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, k, t
from proveit.linear_algebra import ScalarMult
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
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 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), NumKet(sub_expr1, Add(t, one)))
expr = Equals(Conditional(sub_expr2, InSet(k, Interval(zero, subtract(two_pow_t, one)))), sub_expr2)

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\{\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.. = \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}\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	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operands: 7
5	Operation	operator: 8 operands: 9
6	Literal
7	ExprTuple	10, 11
8	Literal
9	ExprTuple	36, 12
10	Operation	operator: 39 operands: 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	18, 19
14	Literal
15	ExprTuple	30, 20
16	Literal
17	ExprTuple	21, 22
18	Literal
19	Operation	operator: 23 operands: 24
20	Operation	operator: 32 operands: 25
21	Literal
22	Operation	operator: 32 operands: 26
23	Literal
24	ExprTuple	41, 27, 28, 29, 30
25	ExprTuple	42, 38
26	ExprTuple	37, 31
27	Literal
28	Literal
29	Literal
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operand: 38
32	Literal
33	ExprTuple	36, 37
34	Literal
35	ExprTuple	38
36	Variable
37	Operation	operator: 39 operands: 40
38	Literal
39	Literal
40	ExprTuple	41, 42
41	Literal
42	Variable