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
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, Real, RealPos

# build up the expression from sub-expressions
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)

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_{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]

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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 40 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 10
5	Operation	operator: 8 operands: 9
6	Literal
7	ExprTuple	10
8	Literal
9	ExprTuple	11, 12, 13
10	Lambda	parameter: 52 body: 15
11	ExprRange	lambda_map: 16 start_index: 49 end_index: 47
12	Operation	operator: 27 operands: 17
13	ExprRange	lambda_map: 18 start_index: 49 end_index: 50
14	ExprTuple	52
15	Conditional	value: 19 condition: 20
16	Lambda	parameter: 58 body: 21
17	ExprTuple	51, 32
18	Lambda	parameter: 58 body: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 27 operands: 25
21	Operation	operator: 27 operands: 26
22	Operation	operator: 27 operands: 28
23	Literal
24	ExprTuple	29, 30
25	ExprTuple	52, 31
26	ExprTuple	53, 32
27	Literal
28	ExprTuple	54, 32
29	Operation	operator: 34 operands: 33
30	Operation	operator: 34 operands: 35
31	Literal
32	Literal
33	ExprTuple	36, 52
34	Literal
35	ExprTuple	37, 52
36	Operation	operator: 39 operands: 38
37	Operation	operator: 39 operands: 40
38	ExprTuple	42, 41, 43
39	Literal
40	ExprTuple	42, 51, 43
41	Operation	operator: 44 operands: 45
42	ExprRange	lambda_map: 46 start_index: 49 end_index: 47
43	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
44	Literal
45	ExprTuple	51, 52
46	Lambda	parameter: 58 body: 53
47	Variable
48	Lambda	parameter: 58 body: 54
49	Literal
50	Variable
51	Variable
52	Variable
53	IndexedVar	variable: 55 index: 58
54	IndexedVar	variable: 56 index: 58
55	Variable
56	Variable
57	ExprTuple	58
58	Variable