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 ExprRange, IndexedVar, Variable, a, n
from proveit.core_expr_types import a_1_to_n
from proveit.logic import Equals, Forall, InSet, Or
from proveit.numbers import Complex, Mult, Natural, ZeroSet, one, zero

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = InSet(Mult(a_1_to_n), ZeroSet), domain = Complex, condition = Or(ExprRange(sub_expr1, Equals(IndexedVar(a, sub_expr1), zero), one, n))), 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_{n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}~|~\left(a_{1} = 0\right) \lor  \left(a_{2} = 0\right) \lor  \ldots \lor  \left(a_{n} = 0\right)}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{n}\right) \in \left\{0\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	parameter: 37 body: 4
3	ExprTuple	37
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 33 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	37, 11
10	Lambda	parameters: 23 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 33 operands: 15
14	Operation	operator: 16 operands: 17
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 24 operand: 43
20	ExprRange	lambda_map: 26 start_index: 36 end_index: 37
21	Operation	operator: 27 operands: 28
22	Literal
23	ExprTuple	29
24	Literal
25	ExprTuple	43
26	Lambda	parameter: 46 body: 30
27	Literal
28	ExprTuple	31
29	ExprRange	lambda_map: 32 start_index: 36 end_index: 37
30	Operation	operator: 33 operands: 34
31	ExprRange	lambda_map: 35 start_index: 36 end_index: 37
32	Lambda	parameter: 46 body: 42
33	Literal
34	ExprTuple	42, 38
35	Lambda	parameter: 46 body: 39
36	Literal
37	Variable
38	Literal
39	Operation	operator: 40 operands: 41
40	Literal
41	ExprTuple	42, 43
42	IndexedVar	variable: 44 index: 46
43	Literal
44	Variable
45	ExprTuple	46
46	Variable