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, Lambda, Variable, t
from proveit.core_expr_types import Len
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one, two
from proveit.physics.quantum.QPE import _ket_u, _psi_t_ket, _s
from proveit.physics.quantum.circuits import MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = TensorProd(_psi_t_ket, _ket_u)
sub_expr3 = Interval(one, Add(t, _s))
expr = Equals(Len(operands = [Lambda(Variable("_b", latex_format = r"{_{-}b}"), MultiQubitElem(element = Output(state = sub_expr2, part = one), targets = sub_expr3)), Lambda(sub_expr1, MultiQubitElem(element = Output(state = sub_expr2, part = Add(sub_expr1, t)), targets = sub_expr3))]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, two)]))

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({_{-}b} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~{_{-}a} + t~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} 
} \end{array}\right)| = |\left(1, \ldots, 2\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	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10
8	Lambda	parameter: 16 body: 12
9	Lambda	parameter: 40 body: 13
10	ExprRange	lambda_map: 14 start_index: 31 end_index: 15
11	ExprTuple	16
12	Operation	operator: 18 operands: 17
13	Operation	operator: 18 operands: 19
14	Lambda	parameter: 40 body: 40
15	Literal
16	Variable
17	NamedExprs	element: 21 targets: 23
18	Literal
19	NamedExprs	element: 22 targets: 23
20	ExprTuple	40
21	Operation	operator: 25 operands: 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	NamedExprs	state: 29 part: 31
25	Literal
26	NamedExprs	state: 29 part: 30
27	Literal
28	ExprTuple	31, 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 36 operands: 35
31	Literal
32	Operation	operator: 36 operands: 37
33	Literal
34	ExprTuple	38, 39
35	ExprTuple	40, 44
36	Literal
37	ExprTuple	44, 41
38	Operation	operator: 42 operand: 44
39	Literal
40	Variable
41	Literal
42	Literal
43	ExprTuple	44
44	Variable