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Expression of type Lambda

from the theory of proveit.logic.booleans.quantification.existence

In [1]:
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_Px_st_Qx, general_forall__Py_not_T__st_Qy
from proveit.numbers import NaturalPos
In [2]:
# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [P, Q], instance_expr = Implies(general_exists_Px_st_Qx, Not(general_forall__Py_not_T__st_Qy)).with_wrapping_at(2)), InSet(n, NaturalPos)))
expr:
In [3]:
# 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
In [4]:
# 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)}~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)}~\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..
In [5]:
stored_expr.style_options()
no style options
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Lambdaparameter: 46
body: 2
1ExprTuple46
2Conditionalvalue: 3
condition: 4
3Operationoperator: 23
operand: 8
4Operationoperator: 6
operands: 7
5ExprTuple8
6Literal
7ExprTuple46, 9
8Lambdaparameters: 10
body: 11
9Literal
10ExprTuple41, 36
11Operationoperator: 12
operands: 13
12Literal
13ExprTuple14, 15
14Operationoperator: 16
operand: 20
15Operationoperator: 18
operand: 21
16Literal
17ExprTuple20
18Literal
19ExprTuple21
20Lambdaparameters: 28
body: 22
21Operationoperator: 23
operand: 27
22Conditionalvalue: 25
condition: 26
23Literal
24ExprTuple27
25Operationoperator: 41
operands: 28
26Operationoperator: 36
operands: 28
27Lambdaparameters: 42
body: 29
28ExprTuple30
29Conditionalvalue: 31
condition: 32
30ExprRangelambda_map: 33
start_index: 45
end_index: 46
31Operationoperator: 34
operands: 35
32Operationoperator: 36
operands: 42
33Lambdaparameter: 50
body: 37
34Literal
35ExprTuple38, 39
36Variable
37IndexedVarvariable: 40
index: 50
38Operationoperator: 41
operands: 42
39Literal
40Variable
41Variable
42ExprTuple43
43ExprRangelambda_map: 44
start_index: 45
end_index: 46
44Lambdaparameter: 50
body: 47
45Literal
46Variable
47IndexedVarvariable: 48
index: 50
48Variable
49ExprTuple50
50Variable