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 = [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:

# 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_{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]

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: 19 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 24 body: 3
3	Operation	operator: 19 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 38 body: 7
6	ExprTuple	38
7	Operation	operator: 8 operands: 9
8	Literal
9	ExprTuple	10, 11
10	Operation	operator: 41 operands: 12
11	Operation	operator: 30 operands: 13
12	ExprTuple	38, 14
13	ExprTuple	15, 16
14	Operation	operator: 17 operands: 24
15	Operation	operator: 41 operands: 18
16	Operation	operator: 19 operand: 22
17	Literal
18	ExprTuple	38, 21
19	Literal
20	ExprTuple	22
21	Operation	operator: 23 operands: 24
22	Lambda	parameters: 44 body: 25
23	Literal
24	ExprTuple	45, 26
25	Conditional	value: 27 condition: 28
26	Variable
27	Operation	operator: 43 operands: 29
28	Operation	operator: 30 operands: 31
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35, 36
32	Operation	operator: 38 operand: 46
33	Operation	operator: 38 operand: 47
34	Operation	operator: 41 operands: 40
35	Operation	operator: 41 operands: 42
36	Operation	operator: 43 operands: 44
37	ExprTuple	46
38	Variable
39	ExprTuple	47
40	ExprTuple	46, 45
41	Literal
42	ExprTuple	47, 45
43	Literal
44	ExprTuple	46, 47
45	Variable
46	Variable
47	Variable