Expression of type Lambda

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, Lambda, 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
sub_expr1 = [n]
sub_expr2 = Forall(instance_param_or_params = [x_1_to_n], instance_expr = P__x_1_to_n)
expr = Lambda(P, Implies(And(Function(P, []), Forall(instance_param_or_params = sub_expr1, instance_expr = Implies(sub_expr2, Forall(instance_param_or_params = [x_1_to_np1], instance_expr = P__x_1_to_np1)).with_wrapping_at(2), domain = Natural)), Forall(instance_param_or_params = sub_expr1, instance_expr = sub_expr2, domain = Natural)).with_wrapping_at(1))

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

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 35 body: 2
1	ExprTuple	35
2	Operation	operator: 20 operands: 3
3	ExprTuple	4, 5
4	Operation	operator: 6 operands: 7
5	Operation	operator: 28 operand: 11
6	Literal
7	ExprTuple	9, 10
8	ExprTuple	11
9	Operation	operator: 35 operands: 12
10	Operation	operator: 28 operand: 15
11	Lambda	parameter: 46 body: 14
12	ExprTuple
13	ExprTuple	15
14	Conditional	value: 24 condition: 19
15	Lambda	parameter: 46 body: 17
16	ExprTuple	46
17	Conditional	value: 18 condition: 19
18	Operation	operator: 20 operands: 21
19	Operation	operator: 22 operands: 23
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	46, 26
24	Operation	operator: 28 operand: 30
25	Operation	operator: 28 operand: 31
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	31
30	Lambda	parameters: 34 body: 32
31	Lambda	parameters: 36 body: 33
32	Operation	operator: 35 operands: 34
33	Operation	operator: 35 operands: 36
34	ExprTuple	37
35	Variable
36	ExprTuple	38
37	ExprRange	lambda_map: 39 start_index: 47 end_index: 46
38	ExprRange	lambda_map: 39 start_index: 47 end_index: 40
39	Lambda	parameter: 48 body: 41
40	Operation	operator: 42 operands: 43
41	IndexedVar	variable: 44 index: 48
42	Literal
43	ExprTuple	46, 47
44	Variable
45	ExprTuple	48
46	Variable
47	Literal
48	Variable