Expression of type Iff

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, Iff, InSet, Union
from proveit.numbers import ModAbs, greater
from proveit.physics.quantum.QPE import _full_domain, _neg_domain, _pos_domain, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = greater(ModAbs(l, _two_pow_t), e)
expr = Iff(And(InSet(l, Union(_neg_domain, _pos_domain)), sub_expr1), And(InSet(l, _full_domain), sub_expr1)).with_wrapping_at(1)

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(\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) \\  \Leftrightarrow \left(\left(l \in \{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\}\right) \land \left(\left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right)\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.	()	(1)	('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 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 10
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 12 operands: 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	26, 16
12	Literal
13	ExprTuple	26, 17
14	Literal
15	ExprTuple	41, 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 29 operands: 21
18	Operation	operator: 22 operands: 23
19	Literal
20	ExprTuple	24, 25
21	ExprTuple	32, 40
22	Literal
23	ExprTuple	26, 27
24	Operation	operator: 29 operands: 28
25	Operation	operator: 29 operands: 30
26	Variable
27	Operation	operator: 42 operands: 31
28	ExprTuple	32, 33
29	Literal
30	ExprTuple	37, 40
31	ExprTuple	44, 48
32	Operation	operator: 46 operands: 34
33	Operation	operator: 50 operand: 37
34	ExprTuple	36, 52
35	ExprTuple	37
36	Operation	operator: 50 operand: 40
37	Operation	operator: 46 operands: 39
38	ExprTuple	40
39	ExprTuple	41, 52
40	Operation	operator: 42 operands: 43
41	Variable
42	Literal
43	ExprTuple	44, 45
44	Literal
45	Operation	operator: 46 operands: 47
46	Literal
47	ExprTuple	48, 49
48	Literal
49	Operation	operator: 50 operand: 52
50	Literal
51	ExprTuple	52
52	Literal