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 = [l], instance_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()).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_{l}~\\
\left[\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}\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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 44 body: 4
3	ExprTuple	44
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameter: 59 body: 9
8	ExprTuple	59
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 27 operands: 14
12	Literal
13	ExprTuple	15, 16
14	ExprTuple	59, 17
15	Operation	operator: 19 operands: 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 47 operands: 21
18	ExprTuple	22, 24
19	Literal
20	ExprTuple	23, 24
21	ExprTuple	70, 25
22	Operation	operator: 27 operands: 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	Operation	operator: 64 operands: 31
26	ExprTuple	44, 32
27	Literal
28	ExprTuple	44, 33
29	Literal
30	ExprTuple	59, 34
31	ExprTuple	58, 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 47 operands: 38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 68 operand: 62
36	Literal
37	ExprTuple	42, 43
38	ExprTuple	50, 58
39	Literal
40	ExprTuple	44, 45
41	ExprTuple	62
42	Operation	operator: 47 operands: 46
43	Operation	operator: 47 operands: 48
44	Variable
45	Operation	operator: 60 operands: 49
46	ExprTuple	50, 51
47	Literal
48	ExprTuple	55, 58
49	ExprTuple	62, 66
50	Operation	operator: 64 operands: 52
51	Operation	operator: 68 operand: 55
52	ExprTuple	54, 70
53	ExprTuple	55
54	Operation	operator: 68 operand: 58
55	Operation	operator: 64 operands: 57
56	ExprTuple	58
57	ExprTuple	59, 70
58	Operation	operator: 60 operands: 61
59	Variable
60	Literal
61	ExprTuple	62, 63
62	Literal
63	Operation	operator: 64 operands: 65
64	Literal
65	ExprTuple	66, 67
66	Literal
67	Operation	operator: 68 operand: 70
68	Literal
69	ExprTuple	70
70	Literal