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, two_pow_t

# 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))
sub_expr4 = ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))
expr = Equals([ExprRange(sub_expr1, sub_expr4, sub_expr2, zero)], [ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, two_pow_t, _phase)), ket1))), ExprRange(sub_expr1, sub_expr4, Add(sub_expr2, one), zero)]).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t\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 + 1\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(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t} \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 + 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) \end{array} \end{array}

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.	()	(2)	('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
4	ExprTuple	6, 7
5	ExprRange	lambda_map: 9 start_index: 17 end_index: 37
6	Operation	operator: 33 operands: 8
7	ExprRange	lambda_map: 9 start_index: 10 end_index: 37
8	ExprTuple	19, 11
9	Lambda	parameter: 64 body: 12
10	Operation	operator: 13 operands: 14
11	Operation	operator: 24 operands: 15
12	Operation	operator: 33 operands: 16
13	Literal
14	ExprTuple	17, 49
15	ExprTuple	28, 18
16	ExprTuple	19, 20
17	Operation	operator: 62 operand: 53
18	Operation	operator: 33 operands: 22
19	Operation	operator: 41 operands: 23
20	Operation	operator: 24 operands: 25
21	ExprTuple	53
22	ExprTuple	26, 39
23	ExprTuple	49, 27
24	Literal
25	ExprTuple	28, 29
26	Operation	operator: 58 operands: 30
27	Operation	operator: 58 operands: 31
28	Operation	operator: 44 operand: 37
29	Operation	operator: 33 operands: 34
30	ExprTuple	47, 35
31	ExprTuple	60, 36
32	ExprTuple	37
33	Literal
34	ExprTuple	38, 39
35	Operation	operator: 51 operands: 40
36	Operation	operator: 41 operands: 42
37	Literal
38	Operation	operator: 58 operands: 43
39	Operation	operator: 44 operand: 49
40	ExprTuple	60, 54, 55, 46, 57
41	Literal
42	ExprTuple	49, 60
43	ExprTuple	47, 48
44	Literal
45	ExprTuple	49
46	Operation	operator: 58 operands: 50
47	Literal
48	Operation	operator: 51 operands: 52
49	Literal
50	ExprTuple	60, 53
51	Literal
52	ExprTuple	60, 54, 55, 56, 57
53	Variable
54	Literal
55	Literal
56	Operation	operator: 58 operands: 59
57	Literal
58	Literal
59	ExprTuple	60, 61
60	Literal
61	Operation	operator: 62 operand: 64
62	Literal
63	ExprTuple	64
64	Variable