Expression of type Forall

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

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = 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), domain = 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())

\forall_{i, j \in \mathbb{N}}~\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)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	60, 63
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 22 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 13 body: 14
11	Operation	operator: 46 operands: 15
12	Operation	operator: 46 operands: 16
13	ExprTuple	56, 49, 57, 58
14	Conditional	value: 17 condition: 18
15	ExprTuple	60, 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, 29
24	Operation	operator: 54 operands: 30
25	Operation	operator: 41 operands: 31
26	ExprRange	lambda_map: 32 start_index: 62 end_index: 60
27	Operation	operator: 46 operands: 33
28	Operation	operator: 46 operands: 34
29	ExprRange	lambda_map: 35 start_index: 62 end_index: 63
30	ExprTuple	56, 36, 58
31	ExprTuple	37, 38
32	Lambda	parameter: 69 body: 39
33	ExprTuple	49, 51
34	ExprTuple	57, 51
35	Lambda	parameter: 69 body: 40
36	Operation	operator: 41 operands: 42
37	Operation	operator: 54 operands: 43
38	Operation	operator: 52 operand: 50
39	Operation	operator: 46 operands: 45
40	Operation	operator: 46 operands: 47
41	Literal
42	ExprTuple	49, 48
43	ExprTuple	56, 49, 58
44	ExprTuple	50
45	ExprTuple	64, 51
46	Literal
47	ExprTuple	65, 51
48	Operation	operator: 52 operand: 57
49	Variable
50	Operation	operator: 54 operands: 55
51	Literal
52	Literal
53	ExprTuple	57
54	Literal
55	ExprTuple	56, 57, 58
56	ExprRange	lambda_map: 59 start_index: 62 end_index: 60
57	Variable
58	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
59	Lambda	parameter: 69 body: 64
60	Variable
61	Lambda	parameter: 69 body: 65
62	Literal
63	Variable
64	IndexedVar	variable: 66 index: 69
65	IndexedVar	variable: 67 index: 69
66	Variable
67	Variable
68	ExprTuple	69
69	Variable