Expression of type Forall

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

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, n
from proveit.core_expr_types import P__y_1_to_n, y_1_to_n
from proveit.logic import Forall
from proveit.logic.booleans.quantification import general_exists_Px
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [P], instance_expr = Forall(instance_param_or_params = [y_1_to_n], instance_expr = general_exists_Px, condition = P__y_1_to_n)), domain = NaturalPos)

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