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

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

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\{\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} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. = \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)

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.	()	()	('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: 4 condition: 5
4	Operation	operator: 35 operands: 6
5	Operation	operator: 7 operands: 8
6	ExprTuple	9, 10
7	Literal
8	ExprTuple	41, 11
9	Operation	operator: 37 operands: 12
10	Operation	operator: 37 operands: 13
11	Operation	operator: 14 operands: 15
12	ExprTuple	17, 16
13	ExprTuple	17, 18
14	Literal
15	ExprTuple	19, 20
16	Operation	operator: 30 operand: 25
17	Literal
18	Operation	operator: 35 operands: 22
19	Literal
20	Operation	operator: 23 operands: 24
21	ExprTuple	25
22	ExprTuple	43, 39, 40, 26, 41
23	Literal
24	ExprTuple	33, 27
25	Operation	operator: 28 operands: 29
26	Literal
27	Operation	operator: 30 operand: 34
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34
32	Operation	operator: 35 operands: 36
33	Operation	operator: 37 operands: 38
34	Literal
35	Literal
36	ExprTuple	43, 39, 40, 41, 42
37	Literal
38	ExprTuple	43, 44
39	Literal
40	Literal
41	Variable
42	Variable
43	Literal
44	Literal