Expression of type Conditional

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, 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 = 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\{\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()

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