Humaneval Nagini chain of though prompt with instructions

prompts/humaneval-nagini-cot-instruct/ask_for_fixed.txt

@@ -0,0 +1,6 @@
+The following errors occurred during verification:
+{error}
+
+Please fix the error by adding, removing or modifying the invariants and return the fixed program.
+Don't add any additional text comments, your response must contain only program with invariants.
+Do not provide ANY explanations. Don't include markdown backticks. Respond only in Python code, nothing else.

prompts/humaneval-nagini-cot-instruct/ask_for_fixed_had_errors.txt

@@ -0,0 +1,6 @@
+There are still some errors:
+{error}
+
+Could you please fix them?
+Don't add any additional text comments, your response must contain only program with invariants.
+Do not provide ANY explanations. Don't include markdown backticks. Respond only in Python code, nothing else.

prompts/humaneval-nagini-cot-instruct/rewrite/examples/001/answer.txt

@@ -0,0 +1,103 @@
+During the previous steps, we've found that the `sum_loop` function has a loop that goes over the elements of the array `numbers` and adds them to the `s` variable, which is equal to the `Sum(numbers, 0, i)` for all `i`, such that `0 <= i and i <= len(numbers)`. Also we need list access to numbers. This means that during the execution of the loop the following invariant holds:
+1. `Acc(list_pred(numbers))`
+2. `0 <= i and i <= len(numbers)`
+3. `s == Sum(numbers, 0, i)`
+
+Let's put them in:
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```
+
+We've also worked out how to prove the third invariant. For that we need to prove to the verifier that `Sum(numbers, 0, i + 1)== Sum(numbers, 0, i) + numbers[i]`. For that we need to add the following invariants and asserts:
+```
+Invariant(Forall(int, lambda d_1_p_:
+    (Implies(0 <= d_1_p_ and d_1_p_ < len(numbers), Sum(numbers, 0, d_1_p_ + 1) == Sum(numbers, 0, d_1_p_) + numbers[d_1_p_]), [[Sum(numbers, 0, d_1_p_ + 1)]])))
+Assert(Sum(numbers, 0, i + 1) == Sum(numbers, 0, i) + numbers[i])
+```
+Let's add these lines to the code:
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        Invariant(Acc(list_pred(numbers)))
+        Invariant(0 <= i and i <= len(numbers))
+        Invariant(Forall(int, lambda d_1_p_:
+            (Implies(0 <= d_1_p_ and d_1_p_ < len(numbers), Sum(numbers, 0, d_1_p_ + 1) == Sum(numbers, 0, d_1_p_) + numbers[d_1_p_]), [[Sum(numbers, 0, d_1_p_ + 1)]])))
+        Invariant(s == Sum(numbers, 0, i))
+        Assert(Sum(numbers, 0, i + 1) == Sum(numbers, 0, i) + numbers[i])
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```
+
+Other than that, we didn't find any other invariants or patterns that hold during the execution of the program. Therefore, the resulting program is (notice the lack of backticks):
+<answer>
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        Invariant(Acc(list_pred(numbers)))
+        Invariant(0 <= i and i <= len(numbers))
+        Invariant(Forall(int, lambda d_1_p_:
+            (Implies(0 <= d_1_p_ and d_1_p_ < len(numbers), Sum(numbers, 0, d_1_p_ + 1) == Sum(numbers, 0, d_1_p_) + numbers[d_1_p_]), [[Sum(numbers, 0, d_1_p_ + 1)]])))
+        Invariant(s == Sum(numbers, 0, i))
+        Assert(Sum(numbers, 0, i + 1) == Sum(numbers, 0, i) + numbers[i])
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+</answer>

prompts/humaneval-nagini-cot-instruct/rewrite/examples/001/question.txt

@@ -0,0 +1,27 @@
+It's time to rewrite the program to contain the invariants and assertions that we've found. Work on the following program and go step by step through the reasoning process. After that, output the rewritten program in the `<answer>` tag without backticks.
+
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```

