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
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

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q], 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))

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_{P, Q}~\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)

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: 13 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	31, 26
4	Operation	operator: 24 operands: 5
5	ExprTuple	6, 29
6	Operation	operator: 7 operands: 8
7	Literal
8	ExprTuple	9, 10
9	Operation	operator: 11 operand: 15
10	Operation	operator: 13 operand: 16
11	Literal
12	ExprTuple	15
13	Literal
14	ExprTuple	16
15	Lambda	parameters: 23 body: 17
16	Lambda	parameters: 32 body: 18
17	Conditional	value: 19 condition: 20
18	Conditional	value: 21 condition: 22
19	Operation	operator: 31 operands: 23
20	Operation	operator: 26 operands: 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 32
23	ExprTuple	27
24	Literal
25	ExprTuple	28, 29
26	Variable
27	ExprRange	lambda_map: 30 start_index: 37 end_index: 38
28	Operation	operator: 31 operands: 32
29	Variable
30	Lambda	parameter: 42 body: 33
31	Variable
32	ExprTuple	34
33	IndexedVar	variable: 35 index: 42
34	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
35	Variable
36	Lambda	parameter: 42 body: 39
37	Literal
38	Variable
39	IndexedVar	variable: 40 index: 42
40	Variable
41	ExprTuple	42
42	Variable