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 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
sub_expr3 = InSet(k, _m_domain)
expr = Equals(Conditional(Mult(sub_expr1, sub_expr2), sub_expr3), Conditional(Mult(sub_expr2, sub_expr1), sub_expr3)).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}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \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} \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: 7
4	Conditional	value: 6 condition: 7
5	Operation	operator: 38 operands: 8
6	Operation	operator: 38 operands: 9
7	Operation	operator: 10 operands: 11
8	ExprTuple	13, 12
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	44, 14
12	Operation	operator: 40 operands: 15
13	Operation	operator: 40 operands: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	20, 19
16	ExprTuple	20, 21
17	Literal
18	ExprTuple	22, 23
19	Operation	operator: 33 operand: 28
20	Literal
21	Operation	operator: 38 operands: 25
22	Literal
23	Operation	operator: 26 operands: 27
24	ExprTuple	28
25	ExprTuple	46, 42, 43, 29, 44
26	Literal
27	ExprTuple	36, 30
28	Operation	operator: 31 operands: 32
29	Literal
30	Operation	operator: 33 operand: 37
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37
35	Operation	operator: 38 operands: 39
36	Operation	operator: 40 operands: 41
37	Literal
38	Literal
39	ExprTuple	46, 42, 43, 44, 45
40	Literal
41	ExprTuple	46, 47
42	Literal
43	Literal
44	Variable
45	Variable
46	Literal
47	Literal