Expression of type Implies

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 k, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals, Implies, InClass
from proveit.numbers import Exp, Mult, Sum, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult, QmultCodomain
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 = [k]
sub_expr2 = NumBra(m, _t)
sub_expr3 = InverseFourierTransform(_t)
sub_expr4 = NumKet(k, _t)
sub_expr5 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr6 = Qmult(sub_expr2, sub_expr3, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr5, sub_expr4), domain = _m_domain))
expr = Implies(InClass(sub_expr6, QmultCodomain), Equals(sub_expr6, Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr5, Qmult(sub_expr2, sub_expr3, sub_expr4)), domain = _m_domain)).with_wrapping_at(1)).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(\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) \\  = \left(\sum_{k = 0}^{2^{t} - 1} \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)\right)\right) \end{array} \end{array}\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	Operation	operator: 5 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	10, 9
7	Literal
8	ExprTuple	10, 11
9	Literal
10	Operation	operator: 32 operands: 12
11	Operation	operator: 13 operand: 16
12	ExprTuple	36, 37, 15
13	Literal
14	ExprTuple	16
15	Operation	operator: 17 operand: 20
16	Lambda	parameter: 60 body: 19
17	Literal
18	ExprTuple	20
19	Conditional	value: 21 condition: 26
20	Lambda	parameter: 60 body: 23
21	Operation	operator: 53 operands: 24
22	ExprTuple	60
23	Conditional	value: 25 condition: 26
24	ExprTuple	34, 27
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operands: 31
27	Operation	operator: 32 operands: 33
28	Literal
29	ExprTuple	34, 38
30	Literal
31	ExprTuple	60, 35
32	Literal
33	ExprTuple	36, 37, 38
34	Operation	operator: 63 operands: 39
35	Operation	operator: 40 operands: 41
36	Operation	operator: 42 operands: 43
37	Operation	operator: 44 operand: 68
38	Operation	operator: 46 operands: 47
39	ExprTuple	48, 49
40	Literal
41	ExprTuple	50, 51
42	Literal
43	ExprTuple	52, 68
44	Literal
45	ExprTuple	68
46	Literal
47	ExprTuple	60, 68
48	Literal
49	Operation	operator: 53 operands: 54
50	Literal
51	Operation	operator: 55 operands: 56
52	Variable
53	Literal
54	ExprTuple	67, 57, 58, 59, 60
55	Literal
56	ExprTuple	61, 62
57	Literal
58	Literal
59	Literal
60	Variable
61	Operation	operator: 63 operands: 64
62	Operation	operator: 65 operand: 69
63	Literal
64	ExprTuple	67, 68
65	Literal
66	ExprTuple	69
67	Literal
68	Literal
69	Literal