… reaches

The partially ordered set uses a method named 'dominates' to determine whether
two nodes are partially ordered. Dominates does a depth-first search of the
DAG, beginning at the source node, and returns true as soon as it finds a path
to the target node. In the context of the forest-of-DAGs that makes up the
poset, dominates is not necessarily checking dominance, but is checking
reachability. See the issue tracker for a more detailed discussion of the
difference.

Fortunately, reachability is logically correct everywhere dominates is currently
used in poset.go. Reachability within a DAG is sufficient to establish the
partial ordering (source < target).

This CL changes the name of the method (dominates -> reaches) and updates
all the comments in the file accordingly.

Updates #34807

Change-Id: Ia3a34f7b14b363801d75b05099cfc686035f7d96
Reviewed-on: https://go-review.googlesource.com/c/go/+/192617
Reviewed-by: Giovanni Bajo <rasky@develer.com>
Run-TryBot: Giovanni Bajo <rasky@develer.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-on: https://go-review.googlesource.com/c/go/+/201059
Run-TryBot: Alexander Rakoczy <alex@golang.org>
Reviewed-by: Keith Randall <khr@golang.org>

…ions for OrderedOrEqual

When assessing whether A <= B, the poset's OrderedOrEqual has a passing
condition which permits A <= B, but is not sufficient to infer that A <= B.
This CL removes that incorrect passing condition.

Having identified that A and B are in the poset, the method will report that
A <= B if any of these three conditions are true:
 (1) A and B are the same node in the poset.
 	- This means we know that A == B.
 (2) There is a directed path, strict or not, from A -> B
 	- This means we know that, at least, A <= B, but A < B is possible.
 (3) There is a directed path from B -> A, AND that path has no strict edges.
 	- This means we know that B <= A, but do not know that B < A.

In condition (3), we do not have enough information to say that A <= B, rather
we only know that B == A (which satisfies A <= B) is possible. The way I
understand it, a strict edge shows a known, strictly-ordered relation (<) but
the lack of a strict edge does not show the lack of a strictly-ordered relation.

The difference is highlighted by the example in #34802, where a bounds check is
incorrectly removed by prove, such that negative indexes into a slice
succeed:

	n := make([]int, 1)
	for i := -1; i <= 0; i++ {
	    fmt.Printf("i is %d\n", i)
	    n[i] = 1  // No Bounds check, program runs, assignment to n[-1] succeeds!!
	}

When prove is checking the negative/failed branch from the bounds check at n[i],
in the signed domain we learn (0 > i || i >= len(n)). Because prove can't learn
the OR condition, we check whether we know that i is non-negative so we can
learn something, namely that i >= len(n). Prove uses the poset to check whether
we know that i is non-negative.  At this point the poset holds the following
relations as a directed graph:

	-1 <= i <= 0
	-1 < 0

In poset.OrderedOrEqual, we are testing for 0 <= i. In this case, condition (3)
above is true because there is a non-strict path from i -> 0, and that path
does NOT have any strict edges. Because this condition is true, the poset
reports to prove that i is known to be >= 0. Knowing, incorrectly, that i >= 0,
prove learns from the failed bounds check that i >= len(n) in the signed domain.

When the slice, n, was created, prove learned that len(n) == 1. Because i is
also the induction variable for the loop, upon entering the loop, prove previously
learned that i is in [-1,0]. So when prove attempts to learn from the failed
bounds check, it finds the new fact, i > len(n), unsatisfiable given that it
previously learned that i <= 0 and len(n) = 1.

Fixes #34807

Change-Id: I235f4224bef97700c3aa5c01edcc595eb9f13afc
Reviewed-on: https://go-review.googlesource.com/c/go/+/200759
Run-TryBot: Zach Jones <zachj1@gmail.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Giovanni Bajo <rasky@develer.com>
Reviewed-by: Keith Randall <khr@golang.org>
Reviewed-on: https://go-review.googlesource.com/c/go/+/201060
Run-TryBot: Alexander Rakoczy <alex@golang.org>

…inates to reaches

The partially ordered set uses a method named 'dominates' to determine whether
two nodes are partially ordered. Dominates does a depth-first search of the
DAG, beginning at the source node, and returns true as soon as it finds a path
to the target node. In the context of the forest-of-DAGs that makes up the
poset, dominates is not necessarily checking dominance, but is checking
reachability. See the issue tracker for a more detailed discussion of the
difference.

