Expression of type Lambda

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, Lambda, 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 = Lambda(S, 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())

S \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 30 body: 2
1	ExprTuple	30
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 7 operands: 13
5	Literal
6	ExprTuple	8, 9
7	Literal
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14, 15
12	Literal
13	ExprTuple	30, 16
14	Operation	operator: 27 operands: 17
15	Operation	operator: 18 operand: 21
16	Literal
17	ExprTuple	20, 30
18	Literal
19	ExprTuple	21
20	Literal
21	Lambda	parameter: 33 body: 23
22	ExprTuple	33
23	Conditional	value: 24 condition: 25
24	Operation	operator: 27 operands: 26
25	Operation	operator: 27 operands: 28
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	33, 30
29	Operation	operator: 31 operands: 32
30	Variable
31	Literal
32	ExprTuple	33, 34
33	Variable
34	Literal