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.logic import Implies, NotEquals
from proveit.numbers import Abs, Exp, Mod, four, frac, greater, pi, two, zero
from proveit.physics.quantum.QPE import SubIndexed, _alpha, _b_round, _delta_b_round, _two_pow_t

# build up the expression from sub-expressions
expr = Implies(NotEquals(_delta_b_round, zero), greater(Exp(Abs(SubIndexed(_alpha, [Mod(_b_round, _two_pow_t)])), two), frac(four, Exp(pi, two))))

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(\delta_{b_{\textit{r}}} \neq 0\right) \Rightarrow \left(\left|\alpha_{b_{\textit{r}} ~\textup{mod}~ 2^{t}}\right|^{2} > \frac{4}{\pi^{2}}\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 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9, 10
7	Literal
8	ExprTuple	11, 12
9	Operation	operator: 13 operand: 31
10	Literal
11	Operation	operator: 15 operands: 16
12	Operation	operator: 33 operands: 17
13	Literal
14	ExprTuple	31
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	20, 35
18	Literal
19	Operation	operator: 33 operands: 21
20	Operation	operator: 22 operand: 25
21	ExprTuple	24, 35
22	Literal
23	ExprTuple	25
24	Literal
25	Operation	operator: 26 operand: 28
26	Literal
27	ExprTuple	28
28	Operation	operator: 29 operands: 30
29	Literal
30	ExprTuple	31, 32
31	Literal
32	Operation	operator: 33 operands: 34
33	Literal
34	ExprTuple	35, 36
35	Literal
36	Literal