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 VecSum
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, Sum, 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 = [k]
sub_expr2 = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Qmult(NumBra(m, _t), InverseFourierTransform(_t), NumKet(k, _t)))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Complex), domain = _m_domain), Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _m_domain), Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _m_domain))).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[\forall_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\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) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left(\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) = \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)\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 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 56 body: 12
10	Operation	operator: 13 operand: 17
11	Operation	operator: 14 operand: 17
12	Conditional	value: 16 condition: 23
13	Literal
14	Literal
15	ExprTuple	17
16	Operation	operator: 25 operands: 18
17	Lambda	parameter: 56 body: 20
18	ExprTuple	22, 21
19	ExprTuple	56
20	Conditional	value: 22 condition: 23
21	Literal
22	Operation	operator: 42 operands: 24
23	Operation	operator: 25 operands: 26
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	56, 29
27	Operation	operator: 59 operands: 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	37, 38, 39
33	Literal
34	ExprTuple	40, 41
35	Literal
36	Operation	operator: 42 operands: 43
37	Operation	operator: 44 operands: 45
38	Operation	operator: 46 operand: 64
39	Operation	operator: 48 operands: 49
40	Literal
41	Operation	operator: 50 operands: 51
42	Literal
43	ExprTuple	63, 52, 53, 54, 56
44	Literal
45	ExprTuple	55, 64
46	Literal
47	ExprTuple	64
48	Literal
49	ExprTuple	56, 64
50	Literal
51	ExprTuple	57, 58
52	Literal
53	Literal
54	Literal
55	Variable
56	Variable
57	Operation	operator: 59 operands: 60
58	Operation	operator: 61 operand: 65
59	Literal
60	ExprTuple	63, 64
61	Literal
62	ExprTuple	65
63	Literal
64	Literal
65	Literal