Expression of type Conditional

from the theory of proveit.physics.quantum.QPE

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, k, m
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Integer, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Conditional(Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), domain = _m_domain))), InSet(m, Integer))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\right)\right) \textrm{ if } m \in \mathbb{Z}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 26 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	61, 8
6	Operation	operator: 9 operand: 12
7	Operation	operator: 54 operands: 11
8	Literal
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 47 operands: 17
14	Operation	operator: 18 operand: 20
15	Literal
16	ExprTuple	61, 52
17	ExprTuple	53, 52
18	Literal
19	ExprTuple	20
20	Lambda	parameter: 60 body: 22
21	ExprTuple	60
22	Conditional	value: 23 condition: 24
23	Operation	operator: 54 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	60, 30
28	Operation	operator: 56 operands: 31
29	Operation	operator: 56 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprTuple	36, 35
32	ExprTuple	36, 37
33	Literal
34	ExprTuple	38, 39
35	Operation	operator: 49 operand: 44
36	Literal
37	Operation	operator: 54 operands: 41
38	Literal
39	Operation	operator: 42 operands: 43
40	ExprTuple	44
41	ExprTuple	62, 58, 59, 45, 60
42	Literal
43	ExprTuple	52, 46
44	Operation	operator: 47 operands: 48
45	Literal
46	Operation	operator: 49 operand: 53
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53
51	Operation	operator: 54 operands: 55
52	Operation	operator: 56 operands: 57
53	Literal
54	Literal
55	ExprTuple	62, 58, 59, 60, 61
56	Literal
57	ExprTuple	62, 63
58	Literal
59	Literal
60	Variable
61	Variable
62	Literal
63	Literal