Expression of type Lambda

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, Lambda, 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 = Lambda([n, alpha], 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(n, \alpha\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	50, 41
2	Conditional	value: 3 condition: 4
3	Operation	operator: 25 operand: 7
4	Operation	operator: 19 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 13 operands: 12
9	Operation	operator: 13 operands: 14
10	ExprTuple	43, 38
11	Operation	operator: 36 operands: 15
12	ExprTuple	50, 16
13	Literal
14	ExprTuple	41, 17
15	ExprTuple	18, 41
16	Literal
17	Literal
18	Operation	operator: 19 operands: 20
19	Literal
20	ExprTuple	21, 22
21	Operation	operator: 23 operand: 27
22	Operation	operator: 25 operand: 28
23	Literal
24	ExprTuple	27
25	Literal
26	ExprTuple	28
27	Lambda	parameters: 35 body: 29
28	Lambda	parameters: 44 body: 30
29	Conditional	value: 31 condition: 32
30	Conditional	value: 33 condition: 34
31	Operation	operator: 43 operands: 35
32	Operation	operator: 38 operands: 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 44
35	ExprTuple	39
36	Literal
37	ExprTuple	40, 41
38	Variable
39	ExprRange	lambda_map: 42 start_index: 49 end_index: 50
40	Operation	operator: 43 operands: 44
41	Variable
42	Lambda	parameter: 54 body: 45
43	Variable
44	ExprTuple	46
45	IndexedVar	variable: 47 index: 54
46	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
47	Variable
48	Lambda	parameter: 54 body: 51
49	Literal
50	Variable
51	IndexedVar	variable: 52 index: 54
52	Variable
53	ExprTuple	54
54	Variable