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, Lambda, k, m
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = NumKet(k, _t)
sub_expr4 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr5 = InSet(k, _m_domain)
expr = Equals(Lambda(k, Conditional(Qmult(sub_expr1, sub_expr2, ScalarMult(sub_expr4, sub_expr3)), sub_expr5)), Lambda(k, Conditional(Mult(sub_expr4, Qmult(sub_expr1, sub_expr2, sub_expr3)), sub_expr5))).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[k \mapsto \left\{{_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] \end{array} \end{array}

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.	()	(2)	('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	Lambda	parameter: 53 body: 5
4	Lambda	parameter: 53 body: 7
5	Conditional	value: 8 condition: 10
6	ExprTuple	53
7	Conditional	value: 9 condition: 10
8	Operation	operator: 20 operands: 11
9	Operation	operator: 44 operands: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	25, 26, 15
12	ExprTuple	24, 16
13	Literal
14	ExprTuple	53, 17
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 22 operands: 23
18	Literal
19	ExprTuple	24, 27
20	Literal
21	ExprTuple	25, 26, 27
22	Literal
23	ExprTuple	28, 29
24	Operation	operator: 46 operands: 30
25	Operation	operator: 31 operands: 32
26	Operation	operator: 33 operand: 55
27	Operation	operator: 35 operands: 36
28	Literal
29	Operation	operator: 37 operands: 38
30	ExprTuple	39, 40
31	Literal
32	ExprTuple	41, 55
33	Literal
34	ExprTuple	55
35	Literal
36	ExprTuple	53, 55
37	Literal
38	ExprTuple	42, 43
39	Literal
40	Operation	operator: 44 operands: 45
41	Variable
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operand: 56
44	Literal
45	ExprTuple	54, 50, 51, 52, 53
46	Literal
47	ExprTuple	54, 55
48	Literal
49	ExprTuple	56
50	Literal
51	Literal
52	Literal
53	Variable
54	Literal
55	Literal
56	Literal