prompts/humaneval-nagini-cot-instruct/rewrite/question.txt

@@ -0,0 +1,3 @@
+It's time to rewrite the program to contain the invariants and assertions that we've found. Work on the following program and go step by step through the reasoning process. After that, output the rewritten program in the `<answer>` tag without backticks.
+
+{program}

prompts/humaneval-nagini-cot-instruct/steps/001/examples/001/answer.txt

@@ -0,0 +1 @@
+understood

prompts/humaneval-nagini-cot-instruct/steps/001/examples/001/question.txt

@@ -0,0 +1,40 @@
+You are an expert in Nagini, a verification framework in Python.
+You remember the following aspects of Nagini syntax:
+
+1. Nagini DOES NOT SUPPORT some Python features as list comprehensions (k + 1 for k in range(5)), as double inequalities (a <= b <= c).
+Instead of double inequalities it's customary to use two separate inequalities (a <= b and b <= c).
+
+2. In Nagini method preconditions (Requires) and postconditions (Ensures) placed right after method signature, like here:
+"
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+    Ensures(Acc(list_pred(a)))
+    ...
+"
+
+3. Invariant are placed right after `while` statement and before the code of `while` body:
+"
+    while i < len(numbers):
+        Invariant(Acc(list_pred(numbers)))
+        Invariant(0 <= i and i <= len(numbers))
+        s = s + numbers[i]
+"
+Invariants CANNOT be placed in any other position.
+You remember that each invariant (and each expression) should contain equal number of opening and closing brackets, so that it is valid.
+You should sustain balanced parentheses.
+
+4. Nagini requires special annotations for working with lists `Acc(list_pred(..))`. You can use these constructs only inside `Invariant`,
+anywhere else you should not use `Acc()` or `list_pred()`:
+"
+    while i < len(numbers):
+        Invariant(Acc(list_pred(numbers)))
+"
+
+5. Nagini contains `Forall` and `Exists` constructs that can be used in invariants. First argument of Forall/Exists is typically a type (i.e `int`),
+second argument is a lambda. `Forall(type, lambda x : a)` denotes that assertion `a` is true for every element `x` of type `type`.
+
+6. In Nagini `Implies(e1, a2)` plays role of implication. `Implies(e1, a2)` denotes that assertion a2 holds if boolean expression e1 is true.
+You can use it inside invariants and asserts.
+
+Respond with 'understood' to this message. Remember this knowledge to solve the task that will be given further

prompts/humaneval-nagini-cot-instruct/steps/001/question.txt

@@ -0,0 +1,40 @@
+You are an expert in Nagini, a verification framework in Python.
+You remember the following aspects of Nagini syntax:
+
+1. Nagini DOES NOT SUPPORT some Python features as list comprehensions (k + 1 for k in range(5)), as double inequalities (a <= b <= c).
+Instead of double inequalities it's customary to use two separate inequalities (a <= b and b <= c).
+
+2. In Nagini method preconditions (Requires) and postconditions (Ensures) placed right after method signature, like here:
+"
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+    Ensures(Acc(list_pred(a)))
+    ...
+"
+
+3. Invariant are placed right after `while` statement and before the code of `while` body:
+"
+    while i < len(numbers):
+        Invariant(Acc(list_pred(numbers)))
+        Invariant(0 <= i and i <= len(numbers))
+        s = s + numbers[i]
+"
+Invariants CANNOT be placed in any other position.
+You remember that each invariant (and each expression) should contain equal number of opening and closing brackets, so that it is valid.
+You should sustain balanced parentheses.
+
+4. Nagini requires special annotations for working with lists `Acc(list_pred(..))`. You can use these constructs only inside `Invariant`,
+anywhere else you should not use `Acc()` or `list_pred()`:
+"
+    while i < len(numbers):
+        Invariant(Acc(list_pred(numbers)))
+"
+
+5. Nagini contains `Forall` and `Exists` constructs that can be used in invariants. First argument of Forall/Exists is typically a type (i.e `int`),
+second argument is a lambda. `Forall(type, lambda x : a)` denotes that assertion `a` is true for every element `x` of type `type`.
+
+6. In Nagini `Implies(e1, a2)` plays role of implication. `Implies(e1, a2)` denotes that assertion a2 holds if boolean expression e1 is true.
+You can use it inside invariants and asserts.
+
+Respond with 'understood' to this message. Remember this knowledge to solve the task that will be given further

