In [1]:
import proveit
from proveit.logic.booleans import true_is_bool, false_is_bool, in_bool_if_true
from proveit.logic.booleans.conjunction import and_if_both
from proveit.logic.booleans.negation import not_f, not_t
from proveit.logic.equality import substitute_truth
from proveit import defaults
in_bool_if_true.proof().disable(); substitute_truth.proof().disable() # force a more short, symmetric proof
theory = proveit.Theory() # the theorem's theory
In [2]:
%proving closure
With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
closure:
(see dependencies)
closure:
(see dependencies)
In [3]:
true_is_bool
In [4]:
not_f
In [5]:
not_f.sub_left_side_into(true_is_bool)
In [6]:
false_is_bool
In [7]:
not_t
In [8]:
not_t.sub_left_side_into(false_is_bool)
In [9]:
%qed
Out[9]:
| step type | requirements | statement | ||
|---|---|---|---|---|
| 0 | instantiation | 1, 2, 3 | ⊢ | |
 :  ,  : | ||||
| 1 | conjecture | ⊢  | ||
| proveit.logic.booleans.forall_over_bool_by_cases | ||||
| 2 | instantiation | 6, 4, 5 | ⊢  | |
 :  ,  :  ,  :  | ||||
| 3 | instantiation | 6, 7, 8 | ⊢  | |
: , :  , :  | ||||
| 4 | conjecture | ⊢  | ||
| proveit.logic.booleans.false_is_bool | ||||
| 5 | axiom | ⊢  | ||
| proveit.logic.booleans.negation.not_t | ||||
| 6 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 7 | conjecture | ⊢  | ||
| proveit.logic.booleans.true_is_bool | ||||
| 8 | axiom | ⊢  | ||
| proveit.logic.booleans.negation.not_f | ||||