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 Conditional, Lambda, k, t
from proveit.linear_algebra import ScalarMult, TensorProd
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(k, two_pow_t)
sub_expr2 = InSet(k, Interval(zero, subtract(two_pow_t, one)))
expr = Equals(Lambda(k, Conditional(ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), NumKet(sub_expr1, Add(t, one))), sub_expr2)), Lambda(k, Conditional(ScalarMult(Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Mult(two, pi, i, _phase, two_pow_t))), TensorProd(ket1, NumKet(k, t))), sub_expr2))).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[k \mapsto \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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	Lambda	parameter: 56 body: 5
4	Lambda	parameter: 56 body: 7
5	Conditional	value: 8 condition: 10
6	ExprTuple	56
7	Conditional	value: 9 condition: 10
8	Operation	operator: 12 operands: 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16, 17
12	Literal
13	ExprTuple	18, 19
14	Literal
15	ExprTuple	56, 20
16	Operation	operator: 62 operands: 21
17	Operation	operator: 41 operands: 22
18	Operation	operator: 52 operands: 23
19	Operation	operator: 24 operands: 25
20	Operation	operator: 26 operands: 27
21	ExprTuple	46, 28
22	ExprTuple	44, 29
23	ExprTuple	30, 31
24	Literal
25	ExprTuple	32, 33
26	Literal
27	ExprTuple	34, 35
28	Operation	operator: 52 operands: 36
29	Operation	operator: 49 operands: 37
30	Operation	operator: 62 operands: 38
31	Operation	operator: 62 operands: 39
32	Operation	operator: 40 operand: 61
33	Operation	operator: 41 operands: 42
34	Literal
35	Operation	operator: 49 operands: 43
36	ExprTuple	64, 57, 58, 59, 44
37	ExprTuple	65, 61
38	ExprTuple	46, 45
39	ExprTuple	46, 47
40	Literal
41	Literal
42	ExprTuple	56, 65
43	ExprTuple	60, 48
44	Operation	operator: 49 operands: 50
45	Operation	operator: 52 operands: 51
46	Literal
47	Operation	operator: 52 operands: 53
48	Operation	operator: 54 operand: 61
49	Literal
50	ExprTuple	56, 60
51	ExprTuple	64, 57, 58, 59, 56
52	Literal
53	ExprTuple	64, 57, 58, 59, 60
54	Literal
55	ExprTuple	61
56	Variable
57	Literal
58	Literal
59	Literal
60	Operation	operator: 62 operands: 63
61	Literal
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Variable