prompts/humaneval-nagini-cot-instruct/steps/002/examples/001/answer.txt

@@ -0,0 +1,7 @@
+Let's go function by function. The first function, `Sum` has @Pure annotation. This means that this function can be probably used in specifications. We can see that the aruguments to this function are a sequence of integers and two indices. We can see that the function returns an integer. We can conclude that the function does some accumulation of integers from the subsequence [l;r) of the given sequence. Looking at the body, we can see that the function takes the first element of the subsequence and adds it to the accumulated result of evaluating the `sum` function on the rest of the subsequence. This means the function is a specification for the sum of the elements of the subsequence.
+
+Finally, we have the `sum_loop` function. This function has a specification, as indicated by the `Requires` and `Ensures` clauses. The specification states that the returned value should be equal to the sum of the input sequence `numbers`, as computed by the `Sum` function, as seen by the `Result() == Sum(numbers, 0, len(numbers))` clause.
+
+In summary:
+1. The `sum` function is used in a specification for computing the sum of a sequence of integers.
+2. The `sum_loop` function has a specification that ensures it correctly computes the sum of a sequence of integers, matching the result of the `Sum` function.

prompts/humaneval-nagini-cot-instruct/steps/002/examples/001/question.txt

@@ -0,0 +1,26 @@
+Given the following program, for each function, determine if it has a specification. If it does not, is it itself a specification? If it is a specification, what does it specify?
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```

prompts/humaneval-nagini-cot-instruct/steps/002/question.txt

@@ -0,0 +1,5 @@
+Given the following program, for each function, determine if it has a specification. Determine also functions, that can be used in specifications.
+
+```
+{program}
+```

prompts/humaneval-nagini-cot-instruct/steps/003/examples/001/answer.txt

@@ -0,0 +1,6 @@
+Looking at the functions, we can see that the 'sum' function uses recursion and thus contains no loops.
+Now, let's examine the 'sum_loop' function. This method contains a while loop that iterates through the elements of the 'numbers' list. The loop starts with 's' and 'i' initialized to 0, and continues as long as 'i' is less than the length of 'numbers'. Let's try to understand what 's' and 'i' represent in this context.
+
+Looking at the body of the loop, we can see that it adds the current element of 'numbers' to 's'. This means that 's' is the running sum of the elements in 'numbers', namely 's = Sum(numbers, 0, i)'. 'i' is easier to analyze. It is the index of the current element being processed. In each iteration of the loop, 'i' is incremented by 1, so 'i' represents the index of the next element to be processed.
+
+Each iteration of the loop adds the current element to 's', and then increments 'i' to move to the next element. This process continues until 'i' reaches the length of 'numbers', at which point the loop terminates and 's' holds the final sum of the elements in 'numbers'.

prompts/humaneval-nagini-cot-instruct/steps/003/examples/001/question.txt

@@ -0,0 +1,27 @@
+Now we are finished with the specifications. Let's move onto the loops.
+Given the following program, for each function, determine if it has loops. If it does, try to determine what the loop does in detail. Go through all your thinking and reasoning process step by step.
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```

prompts/humaneval-nagini-cot-instruct/steps/003/question.txt

@@ -0,0 +1,4 @@
+Given the following program, for each function, determine if it has loops. If it does, try to determine what the loop does in detail. Go through all your thinking and reasoning process step by step.
+```
+{program}
+```

