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, Boolean, 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], 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)), And(InSet(n, NaturalPos), InSet(alpha, Boolean)))

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}~\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}^+ ,  \alpha \in \mathbb{B}\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: 23 operand: 5
2	Operation	operator: 17 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 8 body: 9
6	Operation	operator: 11 operands: 10
7	Operation	operator: 11 operands: 12
8	ExprTuple	41, 36
9	Operation	operator: 34 operands: 13
10	ExprTuple	48, 14
11	Literal
12	ExprTuple	39, 15
13	ExprTuple	16, 39
14	Literal
15	Literal
16	Operation	operator: 17 operands: 18
17	Literal
18	ExprTuple	19, 20
19	Operation	operator: 21 operand: 25
20	Operation	operator: 23 operand: 26
21	Literal
22	ExprTuple	25
23	Literal
24	ExprTuple	26
25	Lambda	parameters: 33 body: 27
26	Lambda	parameters: 42 body: 28
27	Conditional	value: 29 condition: 30
28	Conditional	value: 31 condition: 32
29	Operation	operator: 41 operands: 33
30	Operation	operator: 36 operands: 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 42
33	ExprTuple	37
34	Literal
35	ExprTuple	38, 39
36	Variable
37	ExprRange	lambda_map: 40 start_index: 47 end_index: 48
38	Operation	operator: 41 operands: 42
39	Variable
40	Lambda	parameter: 52 body: 43
41	Variable
42	ExprTuple	44
43	IndexedVar	variable: 45 index: 52
44	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
45	Variable
46	Lambda	parameter: 52 body: 49
47	Literal
48	Variable
49	IndexedVar	variable: 50 index: 52
50	Variable
51	ExprTuple	52
52	Variable