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 ExprRange, IndexedVar, Variable, a, c, k, m
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, Neg, Sum, e, frac, i, one, pi, two, zero
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [k]
sub_expr3 = IndexedVar(a, one)
sub_expr4 = ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, zero)
sub_expr5 = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))
expr = Implies(Forall(instance_param_or_params = sub_expr2, instance_expr = InSet(sub_expr5, Complex), domain = _m_domain), Forall(instance_param_or_params = [sub_expr3, sub_expr4], instance_expr = Equals(Mult(sub_expr3, Sum(index_or_indices = sub_expr2, summand = sub_expr5, domain = _m_domain), sub_expr4), Sum(index_or_indices = sub_expr2, summand = Mult(sub_expr3, sub_expr5, sub_expr4), domain = _m_domain)).with_wrapping_at(2), domain = Complex).with_wrapping()).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 \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, c_{1}, c_{2}, \ldots, c_{0} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right) =  \\ \left(\sum_{k = 0}^{2^{t} - 1} \left(a_{1} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right)\right) \end{array} \end{array}\right)\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: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 87 body: 10
9	Lambda	parameters: 11 body: 12
10	Conditional	value: 13 condition: 42
11	ExprTuple	43, 45
12	Conditional	value: 14 condition: 15
13	Operation	operator: 46 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	44, 38
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 81 operands: 25
22	Operation	operator: 32 operand: 30
23	Operation	operator: 46 operands: 27
24	ExprRange	lambda_map: 28 start_index: 78 end_index: 61
25	ExprTuple	43, 29, 45
26	ExprTuple	30
27	ExprTuple	43, 38
28	Lambda	parameter: 66 body: 31
29	Operation	operator: 32 operand: 36
30	Lambda	parameter: 87 body: 34
31	Operation	operator: 46 operands: 35
32	Literal
33	ExprTuple	36
34	Conditional	value: 37 condition: 42
35	ExprTuple	54, 38
36	Lambda	parameter: 87 body: 40
37	Operation	operator: 81 operands: 41
38	Literal
39	ExprTuple	87
40	Conditional	value: 44 condition: 42
41	ExprTuple	43, 44, 45
42	Operation	operator: 46 operands: 47
43	IndexedVar	variable: 48 index: 78
44	Operation	operator: 81 operands: 49
45	ExprRange	lambda_map: 50 start_index: 78 end_index: 61
46	Literal
47	ExprTuple	87, 51
48	Variable
49	ExprTuple	52, 53
50	Lambda	parameter: 66 body: 54
51	Operation	operator: 55 operands: 56
52	Operation	operator: 83 operands: 57
53	Operation	operator: 83 operands: 58
54	IndexedVar	variable: 59 index: 66
55	Literal
56	ExprTuple	61, 62
57	ExprTuple	64, 63
58	ExprTuple	64, 65
59	Variable
60	ExprTuple	66
61	Literal
62	Operation	operator: 67 operands: 68
63	Operation	operator: 81 operands: 69
64	Literal
65	Operation	operator: 74 operand: 73
66	Variable
67	Literal
68	ExprTuple	80, 71
69	ExprTuple	89, 85, 86, 72, 87
70	ExprTuple	73
71	Operation	operator: 74 operand: 78
72	Literal
73	Operation	operator: 76 operands: 77
74	Literal
75	ExprTuple	78
76	Literal
77	ExprTuple	79, 80
78	Literal
79	Operation	operator: 81 operands: 82
80	Operation	operator: 83 operands: 84
81	Literal
82	ExprTuple	89, 85, 86, 87, 88
83	Literal
84	ExprTuple	89, 90
85	Literal
86	Literal
87	Variable
88	Variable
89	Literal
90	Literal