Expression of type Forall

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, Forall, Iff, InSet, Union
from proveit.numbers import ModAbs, greater
from proveit.physics.quantum.QPE import _e_domain, _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 = Forall(instance_param_or_params = [e], instance_expr = Iff(And(InSet(l, Union(_neg_domain, _pos_domain)), sub_expr1), And(InSet(l, _full_domain), sub_expr1)).with_wrapping_at(1), domain = _e_domain).with_wrapping()

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}{l}\forall_{e \in \{1~\ldotp \ldotp~2^{t - 1} - 2\}}~\\
\left(\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}\right)\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 55 body: 5
4	ExprTuple	55
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 23 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	55, 13
11	Operation	operator: 15 operands: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 43 operands: 17
14	ExprTuple	18, 20
15	Literal
16	ExprTuple	19, 20
17	ExprTuple	66, 21
18	Operation	operator: 23 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 60 operands: 27
22	ExprTuple	40, 28
23	Literal
24	ExprTuple	40, 29
25	Literal
26	ExprTuple	55, 30
27	ExprTuple	54, 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 43 operands: 34
30	Operation	operator: 35 operands: 36
31	Operation	operator: 64 operand: 58
32	Literal
33	ExprTuple	38, 39
34	ExprTuple	46, 54
35	Literal
36	ExprTuple	40, 41
37	ExprTuple	58
38	Operation	operator: 43 operands: 42
39	Operation	operator: 43 operands: 44
40	Variable
41	Operation	operator: 56 operands: 45
42	ExprTuple	46, 47
43	Literal
44	ExprTuple	51, 54
45	ExprTuple	58, 62
46	Operation	operator: 60 operands: 48
47	Operation	operator: 64 operand: 51
48	ExprTuple	50, 66
49	ExprTuple	51
50	Operation	operator: 64 operand: 54
51	Operation	operator: 60 operands: 53
52	ExprTuple	54
53	ExprTuple	55, 66
54	Operation	operator: 56 operands: 57
55	Variable
56	Literal
57	ExprTuple	58, 59
58	Literal
59	Operation	operator: 60 operands: 61
60	Literal
61	ExprTuple	62, 63
62	Literal
63	Operation	operator: 64 operand: 66
64	Literal
65	ExprTuple	66
66	Literal