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, U, Variable
from proveit.core_expr_types import Len
from proveit.linear_algebra import MatrixMult, ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Interval, IntervalCO, Mult, Real, e, four, i, one, pi, subtract, two, zero
from proveit.physics.quantum import var_ket_u
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}")
expr = Equals(Len(operands = [InSet(phase, Real), InSet(Mult(two_pow_t, phase), Interval(zero, subtract(two_pow_t, one))), Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u)), InSet(phase, IntervalCO(zero, one))]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, four)]))

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