Expression of type Len

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.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 = 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])

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)|

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',)

# 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, 5
3	Operation	operator: 28 operands: 6
4	ExprRange	lambda_map: 7 start_index: 8 end_index: 32
5	Literal
6	ExprTuple	15, 9
7	Lambda	parameter: 63 body: 10
8	Operation	operator: 58 operands: 11
9	Operation	operator: 19 operands: 12
10	Operation	operator: 28 operands: 13
11	ExprTuple	61, 56
12	ExprTuple	23, 14
13	ExprTuple	15, 16
14	Operation	operator: 28 operands: 17
15	Operation	operator: 36 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	21, 34
18	ExprTuple	62, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 53 operands: 25
22	Operation	operator: 53 operands: 26
23	Operation	operator: 39 operand: 32
24	Operation	operator: 28 operands: 29
25	ExprTuple	42, 30
26	ExprTuple	56, 31
27	ExprTuple	32
28	Literal
29	ExprTuple	33, 34
30	Operation	operator: 45 operands: 35
31	Operation	operator: 36 operands: 37
32	Literal
33	Operation	operator: 53 operands: 38
34	Operation	operator: 39 operand: 62
35	ExprTuple	56, 48, 49, 41, 51
36	Literal
37	ExprTuple	62, 56
38	ExprTuple	42, 43
39	Literal
40	ExprTuple	62
41	Operation	operator: 53 operands: 44
42	Literal
43	Operation	operator: 45 operands: 46
44	ExprTuple	56, 47
45	Literal
46	ExprTuple	56, 48, 49, 50, 51
47	Operation	operator: 64 operand: 55
48	Literal
49	Literal
50	Operation	operator: 53 operands: 54
51	Literal
52	ExprTuple	55
53	Literal
54	ExprTuple	56, 57
55	Operation	operator: 58 operands: 59
56	Literal
57	Operation	operator: 64 operand: 63
58	Literal
59	ExprTuple	61, 62
60	ExprTuple	63
61	Operation	operator: 64 operand: 66
62	Literal
63	Variable
64	Literal
65	ExprTuple	66
66	Variable