Expression of type Lambda

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

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

# build up the expression from sub-expressions
expr = Lambda([A, B], 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:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(A, B\right) \mapsto \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]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 22 body: 1
1	Operation	operator: 17 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 36 body: 5
4	ExprTuple	36
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Operation	operator: 39 operands: 10
9	Operation	operator: 28 operands: 11
10	ExprTuple	36, 12
11	ExprTuple	13, 14
12	Operation	operator: 15 operands: 22
13	Operation	operator: 39 operands: 16
14	Operation	operator: 17 operand: 20
15	Literal
16	ExprTuple	36, 19
17	Literal
18	ExprTuple	20
19	Operation	operator: 21 operands: 22
20	Lambda	parameters: 42 body: 23
21	Literal
22	ExprTuple	43, 24
23	Conditional	value: 25 condition: 26
24	Variable
25	Operation	operator: 41 operands: 27
26	Operation	operator: 28 operands: 29
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	32, 33, 34
30	Operation	operator: 36 operand: 44
31	Operation	operator: 36 operand: 45
32	Operation	operator: 39 operands: 38
33	Operation	operator: 39 operands: 40
34	Operation	operator: 41 operands: 42
35	ExprTuple	44
36	Variable
37	ExprTuple	45
38	ExprTuple	44, 43
39	Literal
40	ExprTuple	45, 43
41	Literal
42	ExprTuple	44, 45
43	Variable
44	Variable
45	Variable