# from the theory of proveit.logic.sets.enumeration¶

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.
# import Expression classes needed to build the expression
from proveit import Lambda, P, Px, x
from proveit.core_expr_types import Pk_a1_to_an, a_1_to_n
from proveit.logic import And, Equals, Forall, Set

In [2]:
# build up the expression from sub-expressions
expr = Lambda([a_1_to_n], Forall(instance_param_or_params = [P], instance_expr = Equals(And(Pk_a1_to_an), Forall(instance_param_or_params = [x], instance_expr = Px, domain = Set(a_1_to_n)))))

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())

\left(a_{1}, a_{2}, \ldots, a_{n}\right) \mapsto \left[\forall_{P}~\left(\left(P\left(a_{1}\right) \land  P\left(a_{2}\right) \land  \ldots \land  P\left(a_{n}\right)\right) = \left[\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\right)\right]\right)\right]

In [5]:
stored_expr.style_options()

no style options
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Lambdaparameters: 29
body: 1
1Operationoperator: 12
operand: 3
2ExprTuple3
3Lambdaparameter: 22
body: 5
4ExprTuple22
5Operationoperator: 6
operands: 7
6Literal
7ExprTuple8, 9
8Operationoperator: 10
operands: 11
9Operationoperator: 12
operand: 15
10Literal
11ExprTuple14
12Literal
13ExprTuple15
14ExprRangelambda_map: 16
start_index: 32
end_index: 33
15Lambdaparameter: 26
body: 17
16Lambdaparameter: 37
body: 18
17Conditionalvalue: 19
condition: 20
18Operationoperator: 22
operand: 34
19Operationoperator: 22
operand: 26
20Operationoperator: 24
operands: 25
21ExprTuple34
22Variable
23ExprTuple26
24Literal
25ExprTuple26, 27
26Variable
27Operationoperator: 28
operands: 29
28Literal
29ExprTuple30
30ExprRangelambda_map: 31
start_index: 32
end_index: 33
31Lambdaparameter: 37
body: 34
32Literal
33Variable
34IndexedVarvariable: 35
index: 37
35Variable
36ExprTuple37
37Variable