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 b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Mult, Natural, Neg

# 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, b, c_1_to_j], instance_expr = Equals(Mult(a_1_to_i, Neg(b), c_1_to_j), Neg(Mult(a_1_to_i, b, c_1_to_j))), 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}, b, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left(-b\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{j}\right) = \left(-\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{j}\right)\right)\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	49, 52
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 21 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 40 body: 13
11	Operation	operator: 42 operands: 14
12	Operation	operator: 42 operands: 15
13	Conditional	value: 16 condition: 17
14	ExprTuple	49, 18
15	ExprTuple	52, 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	Literal
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25, 26, 27
23	Operation	operator: 39 operands: 28
24	Operation	operator: 37 operand: 34
25	ExprRange	lambda_map: 30 start_index: 51 end_index: 49
26	Operation	operator: 42 operands: 31
27	ExprRange	lambda_map: 32 start_index: 51 end_index: 52
28	ExprTuple	44, 33, 46
29	ExprTuple	34
30	Lambda	parameter: 58 body: 35
31	ExprTuple	45, 47
32	Lambda	parameter: 58 body: 36
33	Operation	operator: 37 operand: 45
34	Operation	operator: 39 operands: 40
35	Operation	operator: 42 operands: 41
36	Operation	operator: 42 operands: 43
37	Literal
38	ExprTuple	45
39	Literal
40	ExprTuple	44, 45, 46
41	ExprTuple	53, 47
42	Literal
43	ExprTuple	54, 47
44	ExprRange	lambda_map: 48 start_index: 51 end_index: 49
45	Variable
46	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
47	Literal
48	Lambda	parameter: 58 body: 53
49	Variable
50	Lambda	parameter: 58 body: 54
51	Literal
52	Variable
53	IndexedVar	variable: 55 index: 58
54	IndexedVar	variable: 56 index: 58
55	Variable
56	Variable
57	ExprTuple	58
58	Variable