Theorem exists_in_superset of type Forall

from the theory of proveit.logic.booleans.quantification.existence

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.booleans.quantification.existence import exists_in_superset

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

Passed sanity check: exists_in_superset 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_{P, Q, A, B~|~A \subseteq B}~\left(\begin{array}{c} \begin{array}{l} \left[\exists_{x_{1}, x_{2}, \ldots, x_{n} \in A~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left[\exists_{y_{1}, y_{2}, \ldots, y_{n} \in B~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\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	parameter: 56 body: 4
3	ExprTuple	56
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 52 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	56, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	34, 44, 57, 59
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	57, 59
20	Operation	operator: 23 operand: 25
21	Operation	operator: 23 operand: 26
22	ExprTuple	25
23	Literal
24	ExprTuple	26
25	Lambda	parameters: 42 body: 27
26	Lambda	parameters: 45 body: 28
27	Conditional	value: 29 condition: 30
28	Conditional	value: 31 condition: 32
29	Operation	operator: 34 operands: 42
30	Operation	operator: 35 operands: 33
31	Operation	operator: 34 operands: 45
32	Operation	operator: 35 operands: 36
33	ExprTuple	37, 38
34	Variable
35	Literal
36	ExprTuple	39, 40
37	ExprRange	lambda_map: 41 start_index: 55 end_index: 56
38	Operation	operator: 44 operands: 42
39	ExprRange	lambda_map: 43 start_index: 55 end_index: 56
40	Operation	operator: 44 operands: 45
41	Lambda	parameter: 64 body: 46
42	ExprTuple	47
43	Lambda	parameter: 64 body: 48
44	Variable
45	ExprTuple	49
46	Operation	operator: 52 operands: 50
47	ExprRange	lambda_map: 51 start_index: 55 end_index: 56
48	Operation	operator: 52 operands: 53
49	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
50	ExprTuple	58, 57
51	Lambda	parameter: 64 body: 58
52	Literal
53	ExprTuple	60, 59
54	Lambda	parameter: 64 body: 60
55	Literal
56	Variable
57	Variable
58	IndexedVar	variable: 61 index: 64
59	Variable
60	IndexedVar	variable: 62 index: 64
61	Variable
62	Variable
63	ExprTuple	64
64	Variable