Expression of type Lambda

from the theory of proveit.logic.sets.cardinality

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 Conditional, Lambda, S, x
from proveit.logic import And, Card, Equals, InSet, NotInSet, Set, Union
from proveit.numbers import Add, Natural, one

# build up the expression from sub-expressions
sub_expr1 = Card(S)
expr = Lambda([x, S], Conditional(Equals(Card(Union(S, Set(x))), Add(sub_expr1, one)), And(InSet(sub_expr1, Natural), NotInSet(x, S))))

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(x, S\right) \mapsto \left\{|S \cup \left\{x\right\}| = \left(|S| + 1\right) \textrm{ if } |S| \in \mathbb{N} ,  x \notin S\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 18 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 6 operands: 7
4	Literal
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 25 operand: 19
9	Operation	operator: 13 operands: 14
10	Operation	operator: 15 operands: 16
11	Operation	operator: 17 operands: 18
12	ExprTuple	19
13	Literal
14	ExprTuple	21, 20
15	Literal
16	ExprTuple	21, 22
17	Literal
18	ExprTuple	31, 28
19	Operation	operator: 23 operands: 24
20	Literal
21	Operation	operator: 25 operand: 28
22	Literal
23	Literal
24	ExprTuple	28, 27
25	Literal
26	ExprTuple	28
27	Operation	operator: 29 operand: 31
28	Variable
29	Literal
30	ExprTuple	31
31	Variable