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, 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 _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([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)], [ExprRange(sub_expr1, sub_expr5, sub_expr3, zero)])

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^{-\left(-t + 2\right)} \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^{-\left(-t + 3\right)} \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)\right) = \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), \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 2\right)} \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)\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	ExprTuple	5, 6
4	ExprTuple	7
5	Operation	operator: 30 operands: 8
6	ExprRange	lambda_map: 10 start_index: 9 end_index: 34
7	ExprRange	lambda_map: 10 start_index: 57 end_index: 34
8	ExprTuple	17, 11
9	Operation	operator: 60 operands: 12
10	Lambda	parameter: 65 body: 13
11	Operation	operator: 21 operands: 14
12	ExprTuple	63, 58
13	Operation	operator: 30 operands: 15
14	ExprTuple	25, 16
15	ExprTuple	17, 18
16	Operation	operator: 30 operands: 19
17	Operation	operator: 38 operands: 20
18	Operation	operator: 21 operands: 22
19	ExprTuple	23, 36
20	ExprTuple	64, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 55 operands: 27
24	Operation	operator: 55 operands: 28
25	Operation	operator: 41 operand: 34
26	Operation	operator: 30 operands: 31
27	ExprTuple	44, 32
28	ExprTuple	58, 33
29	ExprTuple	34
30	Literal
31	ExprTuple	35, 36
32	Operation	operator: 47 operands: 37
33	Operation	operator: 38 operands: 39
34	Literal
35	Operation	operator: 55 operands: 40
36	Operation	operator: 41 operand: 64
37	ExprTuple	58, 50, 51, 43, 53
38	Literal
39	ExprTuple	64, 58
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	64
43	Operation	operator: 55 operands: 46
44	Literal
45	Operation	operator: 47 operands: 48
46	ExprTuple	58, 49
47	Literal
48	ExprTuple	58, 50, 51, 52, 53
49	Operation	operator: 66 operand: 57
50	Literal
51	Literal
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	57
55	Literal
56	ExprTuple	58, 59
57	Operation	operator: 60 operands: 61
58	Literal
59	Operation	operator: 66 operand: 65
60	Literal
61	ExprTuple	63, 64
62	ExprTuple	65
63	Operation	operator: 66 operand: 68
64	Literal
65	Variable
66	Literal
67	ExprTuple	68
68	Variable