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, n
from proveit.logic import Forall, Implies, Not
from proveit.logic.booleans.quantification import general_exists_Px_st_Qx, general_forall__Py_not_T__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], instance_expr = Implies(general_exists_Px_st_Qx, Not(general_forall__Py_not_T__st_Qy)).with_wrapping_at(2)), 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}~\left(\begin{array}{c} \begin{array}{l} \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] \Rightarrow  \\ (\lnot \left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(P\left(y_{1}, y_{2}, \ldots, y_{n}\right) \neq \top\right)\right]) \end{array} \end{array}\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 48 body: 4
3	ExprTuple	48
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	48, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	43, 38
13	Operation	operator: 14 operands: 15
14	Literal
15	ExprTuple	16, 17
16	Operation	operator: 18 operand: 22
17	Operation	operator: 20 operand: 23
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23
22	Lambda	parameters: 30 body: 24
23	Operation	operator: 25 operand: 29
24	Conditional	value: 27 condition: 28
25	Literal
26	ExprTuple	29
27	Operation	operator: 43 operands: 30
28	Operation	operator: 38 operands: 30
29	Lambda	parameters: 44 body: 31
30	ExprTuple	32
31	Conditional	value: 33 condition: 34
32	ExprRange	lambda_map: 35 start_index: 47 end_index: 48
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 44
35	Lambda	parameter: 52 body: 39
36	Literal
37	ExprTuple	40, 41
38	Variable
39	IndexedVar	variable: 42 index: 52
40	Operation	operator: 43 operands: 44
41	Literal
42	Variable
43	Variable
44	ExprTuple	45
45	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
46	Lambda	parameter: 52 body: 49
47	Literal
48	Variable
49	IndexedVar	variable: 50 index: 52
50	Variable
51	ExprTuple	52
52	Variable