationship between the running time and number of inversions:
 ```cpp
-  T(n) = theta(number_of_inversions + n)
+T(n) = theta(number_of_inversions + n)
 ```