Expression of type Forall

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

# build up the expression from sub-expressions
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:

# 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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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