In [1]:
import proveit
theory = proveit.Theory() # the theorem's theory
In [2]:
%proving double_negation_equiv
With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
double_negation_equiv:
(see dependencies)
double_negation_equiv:
(see dependencies)
Automated proof, folding the forall over the Boolean set. This is a shorter proof than via $\lnot (\lnot A) \Leftrightarrow A$.
In [3]:
%qed
Out[3]:
| step type | requirements | statement | ||
|---|---|---|---|---|
| 0 | instantiation | 1, 2, 3 | ⊢ | |
 :  ,  : | ||||
| 1 | conjecture | ⊢  | ||
| proveit.logic.booleans.forall_over_bool_by_cases | ||||
| 2 | instantiation | 4, 5 | ⊢  | |
:  | ||||
| 3 | instantiation | 6, 7, 8 | ⊢  | |
 :  ,  :  ,  :  | ||||
| 4 | axiom | ⊢  | ||
| proveit.logic.booleans.eq_true_intro | ||||
| 5 | instantiation | 9, 10 | ⊢ | |
:  | ||||
| 6 | axiom | ⊢  | ||
| proveit.logic.equality.equals_transitivity | ||||
| 7 | instantiation | 11, 12 | ⊢  | |
 :  , :  , : | ||||
| 8 | axiom | ⊢  | ||
| proveit.logic.booleans.negation.not_t | ||||
| 9 | theorem | ⊢  | ||
| proveit.logic.booleans.negation.double_negation_intro | ||||
| 10 | axiom | ⊢ | ||
| proveit.logic.booleans.true_axiom | ||||
| 11 | axiom | ⊢  | ||
| proveit.logic.equality.substitution | ||||
| 12 | axiom | ⊢  | ||
| proveit.logic.booleans.negation.not_f | ||||