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 k, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = frac(one, Exp(two, frac(_t, two)))
sub_expr2 = Qmult(NumBra(m, _t), InverseFourierTransform(_t), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain))
expr = Equals(ScalarMult(sub_expr1, sub_expr2), Mult(sub_expr1, sub_expr2))

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(\frac{1}{2^{\frac{t}{2}}} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right) = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\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	Operation	operator: 31 operands: 5
4	Operation	operator: 47 operands: 5
5	ExprTuple	6, 7
6	Operation	operator: 25 operands: 8
7	Operation	operator: 9 operands: 10
8	ExprTuple	63, 11
9	Literal
10	ExprTuple	12, 13, 14
11	Operation	operator: 57 operands: 15
12	Operation	operator: 16 operands: 17
13	Operation	operator: 18 operand: 62
14	Operation	operator: 20 operand: 24
15	ExprTuple	61, 22
16	Literal
17	ExprTuple	23, 62
18	Literal
19	ExprTuple	62
20	Literal
21	ExprTuple	24
22	Operation	operator: 25 operands: 26
23	Variable
24	Lambda	parameter: 54 body: 28
25	Literal
26	ExprTuple	62, 61
27	ExprTuple	54
28	Conditional	value: 29 condition: 30
29	Operation	operator: 31 operands: 32
30	Operation	operator: 33 operands: 34
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	54, 37
35	Operation	operator: 57 operands: 38
36	Operation	operator: 39 operands: 40
37	Operation	operator: 41 operands: 42
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	54, 62
41	Literal
42	ExprTuple	45, 46
43	Literal
44	Operation	operator: 47 operands: 48
45	Literal
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	61, 51, 52, 53, 54
49	Literal
50	ExprTuple	55, 56
51	Literal
52	Literal
53	Literal
54	Variable
55	Operation	operator: 57 operands: 58
56	Operation	operator: 59 operand: 63
57	Literal
58	ExprTuple	61, 62
59	Literal
60	ExprTuple	63
61	Literal
62	Literal
63	Literal