Expression of type Lambda

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

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, P, Q, R, n
from proveit.logic import Forall, Implies, InSet
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy, general_exists_Rz_st_Qz, general_forall_st_Qx__Px_implies_Rx
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [P, Q, R], instance_expr = Implies(general_forall_st_Qx__Px_implies_Rx, Implies(general_exists_Py_st_Qy, general_exists_Rz_st_Qz).with_wrapping_at(2)).with_wrapping_at(2)), InSet(n, NaturalPos)))

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())

n \mapsto \left\{\forall_{P, Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~\left(P\left(x_{1}, x_{2}, \ldots, x_{n}\right) \Rightarrow R\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right] \Rightarrow  \\ \left[\exists_{z_{1}, z_{2}, \ldots, z_{n}~|~Q\left(z_{1}, z_{2}, \ldots, z_{n}\right)}~R\left(z_{1}, z_{2}, \ldots, z_{n}\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right) \textrm{ if } n \in \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: 52 body: 2
1	ExprTuple	52
2	Conditional	value: 3 condition: 4
3	Operation	operator: 15 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	52, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	40, 43, 42
11	Operation	operator: 29 operands: 12
12	ExprTuple	13, 14
13	Operation	operator: 15 operand: 18
14	Operation	operator: 29 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	19, 20
18	Lambda	parameters: 39 body: 21
19	Operation	operator: 23 operand: 27
20	Operation	operator: 23 operand: 28
21	Conditional	value: 25 condition: 26
22	ExprTuple	27
23	Literal
24	ExprTuple	28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 43 operands: 39
27	Lambda	parameters: 41 body: 31
28	Lambda	parameters: 44 body: 32
29	Literal
30	ExprTuple	33, 34
31	Conditional	value: 35 condition: 36
32	Conditional	value: 37 condition: 38
33	Operation	operator: 40 operands: 39
34	Operation	operator: 42 operands: 39
35	Operation	operator: 40 operands: 41
36	Operation	operator: 43 operands: 41
37	Operation	operator: 42 operands: 44
38	Operation	operator: 43 operands: 44
39	ExprTuple	45
40	Variable
41	ExprTuple	46
42	Variable
43	Variable
44	ExprTuple	47
45	ExprRange	lambda_map: 48 start_index: 51 end_index: 52
46	ExprRange	lambda_map: 49 start_index: 51 end_index: 52
47	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
48	Lambda	parameter: 60 body: 53
49	Lambda	parameter: 60 body: 54
50	Lambda	parameter: 60 body: 55
51	Literal
52	Variable
53	IndexedVar	variable: 56 index: 60
54	IndexedVar	variable: 57 index: 60
55	IndexedVar	variable: 58 index: 60
56	Variable
57	Variable
58	Variable
59	ExprTuple	60
60	Variable