Expression of type Equals

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
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = InSet(k, _m_domain)
expr = Equals(Conditional(Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), sub_expr1), Conditional(one, sub_expr1)).with_wrapping_at(1)

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())

\begin{array}{c} \begin{array}{l} \left\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. \\  = \left\{1 \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(1)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Conditional	value: 5 condition: 6
4	Conditional	value: 33 condition: 6
5	Operation	operator: 39 operands: 7
6	Operation	operator: 8 operands: 9
7	ExprTuple	10, 11
8	Literal
9	ExprTuple	37, 12
10	Operation	operator: 43 operands: 13
11	Operation	operator: 43 operands: 14
12	Operation	operator: 15 operands: 16
13	ExprTuple	18, 17
14	ExprTuple	18, 19
15	Literal
16	ExprTuple	20, 21
17	Operation	operator: 30 operand: 26
18	Literal
19	Operation	operator: 39 operands: 23
20	Literal
21	Operation	operator: 24 operands: 25
22	ExprTuple	26
23	ExprTuple	45, 35, 36, 42, 37
24	Literal
25	ExprTuple	41, 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operand: 33
28	Literal
29	ExprTuple	32, 41
30	Literal
31	ExprTuple	33
32	Operation	operator: 39 operands: 34
33	Literal
34	ExprTuple	45, 35, 36, 37, 38
35	Literal
36	Literal
37	Variable
38	Operation	operator: 39 operands: 40
39	Literal
40	ExprTuple	41, 42
41	Operation	operator: 43 operands: 44
42	Literal
43	Literal
44	ExprTuple	45, 46
45	Literal
46	Literal