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 A, B, Conditional, Lambda, P, Q, n
from proveit.logic import Forall, Implies, InSet, SubsetEq
from proveit.logic.booleans.quantification import general_exists_in_A_Px_st_Qx, general_exists_in_B_Py_st_Qy
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [P, Q, A, B], instance_expr = Implies(general_exists_in_A_Px_st_Qx, general_exists_in_B_Py_st_Qy).with_wrapping_at(2), condition = SubsetEq(A, B)), 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, A, B~|~A \subseteq B}~\left(\begin{array}{c} \begin{array}{l} \left[\exists_{x_{1}, x_{2}, \ldots, x_{n} \in A~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left[\exists_{y_{1}, y_{2}, \ldots, y_{n} \in B~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\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: 54 body: 2
1	ExprTuple	54
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 50 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	54, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	32, 42, 55, 57
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operands: 15
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	55, 57
18	Operation	operator: 21 operand: 23
19	Operation	operator: 21 operand: 24
20	ExprTuple	23
21	Literal
22	ExprTuple	24
23	Lambda	parameters: 40 body: 25
24	Lambda	parameters: 43 body: 26
25	Conditional	value: 27 condition: 28
26	Conditional	value: 29 condition: 30
27	Operation	operator: 32 operands: 40
28	Operation	operator: 33 operands: 31
29	Operation	operator: 32 operands: 43
30	Operation	operator: 33 operands: 34
31	ExprTuple	35, 36
32	Variable
33	Literal
34	ExprTuple	37, 38
35	ExprRange	lambda_map: 39 start_index: 53 end_index: 54
36	Operation	operator: 42 operands: 40
37	ExprRange	lambda_map: 41 start_index: 53 end_index: 54
38	Operation	operator: 42 operands: 43
39	Lambda	parameter: 62 body: 44
40	ExprTuple	45
41	Lambda	parameter: 62 body: 46
42	Variable
43	ExprTuple	47
44	Operation	operator: 50 operands: 48
45	ExprRange	lambda_map: 49 start_index: 53 end_index: 54
46	Operation	operator: 50 operands: 51
47	ExprRange	lambda_map: 52 start_index: 53 end_index: 54
48	ExprTuple	56, 55
49	Lambda	parameter: 62 body: 56
50	Literal
51	ExprTuple	58, 57
52	Lambda	parameter: 62 body: 58
53	Literal
54	Variable
55	Variable
56	IndexedVar	variable: 59 index: 62
57	Variable
58	IndexedVar	variable: 60 index: 62
59	Variable
60	Variable
61	ExprTuple	62
62	Variable