Expression of type Lambda

from the theory of proveit.numbers.multiplication

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, Lambda, Variable, b, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Complex, Mult, Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
expr = Lambda([i, j, k], Conditional(Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, Add(b_1_to_j), c_1_to_k), Add(ExprRange(sub_expr1, Mult(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j))).with_wrapping_at(2), domain = Complex).with_wrapping(), And(InSet(i, Natural), InSet(j, Natural), InSet(k, 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, k\right) \mapsto \left\{\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{1}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{2}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \ldots +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right) \end{array} \end{array}\right)\end{array} \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  k \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	58, 50, 63
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 22 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10, 11
8	Lambda	parameters: 12 body: 13
9	Operation	operator: 44 operands: 14
10	Operation	operator: 44 operands: 15
11	Operation	operator: 44 operands: 16
12	ExprTuple	54, 46, 56
13	Conditional	value: 17 condition: 18
14	ExprTuple	58, 19
15	ExprTuple	50, 19
16	ExprTuple	63, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	Literal
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26, 27, 28
24	Operation	operator: 51 operands: 29
25	Operation	operator: 39 operands: 30
26	ExprRange	lambda_map: 31 start_index: 62 end_index: 58
27	ExprRange	lambda_map: 32 start_index: 62 end_index: 50
28	ExprRange	lambda_map: 33 start_index: 62 end_index: 63
29	ExprTuple	54, 34, 56
30	ExprTuple	35
31	Lambda	parameter: 70 body: 36
32	Lambda	parameter: 70 body: 37
33	Lambda	parameter: 70 body: 38
34	Operation	operator: 39 operands: 40
35	ExprRange	lambda_map: 41 start_index: 62 end_index: 50
36	Operation	operator: 44 operands: 42
37	Operation	operator: 44 operands: 43
38	Operation	operator: 44 operands: 45
39	Literal
40	ExprTuple	46
41	Lambda	parameter: 65 body: 47
42	ExprTuple	64, 48
43	ExprTuple	53, 48
44	Literal
45	ExprTuple	66, 48
46	ExprRange	lambda_map: 49 start_index: 62 end_index: 50
47	Operation	operator: 51 operands: 52
48	Literal
49	Lambda	parameter: 70 body: 53
50	Variable
51	Literal
52	ExprTuple	54, 55, 56
53	IndexedVar	variable: 59 index: 70
54	ExprRange	lambda_map: 57 start_index: 62 end_index: 58
55	IndexedVar	variable: 59 index: 65
56	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
57	Lambda	parameter: 70 body: 64
58	Variable
59	Variable
60	ExprTuple	65
61	Lambda	parameter: 70 body: 66
62	Literal
63	Variable
64	IndexedVar	variable: 67 index: 70
65	Variable
66	IndexedVar	variable: 68 index: 70
67	Variable
68	Variable
69	ExprTuple	70
70	Variable