# 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 S, m
from proveit.core_expr_types import A_1_to_m
from proveit.logic import Forall, 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 = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [A_1_to_m, S], instance_expr = SubsetEq(Union(A_1_to_m), S), condition = each_A_in_S), domain = 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())

\forall_{m \in \mathbb{N}^+}~\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)\right]

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 29
body: 4
3ExprTuple29
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple29, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple25, 33
14Conditionalvalue: 15
condition: 16
15Operationoperator: 30
operands: 17
16Operationoperator: 18
operands: 19
17ExprTuple20, 33
18Literal
19ExprTuple21
20Operationoperator: 22
operands: 23
21ExprRangelambda_map: 24
start_index: 28
end_index: 29
22Literal
23ExprTuple25
24Lambdaparameter: 36
body: 26
25ExprRangelambda_map: 27
start_index: 28
end_index: 29
26Operationoperator: 30
operands: 31
27Lambdaparameter: 36
body: 32
28Literal
29Variable
30Literal
31ExprTuple32, 33
32IndexedVarvariable: 34
index: 36
33Variable
34Variable
35ExprTuple36
36Variable