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 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, Qmult(sub_expr2, ScalarMult(sub_expr3, sub_expr4))), Qmult(sub_expr1, ScalarMult(sub_expr3, Qmult(sub_expr2, sub_expr4))))

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({_{t}}\langle m \rvert \thinspace \left({\mathrm {FT}}^{\dag}_{t} \thinspace \left(\frac{1}{2^{\frac{t}{2}}} \cdot \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)\right) = \left({_{t}}\langle m \rvert \thinspace \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({\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)\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: 18 operands: 5
4	Operation	operator: 18 operands: 6
5	ExprTuple	8, 7
6	ExprTuple	8, 9
7	Operation	operator: 18 operands: 10
8	Operation	operator: 11 operands: 12
9	Operation	operator: 38 operands: 13
10	ExprTuple	21, 14
11	Literal
12	ExprTuple	15, 69
13	ExprTuple	20, 16
14	Operation	operator: 38 operands: 17
15	Variable
16	Operation	operator: 18 operands: 19
17	ExprTuple	20, 22
18	Literal
19	ExprTuple	21, 22
20	Operation	operator: 36 operands: 23
21	Operation	operator: 24 operand: 69
22	Operation	operator: 26 operand: 29
23	ExprTuple	70, 28
24	Literal
25	ExprTuple	69
26	Literal
27	ExprTuple	29
28	Operation	operator: 64 operands: 30
29	Lambda	parameter: 61 body: 32
30	ExprTuple	68, 33
31	ExprTuple	61
32	Conditional	value: 34 condition: 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	Literal
37	ExprTuple	69, 68
38	Literal
39	ExprTuple	42, 43
40	Literal
41	ExprTuple	61, 44
42	Operation	operator: 64 operands: 45
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 49
45	ExprTuple	50, 51
46	Literal
47	ExprTuple	61, 69
48	Literal
49	ExprTuple	52, 53
50	Literal
51	Operation	operator: 54 operands: 55
52	Literal
53	Operation	operator: 56 operands: 57
54	Literal
55	ExprTuple	68, 58, 59, 60, 61
56	Literal
57	ExprTuple	62, 63
58	Literal
59	Literal
60	Literal
61	Variable
62	Operation	operator: 64 operands: 65
63	Operation	operator: 66 operand: 70
64	Literal
65	ExprTuple	68, 69
66	Literal
67	ExprTuple	70
68	Literal
69	Literal
70	Literal