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 = [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_{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)

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