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, Q, alpha, n
from proveit.core_expr_types import P__x_1_to_n, Q__x_1_to_n, x_1_to_n
from proveit.logic import And, Forall, Implies
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy
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, Q, alpha], instance_expr = Implies(And(general_exists_Py_st_Qy, Forall(instance_param_or_params = [x_1_to_n], instance_expr = Implies(P__x_1_to_n, alpha), condition = Q__x_1_to_n)), alpha)), 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, Q, \alpha}~\left(\left(\left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right] \land \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~\left(P\left(x_{1}, x_{2}, \ldots, x_{n}\right) \Rightarrow \alpha\right)\right]\right) \Rightarrow \alpha\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: 22 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 47 body: 4
3	ExprTuple	47
4	Conditional	value: 5 condition: 6
5	Operation	operator: 22 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	47, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	40, 35, 38
13	Operation	operator: 33 operands: 14
14	ExprTuple	15, 38
15	Operation	operator: 16 operands: 17
16	Literal
17	ExprTuple	18, 19
18	Operation	operator: 20 operand: 24
19	Operation	operator: 22 operand: 25
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	25
24	Lambda	parameters: 32 body: 26
25	Lambda	parameters: 41 body: 27
26	Conditional	value: 28 condition: 29
27	Conditional	value: 30 condition: 31
28	Operation	operator: 40 operands: 32
29	Operation	operator: 35 operands: 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 41
32	ExprTuple	36
33	Literal
34	ExprTuple	37, 38
35	Variable
36	ExprRange	lambda_map: 39 start_index: 46 end_index: 47
37	Operation	operator: 40 operands: 41
38	Variable
39	Lambda	parameter: 51 body: 42
40	Variable
41	ExprTuple	43
42	IndexedVar	variable: 44 index: 51
43	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
44	Variable
45	Lambda	parameter: 51 body: 48
46	Literal
47	Variable
48	IndexedVar	variable: 49 index: 51
49	Variable
50	ExprTuple	51
51	Variable