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, 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 = And(Pk_a1_to_an), condition = 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~|~\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\right)}~\left(P\left(a_{1}\right) \land  P\left(a_{2}\right) \land  \ldots \land  P\left(a_{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: 12 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 29 body: 3
3	Operation	operator: 12 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 22 body: 7
6	ExprTuple	22
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operand: 15
10	Literal
11	ExprTuple	14
12	Literal
13	ExprTuple	15
14	ExprRange	lambda_map: 16 start_index: 32 end_index: 33
15	Lambda	parameter: 26 body: 17
16	Lambda	parameter: 37 body: 18
17	Conditional	value: 19 condition: 20
18	Operation	operator: 22 operand: 34
19	Operation	operator: 22 operand: 26
20	Operation	operator: 24 operands: 25
21	ExprTuple	34
22	Variable
23	ExprTuple	26
24	Literal
25	ExprTuple	26, 27
26	Variable
27	Operation	operator: 28 operands: 29
28	Literal
29	ExprTuple	30
30	ExprRange	lambda_map: 31 start_index: 32 end_index: 33
31	Lambda	parameter: 37 body: 34
32	Literal
33	Variable
34	IndexedVar	variable: 35 index: 37
35	Variable
36	ExprTuple	37
37	Variable