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 Forall, Implies, InSet, Not
from proveit.logic.booleans.quantification import general_exists_notPx_st_Qx, general_forall_Py_if_Qy
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 = Implies(general_exists_notPx_st_Qx, Not(general_forall_Py_if_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[\exists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~(\lnot P\left(x_{1}, x_{2}, \ldots, x_{n}\right))\right] \Rightarrow  \\ (\lnot \left[\forall_{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: 42 body: 2
1	ExprTuple	42
2	Conditional	value: 3 condition: 4
3	Operation	operator: 22 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	42, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	34, 35
11	Operation	operator: 12 operands: 13
12	Literal
13	ExprTuple	14, 15
14	Operation	operator: 16 operand: 19
15	Operation	operator: 27 operand: 20
16	Literal
17	ExprTuple	19
18	ExprTuple	20
19	Lambda	parameters: 33 body: 21
20	Operation	operator: 22 operand: 26
21	Conditional	value: 24 condition: 25
22	Literal
23	ExprTuple	26
24	Operation	operator: 27 operand: 30
25	Operation	operator: 35 operands: 33
26	Lambda	parameters: 36 body: 29
27	Literal
28	ExprTuple	30
29	Conditional	value: 31 condition: 32
30	Operation	operator: 34 operands: 33
31	Operation	operator: 34 operands: 36
32	Operation	operator: 35 operands: 36
33	ExprTuple	37
34	Variable
35	Variable
36	ExprTuple	38
37	ExprRange	lambda_map: 39 start_index: 41 end_index: 42
38	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
39	Lambda	parameter: 48 body: 43
40	Lambda	parameter: 48 body: 44
41	Literal
42	Variable
43	IndexedVar	variable: 45 index: 48
44	IndexedVar	variable: 46 index: 48
45	Variable
46	Variable
47	ExprTuple	48
48	Variable