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, Lambda, i, j, x, y
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Complex, Mult, Natural, subtract

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