# 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 Conditional, P, Px, n, x
from proveit.core_expr_types import Pk_a1_to_an, a_1_to_n
from proveit.logic import And, Equals, Forall, InSet, Set
from proveit.numbers import Natural

In [2]:
# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [a_1_to_n], instance_expr = 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))))), InSet(n, Natural))

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\{\forall_{a_{1}, a_{2}, \ldots, a_{n}}~\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] \textrm{ if } n \in \mathbb{N}\right..

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 18
operand: 5
2Operationoperator: 30
operands: 4
3ExprTuple5
4ExprTuple39, 6
5Lambdaparameters: 35
body: 7
6Literal
7Operationoperator: 18
operand: 9
8ExprTuple9
9Lambdaparameter: 28
body: 11
10ExprTuple28
11Operationoperator: 12
operands: 13
12Literal
13ExprTuple14, 15
14Operationoperator: 16
operands: 17
15Operationoperator: 18
operand: 21
16Literal
17ExprTuple20
18Literal
19ExprTuple21
20ExprRangelambda_map: 22
start_index: 38
end_index: 39
21Lambdaparameter: 32
body: 23
22Lambdaparameter: 43
body: 24
23Conditionalvalue: 25
condition: 26
24Operationoperator: 28
operand: 40
25Operationoperator: 28
operand: 32
26Operationoperator: 30
operands: 31
27ExprTuple40
28Variable
29ExprTuple32
30Literal
31ExprTuple32, 33
32Variable
33Operationoperator: 34
operands: 35
34Literal
35ExprTuple36
36ExprRangelambda_map: 37
start_index: 38
end_index: 39
37Lambdaparameter: 43
body: 40
38Literal
39Variable
40IndexedVarvariable: 41
index: 43
41Variable
42ExprTuple43
43Variable