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, TensorProd, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, subtract, two, zero
from proveit.physics.quantum import 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}")
expr = Equals(Len(operands = [frac(one, Exp(two, subtract(frac(Add(t, one), two), 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())]), 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(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}}, \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)| = |\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	Operation	operator: 62 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprRange	lambda_map: 14 start_index: 70 end_index: 79
11	ExprTuple	70, 15
12	Literal
13	ExprTuple	16, 17
14	Lambda	parameter: 83 body: 83
15	Operation	operator: 77 operands: 18
16	Operation	operator: 40 operands: 19
17	ExprRange	lambda_map: 20 start_index: 21 end_index: 58
18	ExprTuple	79, 22
19	ExprTuple	46, 23
20	Lambda	parameter: 83 body: 24
21	Operation	operator: 48 operands: 25
22	Operation	operator: 48 operands: 26
23	Operation	operator: 54 operands: 27
24	Operation	operator: 54 operands: 28
25	ExprTuple	29, 70
26	ExprTuple	30, 31
27	ExprTuple	32, 60
28	ExprTuple	33, 34
29	Operation	operator: 81 operand: 67
30	Operation	operator: 62 operands: 36
31	Operation	operator: 81 operand: 43
32	Operation	operator: 77 operands: 38
33	Operation	operator: 62 operands: 39
34	Operation	operator: 40 operands: 41
35	ExprTuple	67
36	ExprTuple	42, 79
37	ExprTuple	43
38	ExprTuple	68, 44
39	ExprTuple	70, 45
40	Literal
41	ExprTuple	46, 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 62 operands: 50
44	Operation	operator: 71 operands: 51
45	Operation	operator: 77 operands: 52
46	Operation	operator: 65 operand: 58
47	Operation	operator: 54 operands: 55
48	Literal
49	ExprTuple	67, 70
50	ExprTuple	67, 79
51	ExprTuple	79, 73, 74, 76, 56
52	ExprTuple	79, 57
53	ExprTuple	58
54	Literal
55	ExprTuple	59, 60
56	Operation	operator: 77 operands: 61
57	Operation	operator: 62 operands: 63
58	Literal
59	Operation	operator: 77 operands: 64
60	Operation	operator: 65 operand: 70
61	ExprTuple	79, 67
62	Literal
63	ExprTuple	70, 79
64	ExprTuple	68, 69
65	Literal
66	ExprTuple	70
67	Variable
68	Literal
69	Operation	operator: 71 operands: 72
70	Literal
71	Literal
72	ExprTuple	79, 73, 74, 75, 76
73	Literal
74	Literal
75	Operation	operator: 77 operands: 78
76	Literal
77	Literal
78	ExprTuple	79, 80
79	Literal
80	Operation	operator: 81 operand: 83
81	Literal
82	ExprTuple	83
83	Variable