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 ExprRange, Lambda, Variable, t
from proveit.core_expr_types import Len
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase
from proveit.physics.quantum.circuits import Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = frac(one, sqrt(two))
expr = Equals(Len(operands = [Lambda(Variable("_b", latex_format = r"{_{-}b}"), Output(state = ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1))))), Lambda(sub_expr1, Output(state = ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))))]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, two)]))

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({_{-}b} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}\right)| = |\left(1, \ldots, 2\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10
8	Lambda	parameter: 15 body: 12
9	Lambda	parameter: 70 body: 13
10	ExprRange	lambda_map: 14 start_index: 74 end_index: 65
11	ExprTuple	15
12	Operation	operator: 17 operands: 16
13	Operation	operator: 17 operands: 18
14	Lambda	parameter: 70 body: 70
15	Variable
16	NamedExprs	state: 19
17	Literal
18	NamedExprs	state: 20
19	Operation	operator: 37 operands: 21
20	Operation	operator: 37 operands: 22
21	ExprTuple	24, 23
22	ExprTuple	24, 25
23	Operation	operator: 28 operands: 26
24	Operation	operator: 45 operands: 27
25	Operation	operator: 28 operands: 29
26	ExprTuple	32, 30
27	ExprTuple	74, 31
28	Literal
29	ExprTuple	32, 33
30	Operation	operator: 37 operands: 34
31	Operation	operator: 62 operands: 35
32	Operation	operator: 48 operand: 41
33	Operation	operator: 37 operands: 38
34	ExprTuple	39, 43
35	ExprTuple	65, 40
36	ExprTuple	41
37	Literal
38	ExprTuple	42, 43
39	Operation	operator: 62 operands: 44
40	Operation	operator: 45 operands: 46
41	Literal
42	Operation	operator: 62 operands: 47
43	Operation	operator: 48 operand: 74
44	ExprTuple	51, 50
45	Literal
46	ExprTuple	74, 65
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	74
50	Operation	operator: 54 operands: 53
51	Literal
52	Operation	operator: 54 operands: 55
53	ExprTuple	65, 57, 58, 56, 60
54	Literal
55	ExprTuple	65, 57, 58, 59, 60
56	Operation	operator: 62 operands: 61
57	Literal
58	Literal
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	65, 64
62	Literal
63	ExprTuple	65, 66
64	Operation	operator: 75 operand: 69
65	Literal
66	Operation	operator: 75 operand: 70
67	ExprTuple	69
68	ExprTuple	70
69	Operation	operator: 71 operands: 72
70	Variable
71	Literal
72	ExprTuple	73, 74
73	Operation	operator: 75 operand: 77
74	Literal
75	Literal
76	ExprTuple	77
77	Variable