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, i, j, k
from proveit.core_expr_types import a_1_to_i, c_1_to_j, d_1_to_k
from proveit.logic import Equals, Forall, Set
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_j, d_1_to_k], instance_expr = Equals(Set(a_1_to_i, b, c_1_to_j, d_1_to_k), Set(a_1_to_i, c_1_to_j, b, d_1_to_k))), 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, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j}, d_{1}, d_{2}, \ldots, d_{k}}~\left(\left\{a_{1}, a_{2}, \ldots, a_{i}, b,c_{1}, c_{2}, \ldots, c_{j},d_{1}, d_{2}, \ldots, d_{k}\right\} = \left\{a_{1}, a_{2}, \ldots, a_{i},c_{1}, c_{2}, \ldots, c_{j}, b,d_{1}, d_{2}, \ldots, d_{k}\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	33, 35, 38
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, 14
11	Lambda	parameters: 25 body: 15
12	Operation	operator: 18 operands: 16
13	Operation	operator: 18 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operands: 21
16	ExprTuple	33, 22
17	ExprTuple	35, 22
18	Literal
19	ExprTuple	38, 22
20	Literal
21	ExprTuple	23, 24
22	Literal
23	Operation	operator: 26 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 30, 29, 31
26	Literal
27	ExprTuple	28, 29, 30, 31
28	ExprRange	lambda_map: 32 start_index: 37 end_index: 33
29	ExprRange	lambda_map: 34 start_index: 37 end_index: 35
30	Variable
31	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
32	Lambda	parameter: 46 body: 39
33	Variable
34	Lambda	parameter: 46 body: 40
35	Variable
36	Lambda	parameter: 46 body: 41
37	Literal
38	Variable
39	IndexedVar	variable: 42 index: 46
40	IndexedVar	variable: 43 index: 46
41	IndexedVar	variable: 44 index: 46
42	Variable
43	Variable
44	Variable
45	ExprTuple	46
46	Variable