prompts/humaneval-nagini-cot-instruct/steps/004/examples/001/answer.txt

@@ -0,0 +1,14 @@
+During the previous steps, we've found that the `sum_loop` function has a loop that goes over the elements of the array `numbers` and adds them to the `s` variable, which is equal to the `Sum(numbers, 0, i)` for all `i`, such that `0 <= i and i <= len(numbers)`. Also we need list access to numbers. This means that during the execution of the loop the following invariant holds:
+1. `Acc(list_pred(numbers))`
+2. `0 <= i and i <= len(numbers)`
+3. `s == Sum(numbers, 0, i)`
+
+The third invariant might be a bit difficult to prove, as it adds the last element of the array to the sum, while, as I've said before, the `Sum` function is defined as `Sum(a, 0, i + 1) == Sum(a, 0, i) + a[i]`. However, as we've seen, it's possible to use additional invariants and assertions to prove this invariants holds.
+Let's work out how to use it. One of the ways to prove it is to add the following invariants and assertions:
+```
+Invariant(Forall(int, lambda d_1_p_:
+    (Implies(0 <= d_1_p_ and d_1_p_ < len(numbers), Sum(numbers, 0, d_1_p_ + 1) == Sum(numbers, 0, d_1_p_) + numbers[d_1_p_]), [[Sum(numbers, 0, d_1_p_ + 1)]])))
+Assert(Sum(numbers, 0, i + 1) == Sum(numbers, 0, i) + numbers[i])
+```
+
+In conclusion, we can say that two invariants hold and the third one could be proved using additional invariants and assertions.

prompts/humaneval-nagini-cot-instruct/steps/004/examples/001/question.txt

@@ -0,0 +1,26 @@
+As we've already analyzed the specifications and the loops, it's time to try to analyze what invariants and patterns hold during the execution of the program. Go over the found loops in the following program in the code and try to find out what invariants and patterns they hold. Go through all your thinking and reasoning process step by step.
+```
+from typing import cast, List, Dict, Set, Optional, Union
+from nagini_contracts.contracts import *
+
+@Pure
+def Sum(a : List[int], s : int, t : int) -> int :
+    Requires(Acc(list_pred(a)))
+    Requires(((0) <= (s)) and ((s) <= (t)) and ((t) <= (len(a))))
+
+    if s == t:
+        return 0
+    else:
+        return (a)[t - 1] + (Sum(a, s, t - 1))
+
+def sum_loop(numbers: List[int]) -> int:
+    Requires(Acc(list_pred(numbers)))
+    Ensures(Acc(list_pred(numbers)))
+    Ensures(Result() == Sum(numbers, 0, len(numbers)))
+    s = int(0)
+    i = int(0)
+    while (i) < (len(numbers)):
+        s = s + (numbers)[i]
+        i = i + 1
+    return s
+```

prompts/humaneval-nagini-cot-instruct/steps/004/question.txt

@@ -0,0 +1,4 @@
+As we've already analyzed the specifications and the loops, it's time to try to analyze what invariants and patterns hold during the execution of the program. Go over the found loops in the following program in the code and try to find out what invariants and patterns they hold. Go through all your thinking and reasoning process step by step.
+```
+{program}
+```

prompts/humaneval-nagini-cot-instruct/sys.txt

@@ -0,0 +1,4 @@
+You are an expert in a Python verification framework Nagini.
+You will be given tasks dealing with Python programs including precise docstrings and specifications.
+Do not provide ANY explanations. Don't include markdown backticks. Respond only in Python code, nothing else.
+Take into account that arrays inside the invariants are indexed by type `int`.

prompts/humaneval-nagini-cot-instruct/timeout.txt

@@ -0,0 +1,3 @@
+The verifier timed out during the verification.
+This usually means that the provided invariants were too broad or were difficult to check.
+Could you please try to improve the invariants and try again?

prompts/humaneval-nagini-few-shot/add.txt

@@ -0,0 +1,6 @@
+Do not provide ANY explanations. Don't include markdown backticks. Respond only in Python code, nothing else.
+The program:
+{program}
+–
+The invariants:
+{checks}