Expression of type And

from the theory of proveit.logic.booleans.quantification.universality

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 Function, P, n
from proveit.core_expr_types import P__x_1_to_n, P__x_1_to_np1, x_1_to_n, x_1_to_np1
from proveit.logic import And, Forall, Implies
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = And(Function(P, []), Forall(instance_param_or_params = [n], instance_expr = Implies(Forall(instance_param_or_params = [x_1_to_n], instance_expr = P__x_1_to_n), Forall(instance_param_or_params = [x_1_to_np1], instance_expr = P__x_1_to_np1)).with_wrapping_at(2), 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())

P\left(\right) \land \left[\forall_{n \in \mathbb{N}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left[\forall_{x_{1}, x_{2}, \ldots, x_{n + 1}}~P\left(x_{1}, x_{2}, \ldots, x_{n + 1}\right)\right] \end{array} \end{array}\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',)

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