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.logic import Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, Neg, Sum, e, frac, i, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = 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_expr1, instance_expr = InSet(sub_expr2, Complex), domain = _m_domain), InSet(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _m_domain), Complex))

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())

\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(\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) \in \mathbb{C}\right)

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.	()	()	('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: 22 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 18
8	Lambda	parameter: 56 body: 10
9	Operation	operator: 11 operand: 14
10	Conditional	value: 13 condition: 20
11	Literal
12	ExprTuple	14
13	Operation	operator: 22 operands: 15
14	Lambda	parameter: 56 body: 17
15	ExprTuple	19, 18
16	ExprTuple	56
17	Conditional	value: 19 condition: 20
18	Literal
19	Operation	operator: 50 operands: 21
20	Operation	operator: 22 operands: 23
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	56, 26
24	Operation	operator: 52 operands: 27
25	Operation	operator: 52 operands: 28
26	Operation	operator: 29 operands: 30
27	ExprTuple	32, 31
28	ExprTuple	32, 33
29	Literal
30	ExprTuple	34, 35
31	Operation	operator: 50 operands: 36
32	Literal
33	Operation	operator: 45 operand: 41
34	Literal
35	Operation	operator: 38 operands: 39
36	ExprTuple	58, 54, 55, 40, 56
37	ExprTuple	41
38	Literal
39	ExprTuple	48, 42
40	Literal
41	Operation	operator: 43 operands: 44
42	Operation	operator: 45 operand: 49
43	Literal
44	ExprTuple	47, 48
45	Literal
46	ExprTuple	49
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operands: 53
49	Literal
50	Literal
51	ExprTuple	58, 54, 55, 56, 57
52	Literal
53	ExprTuple	58, 59
54	Literal
55	Literal
56	Variable
57	Variable
58	Literal
59	Literal