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

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 \in \left[A \xrightarrow[]{\text{1-to-1}} B\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)\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	parameters: 22 body: 3
3	Operation	operator: 16 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 36 body: 7
6	ExprTuple	36
7	Conditional	value: 8 condition: 9
8	Operation	operator: 28 operands: 10
9	Operation	operator: 39 operands: 11
10	ExprTuple	12, 13
11	ExprTuple	36, 14
12	Operation	operator: 39 operands: 15
13	Operation	operator: 16 operand: 20
14	Operation	operator: 18 operands: 22
15	ExprTuple	36, 19
16	Literal
17	ExprTuple	20
18	Literal
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