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.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, 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 = Add(sub_expr2, one)
sub_expr4 = frac(one, sqrt(two))
sub_expr5 = ScalarMult(sub_expr4, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))
expr = Equals(TensorProd(ScalarMult(sub_expr4, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr3)), _phase)), ket1))), ExprRange(sub_expr1, sub_expr5, Add(sub_expr2, two), zero).with_decreasing_order(), _ket_u), TensorProd(ExprRange(sub_expr1, sub_expr5, sub_expr3, 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(\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)\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}  \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) {\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) {\otimes} \lvert u \rangle\right) = \left(\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) {\otimes} \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',)
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, 11
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 34 operands: 12
9	ExprRange	lambda_map: 14 start_index: 13 end_index: 38
10	ExprRange	lambda_map: 14 start_index: 61 end_index: 38
11	Literal
12	ExprTuple	21, 15
13	Operation	operator: 64 operands: 16
14	Lambda	parameter: 69 body: 17
15	Operation	operator: 25 operands: 18
16	ExprTuple	67, 62
17	Operation	operator: 34 operands: 19
18	ExprTuple	29, 20
19	ExprTuple	21, 22
20	Operation	operator: 34 operands: 23
21	Operation	operator: 42 operands: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	27, 40
24	ExprTuple	68, 28
25	Literal
26	ExprTuple	29, 30
27	Operation	operator: 59 operands: 31
28	Operation	operator: 59 operands: 32
29	Operation	operator: 45 operand: 38
30	Operation	operator: 34 operands: 35
31	ExprTuple	48, 36
32	ExprTuple	62, 37
33	ExprTuple	38
34	Literal
35	ExprTuple	39, 40
36	Operation	operator: 51 operands: 41
37	Operation	operator: 42 operands: 43
38	Literal
39	Operation	operator: 59 operands: 44
40	Operation	operator: 45 operand: 68
41	ExprTuple	62, 54, 55, 47, 57
42	Literal
43	ExprTuple	68, 62
44	ExprTuple	48, 49
45	Literal
46	ExprTuple	68
47	Operation	operator: 59 operands: 50
48	Literal
49	Operation	operator: 51 operands: 52
50	ExprTuple	62, 53
51	Literal
52	ExprTuple	62, 54, 55, 56, 57
53	Operation	operator: 70 operand: 61
54	Literal
55	Literal
56	Operation	operator: 59 operands: 60
57	Literal
58	ExprTuple	61
59	Literal
60	ExprTuple	62, 63
61	Operation	operator: 64 operands: 65
62	Literal
63	Operation	operator: 70 operand: 69
64	Literal
65	ExprTuple	67, 68
66	ExprTuple	69
67	Operation	operator: 70 operand: 72
68	Literal
69	Variable
70	Literal
71	ExprTuple	72
72	Variable