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, 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, zero
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 = Neg(t)
sub_expr3 = frac(one, sqrt(two))
expr = Equals(Len(operands = [Output(state = ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(sub_expr2, one))), _phase)), ket1)))), ExprRange(sub_expr1, Output(state = ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), Add(sub_expr2, two), zero).with_decreasing_order()]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, t)]))

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(\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},\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^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}, \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^{t - 3} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}, \ldots, \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^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}\right)| = |\left(1, 2, \ldots, t\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	Operation	operator: 19 operands: 11
9	ExprRange	lambda_map: 12 start_index: 13 end_index: 43
10	ExprRange	lambda_map: 14 start_index: 73 end_index: 77
11	NamedExprs	state: 15
12	Lambda	parameter: 74 body: 16
13	Operation	operator: 69 operands: 17
14	Lambda	parameter: 74 body: 74
15	Operation	operator: 39 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	72, 67
18	ExprTuple	26, 21
19	Literal
20	NamedExprs	state: 22
21	Operation	operator: 30 operands: 23
22	Operation	operator: 39 operands: 24
23	ExprTuple	34, 25
24	ExprTuple	26, 27
25	Operation	operator: 39 operands: 28
26	Operation	operator: 47 operands: 29
27	Operation	operator: 30 operands: 31
28	ExprTuple	32, 45
29	ExprTuple	73, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 64 operands: 36
33	Operation	operator: 64 operands: 37
34	Operation	operator: 50 operand: 43
35	Operation	operator: 39 operands: 40
36	ExprTuple	53, 41
37	ExprTuple	67, 42
38	ExprTuple	43
39	Literal
40	ExprTuple	44, 45
41	Operation	operator: 56 operands: 46
42	Operation	operator: 47 operands: 48
43	Literal
44	Operation	operator: 64 operands: 49
45	Operation	operator: 50 operand: 73
46	ExprTuple	67, 59, 60, 52, 62
47	Literal
48	ExprTuple	73, 67
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	73
52	Operation	operator: 64 operands: 55
53	Literal
54	Operation	operator: 56 operands: 57
55	ExprTuple	67, 58
56	Literal
57	ExprTuple	67, 59, 60, 61, 62
58	Operation	operator: 75 operand: 66
59	Literal
60	Literal
61	Operation	operator: 64 operands: 65
62	Literal
63	ExprTuple	66
64	Literal
65	ExprTuple	67, 68
66	Operation	operator: 69 operands: 70
67	Literal
68	Operation	operator: 75 operand: 74
69	Literal
70	ExprTuple	72, 73
71	ExprTuple	74
72	Operation	operator: 75 operand: 77
73	Literal
74	Variable
75	Literal
76	ExprTuple	77
77	Variable