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, k, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, subtract, two, zero
from proveit.physics.quantum import NumKet, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
sub_expr3 = frac(sub_expr2, two)
expr = Equals(ScalarMult(frac(one, Exp(two, sub_expr3)), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, sub_expr2)), domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))), ScalarMult(frac(one, Exp(two, subtract(sub_expr3, frac(t, two)))), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), ExprRange(sub_expr1, ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Add(Neg(t), one), zero).with_decreasing_order())))

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 + 1}{2}}} \cdot \left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right)\right) = \left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right){\otimes} \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes}  \left(\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)\right) {\otimes}  \ldots {\otimes}  \left(\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)\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: 78 operands: 5
4	Operation	operator: 78 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 10
7	Operation	operator: 88 operands: 11
8	Operation	operator: 12 operand: 18
9	Operation	operator: 88 operands: 14
10	Operation	operator: 15 operands: 16
11	ExprTuple	96, 17
12	Literal
13	ExprTuple	18
14	ExprTuple	96, 19
15	Literal
16	ExprTuple	20, 21
17	Operation	operator: 105 operands: 22
18	Lambda	parameter: 80 body: 24
19	Operation	operator: 105 operands: 25
20	Operation	operator: 60 operands: 26
21	ExprRange	lambda_map: 27 start_index: 28 end_index: 84
22	ExprTuple	107, 45
23	ExprTuple	80
24	Conditional	value: 29 condition: 30
25	ExprTuple	107, 31
26	ExprTuple	68, 32
27	Lambda	parameter: 111 body: 33
28	Operation	operator: 72 operands: 34
29	Operation	operator: 78 operands: 35
30	Operation	operator: 36 operands: 37
31	Operation	operator: 72 operands: 38
32	Operation	operator: 78 operands: 39
33	Operation	operator: 78 operands: 40
34	ExprTuple	41, 96
35	ExprTuple	42, 43
36	Literal
37	ExprTuple	80, 44
38	ExprTuple	45, 46
39	ExprTuple	47, 86
40	ExprTuple	48, 49
41	Operation	operator: 109 operand: 100
42	Operation	operator: 105 operands: 51
43	Operation	operator: 52 operands: 53
44	Operation	operator: 54 operands: 55
45	Operation	operator: 88 operands: 56
46	Operation	operator: 109 operand: 65
47	Operation	operator: 105 operands: 58
48	Operation	operator: 88 operands: 59
49	Operation	operator: 60 operands: 61
50	ExprTuple	100
51	ExprTuple	94, 62
52	Literal
53	ExprTuple	80, 64
54	Literal
55	ExprTuple	84, 63
56	ExprTuple	64, 107
57	ExprTuple	65
58	ExprTuple	94, 66
59	ExprTuple	96, 67
60	Literal
61	ExprTuple	68, 69
62	Operation	operator: 98 operands: 70
63	Operation	operator: 72 operands: 71
64	Operation	operator: 72 operands: 73
65	Operation	operator: 88 operands: 74
66	Operation	operator: 98 operands: 75
67	Operation	operator: 105 operands: 76
68	Operation	operator: 91 operand: 84
69	Operation	operator: 78 operands: 79
70	ExprTuple	107, 101, 102, 104, 80
71	ExprTuple	81, 82
72	Literal
73	ExprTuple	100, 96
74	ExprTuple	100, 107
75	ExprTuple	107, 101, 102, 104, 93
76	ExprTuple	107, 83
77	ExprTuple	84
78	Literal
79	ExprTuple	85, 86
80	Variable
81	Operation	operator: 98 operands: 87
82	Operation	operator: 109 operand: 96
83	Operation	operator: 88 operands: 89
84	Literal
85	Operation	operator: 105 operands: 90
86	Operation	operator: 91 operand: 96
87	ExprTuple	107, 93
88	Literal
89	ExprTuple	96, 107
90	ExprTuple	94, 95
91	Literal
92	ExprTuple	96
93	Operation	operator: 105 operands: 97
94	Literal
95	Operation	operator: 98 operands: 99
96	Literal
97	ExprTuple	107, 100
98	Literal
99	ExprTuple	107, 101, 102, 103, 104
100	Variable
101	Literal
102	Literal
103	Operation	operator: 105 operands: 106
104	Literal
105	Literal
106	ExprTuple	107, 108
107	Literal
108	Operation	operator: 109 operand: 111
109	Literal
110	ExprTuple	111
111	Variable