Expression of type Forall

from the theory of proveit.numbers.modular

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 L, b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Mod, ModAbs, Natural, Real, RealPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_j], instance_expr = Forall(instance_param_or_params = [L], instance_expr = Equals(ModAbs(Add(a_1_to_i, Mod(b, L), c_1_to_j), L), ModAbs(Add(a_1_to_i, b, c_1_to_j), L)).with_wrapping_at(2), domain = RealPos), domain = Real), domain = Natural)

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())

\forall_{i, j \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{R}}~\left[\forall_{L \in \mathbb{R}^+}~\left(\begin{array}{c} \begin{array}{l} \left|a_{1} +  a_{2} +  \ldots +  a_{i} + \left(b ~\textup{mod}~ L\right)+ c_{1} +  c_{2} +  \ldots +  c_{j}\right|_{\textup{mod}\thinspace L} =  \\ \left|a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right|_{\textup{mod}\thinspace L} \end{array} \end{array}\right)\right]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	59, 62
4	Conditional	value: 5 condition: 6
5	Operation	operator: 18 operand: 9
6	Operation	operator: 20 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameters: 52 body: 12
10	Operation	operator: 39 operands: 13
11	Operation	operator: 39 operands: 14
12	Conditional	value: 15 condition: 16
13	ExprTuple	59, 17
14	ExprTuple	62, 17
15	Operation	operator: 18 operand: 22
16	Operation	operator: 20 operands: 21
17	Literal
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23, 24, 25
22	Lambda	parameter: 64 body: 27
23	ExprRange	lambda_map: 28 start_index: 61 end_index: 59
24	Operation	operator: 39 operands: 29
25	ExprRange	lambda_map: 30 start_index: 61 end_index: 62
26	ExprTuple	64
27	Conditional	value: 31 condition: 32
28	Lambda	parameter: 70 body: 33
29	ExprTuple	63, 44
30	Lambda	parameter: 70 body: 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 39 operands: 37
33	Operation	operator: 39 operands: 38
34	Operation	operator: 39 operands: 40
35	Literal
36	ExprTuple	41, 42
37	ExprTuple	64, 43
38	ExprTuple	65, 44
39	Literal
40	ExprTuple	66, 44
41	Operation	operator: 46 operands: 45
42	Operation	operator: 46 operands: 47
43	Literal
44	Literal
45	ExprTuple	48, 64
46	Literal
47	ExprTuple	49, 64
48	Operation	operator: 51 operands: 50
49	Operation	operator: 51 operands: 52
50	ExprTuple	54, 53, 55
51	Literal
52	ExprTuple	54, 63, 55
53	Operation	operator: 56 operands: 57
54	ExprRange	lambda_map: 58 start_index: 61 end_index: 59
55	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
56	Literal
57	ExprTuple	63, 64
58	Lambda	parameter: 70 body: 65
59	Variable
60	Lambda	parameter: 70 body: 66
61	Literal
62	Variable
63	Variable
64	Variable
65	IndexedVar	variable: 67 index: 70
66	IndexedVar	variable: 68 index: 70
67	Variable
68	Variable
69	ExprTuple	70
70	Variable