Expression of type Conditional

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 Conditional, 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, InSet
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(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)), InSet(n, 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())

\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) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 18 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	43, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	36, 31, 34
9	Operation	operator: 29 operands: 10
10	ExprTuple	11, 34
11	Operation	operator: 12 operands: 13
12	Literal
13	ExprTuple	14, 15
14	Operation	operator: 16 operand: 20
15	Operation	operator: 18 operand: 21
16	Literal
17	ExprTuple	20
18	Literal
19	ExprTuple	21
20	Lambda	parameters: 28 body: 22
21	Lambda	parameters: 37 body: 23
22	Conditional	value: 24 condition: 25
23	Conditional	value: 26 condition: 27
24	Operation	operator: 36 operands: 28
25	Operation	operator: 31 operands: 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 37
28	ExprTuple	32
29	Literal
30	ExprTuple	33, 34
31	Variable
32	ExprRange	lambda_map: 35 start_index: 42 end_index: 43
33	Operation	operator: 36 operands: 37
34	Variable
35	Lambda	parameter: 47 body: 38
36	Variable
37	ExprTuple	39
38	IndexedVar	variable: 40 index: 47
39	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
40	Variable
41	Lambda	parameter: 47 body: 44
42	Literal
43	Variable
44	IndexedVar	variable: 45 index: 47
45	Variable
46	ExprTuple	47
47	Variable