Expression of type Lambda

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, Lambda, 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 = Lambda([a_1_to_i, b, c_1_to_j], 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(a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j}\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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