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
from proveit.logic.booleans.quantification import general_forall__Py_not_T__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, general_forall__Py_not_T__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] =  \\ \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) \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: 43 body: 2
1	ExprTuple	43
2	Conditional	value: 3 condition: 4
3	Operation	operator: 18 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	43, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	36, 31
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	Literal
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