# from the theory of proveit.logic.sets.functions.bijections¶

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 A, B, Conditional, Lambda, f, fx, x
from proveit.logic import Bijections, Forall, InSet

In [2]:
# build up the expression from sub-expressions
sub_expr1 = Bijections(A, B)
expr = Forall(instance_param_or_params = [A, B], instance_expr = Forall(instance_param_or_params = [f], instance_expr = InSet(f, sub_expr1), condition = InSet(Lambda(x, Conditional(fx, InSet(x, A))), sub_expr1)))

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_{A, B}~\left[\forall_{f~|~\left[x \mapsto \left\{f\left(x\right) \textrm{ if } x \in A\right..\right] \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]}~\left(f \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]\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: 4
operand: 2
1ExprTuple2
2Lambdaparameters: 17
body: 3
3Operationoperator: 4
operand: 6
4Literal
5ExprTuple6
6Lambdaparameter: 21
body: 8
7ExprTuple21
8Conditionalvalue: 9
condition: 10
9Operationoperator: 23
operands: 11
10Operationoperator: 23
operands: 12
11ExprTuple21, 14
12ExprTuple13, 14
13Lambdaparameter: 25
body: 15
14Operationoperator: 16
operands: 17
15Conditionalvalue: 18
condition: 19
16Literal
17ExprTuple26, 20
18Operationoperator: 21
operand: 25
19Operationoperator: 23
operands: 24
20Variable
21Variable
22ExprTuple25
23Literal
24ExprTuple25, 26
25Variable
26Variable