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 Conditional, Lambda, k, m
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, Forall, Implies, 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 = Mult(sub_expr4, Qmult(sub_expr1, sub_expr2, sub_expr3))
sub_expr6 = Qmult(sub_expr1, sub_expr2, ScalarMult(sub_expr4, sub_expr3))
sub_expr7 = InSet(k, _m_domain)
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr6, sub_expr5), domain = _m_domain), Equals(Lambda(k, Conditional(sub_expr6, sub_expr7)), Lambda(k, Conditional(sub_expr5, sub_expr7))).with_wrapping_at(2)).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({_{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)\right) = \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] \Rightarrow  \\ \left(\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}\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: 8
4	Operation	operator: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 66 body: 11
9	Lambda	parameter: 66 body: 12
10	Lambda	parameter: 66 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	66
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	66, 23
21	Operation	operator: 34 operands: 24
22	Operation	operator: 60 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	39, 40, 28
25	ExprTuple	38, 29
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Literal
31	Operation	operator: 36 operands: 37
32	Literal
33	ExprTuple	38, 41
34	Literal
35	ExprTuple	39, 40, 41
36	Literal
37	ExprTuple	42, 43
38	Operation	operator: 51 operands: 44
39	Operation	operator: 45 operands: 46
40	Operation	operator: 47 operand: 58
41	Operation	operator: 49 operands: 50
42	Operation	operator: 51 operands: 52
43	Operation	operator: 53 operand: 59
44	ExprTuple	55, 56
45	Literal
46	ExprTuple	57, 58
47	Literal
48	ExprTuple	58
49	Literal
50	ExprTuple	66, 58
51	Literal
52	ExprTuple	62, 58
53	Literal
54	ExprTuple	59
55	Literal
56	Operation	operator: 60 operands: 61
57	Variable
58	Literal
59	Literal
60	Literal
61	ExprTuple	62, 63, 64, 65, 66
62	Literal
63	Literal
64	Literal
65	Literal
66	Variable