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, Mult, Real, e, i, one, pi, subtract, three, 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))]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, three)]))

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)\right)| = |\left(1, 2, \ldots, 3\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
5	Literal
6	ExprTuple	10
7	Operation	operator: 12 operands: 11
8	Operation	operator: 12 operands: 13
9	Operation	operator: 14 operands: 15
10	ExprRange	lambda_map: 16 start_index: 51 end_index: 17
11	ExprTuple	55, 18
12	Literal
13	ExprTuple	19, 20
14	Literal
15	ExprTuple	21, 22
16	Lambda	parameter: 31 body: 31
17	Literal
18	Literal
19	Operation	operator: 48 operands: 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 27 operands: 28
22	Operation	operator: 29 operands: 30
23	ExprTuple	31
24	ExprTuple	40, 55
25	Literal
26	ExprTuple	32, 33
27	Literal
28	ExprTuple	34, 36
29	Literal
30	ExprTuple	35, 36
31	Variable
32	Literal
33	Operation	operator: 37 operands: 38
34	Variable
35	Operation	operator: 44 operands: 39
36	Variable
37	Literal
38	ExprTuple	40, 41
39	ExprTuple	42, 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operand: 51
42	Literal
43	Operation	operator: 48 operands: 49
44	Literal
45	ExprTuple	52, 50
46	Literal
47	ExprTuple	51
48	Literal
49	ExprTuple	52, 53, 54, 55
50	Variable
51	Literal
52	Literal
53	Literal
54	Literal
55	Variable