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 P, Px, x
from proveit.core_expr_types import Pk_a1_to_an, a_1_to_n
from proveit.logic import And, Equals, Forall, Set

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [P], instance_expr = Equals(And(Pk_a1_to_an), Forall(instance_param_or_params = [x], instance_expr = Px, domain = Set(a_1_to_n)))))

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_{a_{1}, a_{2}, \ldots, a_{n}}~\left[\forall_{P}~\left(\left(P\left(a_{1}\right) \land  P\left(a_{2}\right) \land  \ldots \land  P\left(a_{n}\right)\right) = \left[\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 31 body: 3
3	Operation	operator: 14 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 24 body: 7
6	ExprTuple	24
7	Operation	operator: 8 operands: 9
8	Literal
9	ExprTuple	10, 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 14 operand: 17
12	Literal
13	ExprTuple	16
14	Literal
15	ExprTuple	17
16	ExprRange	lambda_map: 18 start_index: 34 end_index: 35
17	Lambda	parameter: 28 body: 19
18	Lambda	parameter: 39 body: 20
19	Conditional	value: 21 condition: 22
20	Operation	operator: 24 operand: 36
21	Operation	operator: 24 operand: 28
22	Operation	operator: 26 operands: 27
23	ExprTuple	36
24	Variable
25	ExprTuple	28
26	Literal
27	ExprTuple	28, 29
28	Variable
29	Operation	operator: 30 operands: 31
30	Literal
31	ExprTuple	32
32	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
33	Lambda	parameter: 39 body: 36
34	Literal
35	Variable
36	IndexedVar	variable: 37 index: 39
37	Variable
38	ExprTuple	39
39	Variable