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, L, Lambda, b, 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, Natural, Real, RealPos

# build up the expression from sub-expressions
expr = Lambda([i, j], Conditional(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), And(InSet(i, Natural), InSet(j, Natural))))

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(i, j\right) \mapsto \left\{\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] \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	56, 59
2	Conditional	value: 3 condition: 4
3	Operation	operator: 16 operand: 7
4	Operation	operator: 18 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 49 body: 10
8	Operation	operator: 37 operands: 11
9	Operation	operator: 37 operands: 12
10	Conditional	value: 13 condition: 14
11	ExprTuple	56, 15
12	ExprTuple	59, 15
13	Operation	operator: 16 operand: 20
14	Operation	operator: 18 operands: 19
15	Literal
16	Literal
17	ExprTuple	20
18	Literal
19	ExprTuple	21, 22, 23
20	Lambda	parameter: 61 body: 25
21	ExprRange	lambda_map: 26 start_index: 58 end_index: 56
22	Operation	operator: 37 operands: 27
23	ExprRange	lambda_map: 28 start_index: 58 end_index: 59
24	ExprTuple	61
25	Conditional	value: 29 condition: 30
26	Lambda	parameter: 67 body: 31
27	ExprTuple	60, 42
28	Lambda	parameter: 67 body: 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 37 operands: 35
31	Operation	operator: 37 operands: 36
32	Operation	operator: 37 operands: 38
33	Literal
34	ExprTuple	39, 40
35	ExprTuple	61, 41
36	ExprTuple	62, 42
37	Literal
38	ExprTuple	63, 42
39	Operation	operator: 53 operands: 43
40	Operation	operator: 53 operands: 44
41	Literal
42	Literal
43	ExprTuple	45, 61
44	ExprTuple	46, 61
45	Operation	operator: 48 operands: 47
46	Operation	operator: 48 operands: 49
47	ExprTuple	51, 50, 52
48	Literal
49	ExprTuple	51, 60, 52
50	Operation	operator: 53 operands: 54
51	ExprRange	lambda_map: 55 start_index: 58 end_index: 56
52	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
53	Literal
54	ExprTuple	60, 61
55	Lambda	parameter: 67 body: 62
56	Variable
57	Lambda	parameter: 67 body: 63
58	Literal
59	Variable
60	Variable
61	Variable
62	IndexedVar	variable: 64 index: 67
63	IndexedVar	variable: 65 index: 67
64	Variable
65	Variable
66	ExprTuple	67
67	Variable