Theorem false_for_one_is_false_for_all of type Forall

from the theory of proveit.logic.sets.enumeration

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.logic.sets.enumeration import false_for_one_is_false_for_all

# check that the built expression is the same as the stored expression
assert false_for_one_is_false_for_all.expr == stored_expr
assert false_for_one_is_false_for_all.expr._style_id == stored_expr._style_id
print("Passed sanity check: false_for_one_is_false_for_all matches stored_expr")

Passed sanity check: false_for_one_is_false_for_all 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}}~\left[\forall_{P}~\left(\left(\left(P\left(a_{1}\right) = \bot\right) \lor  \left(P\left(a_{2}\right) = \bot\right) \lor  \ldots \lor  \left(P\left(a_{n}\right) = \bot\right)\right) = \left(\left[\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\right)\right] = \bot\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 48 body: 4
3	ExprTuple	48
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 9
6	Operation	operator: 39 operands: 8
7	ExprTuple	9
8	ExprTuple	48, 10
9	Lambda	parameters: 44 body: 11
10	Literal
11	Operation	operator: 25 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 37 body: 15
14	ExprTuple	37
15	Operation	operator: 29 operands: 16
16	ExprTuple	17, 18
17	Operation	operator: 19 operands: 20
18	Operation	operator: 29 operands: 21
19	Literal
20	ExprTuple	22
21	ExprTuple	23, 33
22	ExprRange	lambda_map: 24 start_index: 47 end_index: 48
23	Operation	operator: 25 operand: 28
24	Lambda	parameter: 52 body: 27
25	Literal
26	ExprTuple	28
27	Operation	operator: 29 operands: 30
28	Lambda	parameter: 41 body: 31
29	Literal
30	ExprTuple	32, 33
31	Conditional	value: 34 condition: 35
32	Operation	operator: 37 operand: 49
33	Literal
34	Operation	operator: 37 operand: 41
35	Operation	operator: 39 operands: 40
36	ExprTuple	49
37	Variable
38	ExprTuple	41
39	Literal
40	ExprTuple	41, 42
41	Variable
42	Operation	operator: 43 operands: 44
43	Literal
44	ExprTuple	45
45	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
46	Lambda	parameter: 52 body: 49
47	Literal
48	Variable
49	IndexedVar	variable: 50 index: 52
50	Variable
51	ExprTuple	52
52	Variable