Expression of type Conditional

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 Conditional, ExprRange, IndexedVar, L, Variable, a, b, c, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Mod, Real, RealPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(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), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), Real), one, i), InSet(b, Real), ExprRange(sub_expr1, InSet(IndexedVar(c, sub_expr1), Real), one, j)))

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())

\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) \textrm{ if } \left(a_{1} \in \mathbb{R}\right) ,  \left(a_{2} \in \mathbb{R}\right) ,  \ldots ,  \left(a_{i} \in \mathbb{R}\right) ,  b \in \mathbb{R}, \left(c_{1} \in \mathbb{R}\right) ,  \left(c_{2} \in \mathbb{R}\right) ,  \ldots ,  \left(c_{j} \in \mathbb{R}\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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