Expression of type Forall

from the theory of proveit.logic.sets.enumeration

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, m, n
from proveit.core_expr_types import a_1_to_m, c_1_to_n
from proveit.logic import Forall, NotInSet, ProperSubset, Set
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = Set(a_1_to_m)
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [a_1_to_m, b, c_1_to_n], instance_expr = ProperSubset(sub_expr1, Set(a_1_to_m, b, c_1_to_n)), condition = NotInSet(b, sub_expr1)), 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_{m, n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m}, b, c_{1}, c_{2}, \ldots, c_{n}~|~b \notin \left\{a_{1}, a_{2}, \ldots, a_{m}\right\}}~\left(\left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \subset \left\{a_{1}, a_{2}, \ldots, a_{m}, b,c_{1}, c_{2}, \ldots, c_{n}\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	37, 34
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13
11	Lambda	parameters: 27 body: 14
12	Operation	operator: 16 operands: 15
13	Operation	operator: 16 operands: 17
14	Conditional	value: 18 condition: 19
15	ExprTuple	37, 20
16	Literal
17	ExprTuple	34, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Literal
21	Literal
22	ExprTuple	26, 25
23	Literal
24	ExprTuple	30, 26
25	Operation	operator: 28 operands: 27
26	Operation	operator: 28 operands: 29
27	ExprTuple	32, 30, 31
28	Literal
29	ExprTuple	32
30	Variable
31	ExprRange	lambda_map: 33 start_index: 36 end_index: 34
32	ExprRange	lambda_map: 35 start_index: 36 end_index: 37
33	Lambda	parameter: 43 body: 38
34	Variable
35	Lambda	parameter: 43 body: 39
36	Literal
37	Variable
38	IndexedVar	variable: 40 index: 43
39	IndexedVar	variable: 41 index: 43
40	Variable
41	Variable
42	ExprTuple	43
43	Variable