In [1]:
import proveit
theory = proveit.Theory() # the theorem's theory
from proveit.logic import InSet
from proveit.numbers import Integer
from proveit.physics.quantum.QPE import _best_floor_def
from proveit.physics.quantum.QPE import _phase_is_real, _two_pow_t_is_nat_pos
In [2]:
%proving _best_floor_is_int
With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
_best_floor_is_int:
(see dependencies)
_best_floor_is_int:
(see dependencies)
In [3]:
_best_floor_def
In [4]:
InSet(_best_floor_def.rhs, Integer).prove()
In [5]:
%qed
Out[5]:
| step type | requirements | statement | ||
|---|---|---|---|---|
| 0 | instantiation | 1, 2, 3 | ⊢ | |
 :  ,  :  ,  :  | ||||
| 1 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 2 | instantiation | 4, 5 | ⊢  | |
:  | ||||
| 3 | axiom | ⊢  | ||
| proveit.physics.quantum.QPE._best_floor_def | ||||
| 4 | axiom | ⊢  | ||
| proveit.numbers.rounding.floor_is_an_int | ||||
| 5 | instantiation | 6, 7, 8 | ⊢  | |
 :  ,  :  | ||||
| 6 | conjecture | ⊢  | ||
| proveit.numbers.multiplication.mult_real_closure_bin | ||||
| 7 | instantiation | 13, 9, 10 | ⊢  | |
 :  ,  :  , : | ||||
| 8 | conjecture | ⊢  | ||
| proveit.physics.quantum.QPE._phase_is_real | ||||
| 9 | conjecture | ⊢  | ||
| proveit.numbers.number_sets.real_numbers.rational_within_real | ||||
| 10 | instantiation | 13, 11, 12 | ⊢  | |
:  , : , : | ||||
| 11 | conjecture | ⊢  | ||
| proveit.numbers.number_sets.rational_numbers.int_within_rational | ||||
| 12 | instantiation | 13, 14, 15 | ⊢  | |
:  , : , : | ||||
| 13 | theorem | ⊢  | ||
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset | ||||
| 14 | conjecture | ⊢  | ||
| proveit.numbers.number_sets.integers.nat_pos_within_int | ||||
| 15 | conjecture | ⊢  | ||
| proveit.physics.quantum.QPE._two_pow_t_is_nat_pos | ||||