Expression of type Conditional

from the theory of proveit.numbers.number_sets.natural_numbers

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, S, x
from proveit.logic import And, Forall, Implies, InSet, SetEquiv, SubsetEq
from proveit.numbers import Add, Natural, one, zero

# build up the expression from sub-expressions
expr = Conditional(Implies(And(InSet(zero, S), Forall(instance_param_or_params = [x], instance_expr = InSet(Add(x, one), S), domain = S)), SetEquiv(S, Natural)), SubsetEq(S, Natural))

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\{\left(\left(0 \in S\right) \land \left[\forall_{x \in S}~\left(\left(x + 1\right) \in S\right)\right]\right) \Rightarrow \left(S \cong \mathbb{N}\right) \textrm{ if } S \subseteq \mathbb{N}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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