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

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, S, m
from proveit.core_expr_types import A_1_to_m
from proveit.logic import Forall, InSet, SubsetEq, Union
from proveit.logic.sets import each_A_in_S
from proveit.numbers import NaturalPos

In [2]:
# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A_1_to_m, S], instance_expr = SubsetEq(Union(A_1_to_m), S), condition = each_A_in_S), InSet(m, 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())

\left\{\forall_{A_{1}, A_{2}, \ldots, A_{m}, S~|~\left(A_{1} \subseteq S\right), \left(A_{2} \subseteq S\right), \ldots, \left(A_{m} \subseteq S\right)}~\left(\left(A_{1} \cup  A_{2} \cup  \ldots \cup  A_{m}\right) \subseteq S\right) \textrm{ if } m \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: 3
operand: 7
2Operationoperator: 5
operands: 6
3Literal
4ExprTuple7
5Literal
6ExprTuple25, 8
7Lambdaparameters: 9
body: 10
8Literal
9ExprTuple21, 29
10Conditionalvalue: 11
condition: 12
11Operationoperator: 26
operands: 13
12Operationoperator: 14
operands: 15
13ExprTuple16, 29
14Literal
15ExprTuple17
16Operationoperator: 18
operands: 19
17ExprRangelambda_map: 20
start_index: 24
end_index: 25
18Literal
19ExprTuple21
20Lambdaparameter: 32
body: 22
21ExprRangelambda_map: 23
start_index: 24
end_index: 25
22Operationoperator: 26
operands: 27
23Lambdaparameter: 32
body: 28
24Literal
25Variable
26Literal
27ExprTuple28, 29
28IndexedVarvariable: 30
index: 32
29Variable
30Variable
31ExprTuple32
32Variable