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, n
from proveit.logic import Equals, Forall, InSet, Not
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy, general_notexists_Px_st_Qx
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [P, Q], instance_expr = Equals(general_notexists_Px_st_Qx, Not(general_exists_Py_st_Qy)).with_wrapping_at(2)), 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())

n \mapsto \left\{\forall_{P, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\nexists_{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] =  \\ (\lnot \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]) \end{array} \end{array}\right) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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