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, 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(Mod(Add(a_1_to_i, Mod(b, L), c_1_to_j), L), Mod(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(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + \left(b ~\textup{mod}~ L\right)+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) ~\textup{mod}~ L\right) =  \\ \left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) ~\textup{mod}~ L\right) \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: 39 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: 51 body: 15
11	ExprRange	lambda_map: 16 start_index: 48 end_index: 46
12	Operation	operator: 27 operands: 17
13	ExprRange	lambda_map: 18 start_index: 48 end_index: 49
14	ExprTuple	51
15	Conditional	value: 19 condition: 20
16	Lambda	parameter: 57 body: 21
17	ExprTuple	50, 32
18	Lambda	parameter: 57 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	51, 31
26	ExprTuple	52, 32
27	Literal
28	ExprTuple	53, 32
29	Operation	operator: 43 operands: 33
30	Operation	operator: 43 operands: 34
31	Literal
32	Literal
33	ExprTuple	35, 51
34	ExprTuple	36, 51
35	Operation	operator: 38 operands: 37
36	Operation	operator: 38 operands: 39
37	ExprTuple	41, 40, 42
38	Literal
39	ExprTuple	41, 50, 42
40	Operation	operator: 43 operands: 44
41	ExprRange	lambda_map: 45 start_index: 48 end_index: 46
42	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
43	Literal
44	ExprTuple	50, 51
45	Lambda	parameter: 57 body: 52
46	Variable
47	Lambda	parameter: 57 body: 53
48	Literal
49	Variable
50	Variable
51	Variable
52	IndexedVar	variable: 54 index: 57
53	IndexedVar	variable: 55 index: 57
54	Variable
55	Variable
56	ExprTuple	57
57	Variable