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
from proveit.core_expr_types import P__y_1_to_n, Q__y_1_to_n, y_1_to_n
from proveit.logic import Forall
from proveit.logic.booleans.quantification import general_exists_Px_st_Qx

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q], instance_expr = Forall(instance_param_or_params = [y_1_to_n], instance_expr = general_exists_Px_st_Qx, conditions = [P__y_1_to_n, Q__y_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_{P, Q}~\left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~P\left(y_{1}, y_{2}, \ldots, y_{n}\right), Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left[\exists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	23, 24
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameters: 19 body: 8
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operand: 15
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	15
13	Literal
14	ExprTuple	16, 17
15	Lambda	parameters: 25 body: 18
16	Operation	operator: 23 operands: 19
17	Operation	operator: 24 operands: 19
18	Conditional	value: 20 condition: 21
19	ExprTuple	22
20	Operation	operator: 23 operands: 25
21	Operation	operator: 24 operands: 25
22	ExprRange	lambda_map: 26 start_index: 30 end_index: 31
23	Variable
24	Variable
25	ExprTuple	27
26	Lambda	parameter: 36 body: 28
27	ExprRange	lambda_map: 29 start_index: 30 end_index: 31
28	IndexedVar	variable: 32 index: 36
29	Lambda	parameter: 36 body: 33
30	Literal
31	Variable
32	Variable
33	IndexedVar	variable: 34 index: 36
34	Variable
35	ExprTuple	36
36	Variable