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, ModAbs, 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(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), 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|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) \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: 49 body: 12
8	ExprRange	lambda_map: 13 start_index: 46 end_index: 44
9	Operation	operator: 24 operands: 14
10	ExprRange	lambda_map: 15 start_index: 46 end_index: 47
11	ExprTuple	49
12	Conditional	value: 16 condition: 17
13	Lambda	parameter: 55 body: 18
14	ExprTuple	48, 29
15	Lambda	parameter: 55 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	49, 28
23	ExprTuple	50, 29
24	Literal
25	ExprTuple	51, 29
26	Operation	operator: 31 operands: 30
27	Operation	operator: 31 operands: 32
28	Literal
29	Literal
30	ExprTuple	33, 49
31	Literal
32	ExprTuple	34, 49
33	Operation	operator: 36 operands: 35
34	Operation	operator: 36 operands: 37
35	ExprTuple	39, 38, 40
36	Literal
37	ExprTuple	39, 48, 40
38	Operation	operator: 41 operands: 42
39	ExprRange	lambda_map: 43 start_index: 46 end_index: 44
40	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
41	Literal
42	ExprTuple	48, 49
43	Lambda	parameter: 55 body: 50
44	Variable
45	Lambda	parameter: 55 body: 51
46	Literal
47	Variable
48	Variable
49	Variable
50	IndexedVar	variable: 52 index: 55
51	IndexedVar	variable: 53 index: 55
52	Variable
53	Variable
54	ExprTuple	55
55	Variable