Expression of type Implies

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 m, n
from proveit.logic import And, Forall, Implies
from proveit.numbers import Natural, P_mAddOne, Pm, Pn, Pzero

# build up the expression from sub-expressions
expr = Implies(And(Pzero, Forall(instance_param_or_params = [m], instance_expr = P_mAddOne, domain = Natural, condition = Pm)), Forall(instance_param_or_params = [n], instance_expr = Pn, domain = 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(P\left(0\right) \land \left[\forall_{m \in \mathbb{N}~|~P\left(m\right)}~P\left(m + 1\right)\right]\right) \Rightarrow \left[\forall_{n \in \mathbb{N}}~P\left(n\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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