How was your exam?

Tom is revising for his exam of tomorrow.

How was your test?

This fact proves her innocence.

This fact proves her innocence.

That fact proves his innocence.

How was your biology exam?

'That proves his guilt,' said the Queen.

We tried them in this way, for they were disobedient.

Many of these theorems are independent of ZFC, requiring stronger axioms for their proof.

Don't worry about the results of your test.

T proves 1. that is, T proves that if T proves P , then T proves ProvA( ( P )).

In other words, T proves that ProvA( ( P )) implies ProvA( (ProvA( (P)))).

Tom studied hard for the test.

In a later tradition, Mary drops her girdle down to the apostle from heaven as testament to the event.

In other words, T proves that ProvA( ( P Q )) and ProvA( ( P )) imply ProvA( ( Q )).

T proves that if T proves that ( P Q ) and T proves P then T proves Q .

You come in, you face the wall, you take your test and you leave.

And there is no way, and no proof of arrival.

What kind of evidence do we have for that?

I've got to prepare for the test.

They were girding for the exam.

You prove one, you prove the other.

In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well ordering theorem.

Category theory There are several results in category theory which invoke the axiom of choice for their proof.

Yet his willingness to finish the race, while struggling mightily earned him a standing ovation from the audience.

Here is the business case for sustainability.

The proof is left to the reader.

I have to study for a biology test.

We have to study for our exam.

The difference is that instead of constructing a new proof, the proof verifier simply checks that a provided formal proof (or, in instructions that can be followed to create a formal proof) is correct.

I mean, if the guy's theory gets a scientific proof, there'd be nothing terrible for the Africans in it.

In Monaco Räikkönen qualified in 8th but finished one place lower in 9th place.

There is no concrete proof of her complicity in Darnley's murder or of a conspiracy with Bothwell.

I should be studying for tomorrow's exam.

I have to study for a biology test.

Soft Proofing

Typically, the computer can recognize the axioms and logical rules for deriving theorems, and the computer can recognize whether a proof is valid, but to determine whether a proof exists for a statement is only soluble by waiting for the proof or disproof to be generated.

Their use in other works such as The Methods of the Sima is considered proof of Sun Tzu's historical priority.

Combinatorial proof A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways.

I have to prepare for the English test.

I have to revise for a biology test.

Proof by contradiction is used in mathematics to construct proofs.

That he did represent the genius of the average is the most convincing proof of his strength.

He wipes sorrow from his face, and puts his thirsty finger to his thirsty tongue, and tastes the salt.