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

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, a, b, f, fa, fb
from proveit.logic import And, Equals, Forall, Functions, InSet, Injections, NotEquals

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 = Equals(InSet(f, Injections(A, B)), And(InSet(f, Functions(A, B)), Forall(instance_param_or_params = [a, b], instance_expr = NotEquals(fa, fb), domain = A, condition = NotEquals(a, b)))).with_wrapping_at(2)))

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(\begin{array}{c} \begin{array}{l} \left(f \in \left[A \xrightarrow[]{\text{1-to-1}} B\right]\right) =  \\ \left(\left(f \in \left[A \rightarrow B\right])\right) \land \left[\forall_{a, b \in A~|~a \neq b}~\left(f\left(a\right) \neq f\left(b\right)\right)\right]\right) \end{array} \end{array}\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: 19
operand: 2
1ExprTuple2
2Lambdaparameters: 24
body: 3
3Operationoperator: 19
operand: 5
4ExprTuple5
5Lambdaparameter: 38
body: 7
6ExprTuple38
7Operationoperator: 8
operands: 9
8Literal
9ExprTuple10, 11
10Operationoperator: 41
operands: 12
11Operationoperator: 30
operands: 13
12ExprTuple38, 14
13ExprTuple15, 16
14Operationoperator: 17
operands: 24
15Operationoperator: 41
operands: 18
16Operationoperator: 19
operand: 22
17Literal
18ExprTuple38, 21
19Literal
20ExprTuple22
21Operationoperator: 23
operands: 24
22Lambdaparameters: 44
body: 25
23Literal
24ExprTuple45, 26
25Conditionalvalue: 27
condition: 28
26Variable
27Operationoperator: 43
operands: 29
28Operationoperator: 30
operands: 31
29ExprTuple32, 33
30Literal
31ExprTuple34, 35, 36
32Operationoperator: 38
operand: 46
33Operationoperator: 38
operand: 47
34Operationoperator: 41
operands: 40
35Operationoperator: 41
operands: 42
36Operationoperator: 43
operands: 44
37ExprTuple46
38Variable
39ExprTuple47
40ExprTuple46, 45
41Literal
42ExprTuple47, 45
43Literal
44ExprTuple46, 47
45Variable
46Variable
47Variable