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 l
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Abs, Exp, Real, Sum, two
from proveit.physics.quantum.QPE import _pos_domain, _rel_indexed_alpha

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Real), domain = _pos_domain), InSet(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _pos_domain), Real))

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_{l \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}}~\left(\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \in \mathbb{R}\right)\right] \Rightarrow \left(\left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) \in \mathbb{R}\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: 47 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: 47 body: 17
15	ExprTuple	19, 18
16	ExprTuple	47
17	Conditional	value: 19 condition: 20
18	Literal
19	Operation	operator: 36 operands: 21
20	Operation	operator: 22 operands: 23
21	ExprTuple	24, 40
22	Literal
23	ExprTuple	47, 25
24	Operation	operator: 26 operand: 30
25	Operation	operator: 28 operands: 29
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	31, 32
30	Operation	operator: 33 operand: 38
31	Operation	operator: 44 operands: 35
32	Operation	operator: 36 operands: 37
33	Literal
34	ExprTuple	38
35	ExprTuple	39, 52
36	Literal
37	ExprTuple	40, 41
38	Operation	operator: 42 operands: 43
39	Variable
40	Literal
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	46, 47
44	Literal
45	ExprTuple	48, 49
46	Literal
47	Variable
48	Literal
49	Operation	operator: 50 operand: 52
50	Literal
51	ExprTuple	52
52	Literal