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
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, two, zero
from proveit.physics.quantum import Ket
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
sub_expr1 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), Ket(k))
expr = Equals(Conditional(sub_expr1, InSet(k, Interval(Add(zero, one), one))), Conditional(sub_expr1, InSet(k, Interval(one, one))))

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 k} \cdot \lvert k \rangle \textrm{ if } k \in \{0 + 1~\ldotp \ldotp~1\}\right.. = \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle \textrm{ if } k \in \{1~\ldotp \ldotp~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: 6 condition: 5
4	Conditional	value: 6 condition: 7
5	Operation	operator: 11 operands: 8
6	Operation	operator: 9 operands: 10
7	Operation	operator: 11 operands: 12
8	ExprTuple	37, 13
9	Literal
10	ExprTuple	14, 15
11	Literal
12	ExprTuple	37, 16
13	Operation	operator: 22 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operand: 37
16	Operation	operator: 22 operands: 23
17	ExprTuple	24, 32
18	Literal
19	ExprTuple	25, 26
20	Literal
21	ExprTuple	37
22	Literal
23	ExprTuple	32, 32
24	Operation	operator: 27 operands: 28
25	Literal
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	31, 32
29	Literal
30	ExprTuple	33, 34, 35, 36, 37
31	Literal
32	Literal
33	Literal
34	Literal
35	Literal
36	Literal
37	Variable