Implies cannot find an answer

Question

I have a simple relation and I try to find if P implies Q or not. I expect to get True because it is somehow obvious for me.

P1 = j == i && j < 0 && t >= -j
Q = t>0
FullSimplify[ Reduce[ForAll[{i, j, t}, Implies[ P1 , Q]]]]

When I run this I get an echo, which means cannot solve it.

But When I remove j == i , it returns True.

can someone explain me why and what is the solution? I am fine to use other functions, I just need to check if P implies Q or not.

P2 = j < 0 && t >= -j
Q = t > 0
FullSimplify[ForAll[{i, j, t}, Implies[P2, Q]]]
True

score 4 · Accepted Answer · answered Apr 07 '23 at 16:05

4

Adding the Reals domain helps (By default, variables are assumed as complexes.).

P1 = j == i && j < 0 && t >= -j;Q = t > 0;
Resolve[ForAll[{i, j, t}, Implies[P1, Q]], Reals]
Reduce[ForAll[{i, j, t}, Implies[P1, Q]], Reals]

True True

answered Apr 07 '23 at 16:05

user64494

Thanks, it was great . But do you mean Complex Numbers? a + i*b? it's very interesting. – Azzurro94 Apr 07 '23 at 16:23
1

@Azzurro94: Yes, I do mean complex numbers. – user64494 Apr 07 '23 at 16:46

1 Answers1