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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, B, f
from proveit.logic import And, Bijections, Forall, InSet, Injections, Surjections

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [A, B], instance_expr = Forall(instance_param_or_params = [f], instance_expr = And(InSet(f, Injections(A, B)), InSet(f, Surjections(A, B))), domain = Bijections(A, B)))

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 \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]}~\left(\left(f \in \left[A \xrightarrow[]{\text{1-to-1}} B\right]\right) \land \left(f \in \left[A \xrightarrow[\text{onto}]{} B\right]\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: 26
body: 3
3Operationoperator: 4
operand: 6
4Literal
5ExprTuple6
6Lambdaparameter: 22
body: 8
7ExprTuple22
8Conditionalvalue: 9
condition: 10
9Operationoperator: 11
operands: 12
10Operationoperator: 18
operands: 13
11Literal
12ExprTuple14, 15
13ExprTuple22, 16
14Operationoperator: 18
operands: 17
15Operationoperator: 18
operands: 19
16Operationoperator: 20
operands: 26
17ExprTuple22, 21
18Literal
19ExprTuple22, 23
20Literal
21Operationoperator: 24
operands: 26
22Variable
23Operationoperator: 25
operands: 26
24Literal
25Literal
26ExprTuple27, 28
27Variable
28Variable