Fortunately, reachability is logically correct everywhere dominates is currently
used in poset.go. Reachability within a DAG is sufficient to establish the
partial ordering (source < target).

This CL changes the name of the method (dominates -> reaches) and updates
all the comments in the file accordingly.

Updates #34807

Change-Id: Ia3a34f7b14b363801d75b05099cfc686035f7d96
Reviewed-on: https://go-review.googlesource.com/c/go/+/192617
Reviewed-by: Giovanni Bajo <rasky@develer.com>
Run-TryBot: Giovanni Bajo <rasky@develer.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-on: https://go-review.googlesource.com/c/go/+/201059
Run-TryBot: Alexander Rakoczy <alex@golang.org>
Reviewed-by: Keith Randall <khr@golang.org>
Reviewed-on: https://team-review.git.corp.google.com/c/golang/go-private/+/575397
Reviewed-by: Filippo Valsorda <valsorda@google.com>

…nt conditions for OrderedOrEqual

When assessing whether A <= B, the poset's OrderedOrEqual has a passing
condition which permits A <= B, but is not sufficient to infer that A <= B.
This CL removes that incorrect passing condition.

Having identified that A and B are in the poset, the method will report that
A <= B if any of these three conditions are true:
 (1) A and B are the same node in the poset.
 	- This means we know that A == B.
 (2) There is a directed path, strict or not, from A -> B
 	- This means we know that, at least, A <= B, but A < B is possible.
 (3) There is a directed path from B -> A, AND that path has no strict edges.
 	- This means we know that B <= A, but do not know that B < A.

In condition (3), we do not have enough information to say that A <= B, rather
we only know that B == A (which satisfies A <= B) is possible. The way I
understand it, a strict edge shows a known, strictly-ordered relation (<) but
the lack of a strict edge does not show the lack of a strictly-ordered relation.

The difference is highlighted by the example in #34802, where a bounds check is
incorrectly removed by prove, such that negative indexes into a slice
succeed:

	n := make([]int, 1)
	for i := -1; i <= 0; i++ {
	    fmt.Printf("i is %d\n", i)
	    n[i] = 1  // No Bounds check, program runs, assignment to n[-1] succeeds!!
	}

When prove is checking the negative/failed branch from the bounds check at n[i],
in the signed domain we learn (0 > i || i >= len(n)). Because prove can't learn
the OR condition, we check whether we know that i is non-negative so we can
learn something, namely that i >= len(n). Prove uses the poset to check whether
we know that i is non-negative.  At this point the poset holds the following
relations as a directed graph:

	-1 <= i <= 0
	-1 < 0

In poset.OrderedOrEqual, we are testing for 0 <= i. In this case, condition (3)
above is true because there is a non-strict path from i -> 0, and that path
does NOT have any strict edges. Because this condition is true, the poset
reports to prove that i is known to be >= 0. Knowing, incorrectly, that i >= 0,
prove learns from the failed bounds check that i >= len(n) in the signed domain.

When the slice, n, was created, prove learned that len(n) == 1. Because i is
also the induction variable for the loop, upon entering the loop, prove previously
learned that i is in [-1,0]. So when prove attempts to learn from the failed
bounds check, it finds the new fact, i > len(n), unsatisfiable given that it
previously learned that i <= 0 and len(n) = 1.

Fixes #34807

Change-Id: I235f4224bef97700c3aa5c01edcc595eb9f13afc
Reviewed-on: https://go-review.googlesource.com/c/go/+/200759
Run-TryBot: Zach Jones <zachj1@gmail.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Giovanni Bajo <rasky@develer.com>
Reviewed-by: Keith Randall <khr@golang.org>
Reviewed-on: https://go-review.googlesource.com/c/go/+/201060
Run-TryBot: Alexander Rakoczy <alex@golang.org>
Reviewed-on: https://team-review.git.corp.google.com/c/golang/go-private/+/575398
Reviewed-by: Filippo Valsorda <valsorda@google.com>