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 _ket_u, _phase

# 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 = [ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(sub_expr2, one))), _phase)), ket1))), ExprRange(sub_expr1, 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(), _ket_u]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, Add(t, one))]))

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}{\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),\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), \left(\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)\right), \ldots, \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), \lvert u \rangle\right)| = |\left(1, 2, \ldots, \left(t + 1\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9, 10
6	Literal
7	ExprTuple	11
8	Operation	operator: 37 operands: 12
9	ExprRange	lambda_map: 13 start_index: 14 end_index: 41
10	Literal
11	ExprRange	lambda_map: 15 start_index: 71 end_index: 16
12	ExprTuple	24, 17
13	Lambda	parameter: 72 body: 18
14	Operation	operator: 67 operands: 19
15	Lambda	parameter: 72 body: 72
16	Operation	operator: 67 operands: 20
17	Operation	operator: 28 operands: 21
18	Operation	operator: 37 operands: 22
19	ExprTuple	70, 65
20	ExprTuple	75, 71
21	ExprTuple	32, 23
22	ExprTuple	24, 25
23	Operation	operator: 37 operands: 26
24	Operation	operator: 45 operands: 27
25	Operation	operator: 28 operands: 29
26	ExprTuple	30, 43
27	ExprTuple	71, 31
28	Literal
29	ExprTuple	32, 33
30	Operation	operator: 62 operands: 34
31	Operation	operator: 62 operands: 35
32	Operation	operator: 48 operand: 41
33	Operation	operator: 37 operands: 38
34	ExprTuple	51, 39
35	ExprTuple	65, 40
36	ExprTuple	41
37	Literal
38	ExprTuple	42, 43
39	Operation	operator: 54 operands: 44
40	Operation	operator: 45 operands: 46
41	Literal
42	Operation	operator: 62 operands: 47
43	Operation	operator: 48 operand: 71
44	ExprTuple	65, 57, 58, 50, 60
45	Literal
46	ExprTuple	71, 65
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	71
50	Operation	operator: 62 operands: 53
51	Literal
52	Operation	operator: 54 operands: 55
53	ExprTuple	65, 56
54	Literal
55	ExprTuple	65, 57, 58, 59, 60
56	Operation	operator: 73 operand: 64
57	Literal
58	Literal
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	64
62	Literal
63	ExprTuple	65, 66
64	Operation	operator: 67 operands: 68
65	Literal
66	Operation	operator: 73 operand: 72
67	Literal
68	ExprTuple	70, 71
69	ExprTuple	72
70	Operation	operator: 73 operand: 75
71	Literal
72	Variable
73	Literal
74	ExprTuple	75
75	Variable