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
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, zero

# 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_1_to_j], instance_expr = Equals(Mult(a_1_to_i, zero, b_1_to_j), zero), 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_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot 0\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right) = 0\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	37, 40
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: 42 operands: 15
12	Operation	operator: 42 operands: 16
13	ExprTuple	31, 33
14	Conditional	value: 17 condition: 18
15	ExprTuple	37, 19
16	ExprTuple	40, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	Literal
20	Literal
21	ExprTuple	24, 32
22	Literal
23	ExprTuple	25, 26
24	Operation	operator: 27 operands: 28
25	ExprRange	lambda_map: 29 start_index: 39 end_index: 37
26	ExprRange	lambda_map: 30 start_index: 39 end_index: 40
27	Literal
28	ExprTuple	31, 32, 33
29	Lambda	parameter: 50 body: 34
30	Lambda	parameter: 50 body: 35
31	ExprRange	lambda_map: 36 start_index: 39 end_index: 37
32	Literal
33	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
34	Operation	operator: 42 operands: 41
35	Operation	operator: 42 operands: 43
36	Lambda	parameter: 50 body: 44
37	Variable
38	Lambda	parameter: 50 body: 45
39	Literal
40	Variable
41	ExprTuple	44, 46
42	Literal
43	ExprTuple	45, 46
44	IndexedVar	variable: 47 index: 50
45	IndexedVar	variable: 48 index: 50
46	Literal
47	Variable
48	Variable
49	ExprTuple	50
50	Variable