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 e, l
from proveit.logic import And, Implies, InSet, Union
from proveit.numbers import Less, ModAbs, greater, zero
from proveit.physics.quantum.QPE import _neg_domain, _pos_domain, _two_pow_t

# build up the expression from sub-expressions
expr = Implies(Less(l, zero), And(InSet(l, Union(_neg_domain, _pos_domain)), greater(ModAbs(l, _two_pow_t), e)))

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(l < 0\right) \Rightarrow \left(\left(l \in \left(\{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\} \cup \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right)\right) \land \left(\left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right)\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: 13 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	23, 8
6	Literal
7	ExprTuple	9, 10
8	Literal
9	Operation	operator: 11 operands: 12
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	23, 15
13	Literal
14	ExprTuple	38, 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 26 operands: 25
22	Operation	operator: 26 operands: 27
23	Variable
24	Operation	operator: 39 operands: 28
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	34, 37
28	ExprTuple	41, 45
29	Operation	operator: 43 operands: 31
30	Operation	operator: 47 operand: 34
31	ExprTuple	33, 49
32	ExprTuple	34
33	Operation	operator: 47 operand: 37
34	Operation	operator: 43 operands: 36
35	ExprTuple	37
36	ExprTuple	38, 49
37	Operation	operator: 39 operands: 40
38	Variable
39	Literal
40	ExprTuple	41, 42
41	Literal
42	Operation	operator: 43 operands: 44
43	Literal
44	ExprTuple	45, 46
45	Literal
46	Operation	operator: 47 operand: 49
47	Literal
48	ExprTuple	49
49	Literal