Expression of type Lambda

from the theory of proveit.logic.sets.cardinality

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, N, S, n
from proveit.core_expr_types import Len, x_1_to_n
from proveit.logic import Card, Distinct, Equals, Exists, Forall, InSet
from proveit.numbers import NaturalPos, greater_eq

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [S, N], instance_expr = Exists(instance_param_or_params = sub_expr1, instance_expr = Distinct(x_1_to_n), domain = S, condition = Equals(Len(operands = sub_expr1), N)), condition = greater_eq(Card(S), N)), 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_{S, N~|~|S| \geq N}~\left[\exists_{x_{1}, x_{2}, \ldots, x_{n} \in S~|~|\left(x_{1}, x_{2}, \ldots, x_{n}\right)| = N}~\textrm{distinct}\left(x_{1}, x_{2}, \ldots, x_{n}\right)\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: 44 body: 2
1	ExprTuple	44
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 36 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	44, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	40, 35
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 18
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18
16	Literal
17	ExprTuple	35, 19
18	Lambda	parameters: 39 body: 20
19	Operation	operator: 21 operand: 40
20	Conditional	value: 23 condition: 24
21	Literal
22	ExprTuple	40
23	Operation	operator: 25 operands: 39
24	Operation	operator: 26 operands: 27
25	Literal
26	Literal
27	ExprTuple	28, 29
28	ExprRange	lambda_map: 30 start_index: 43 end_index: 44
29	Operation	operator: 31 operands: 32
30	Lambda	parameter: 48 body: 33
31	Literal
32	ExprTuple	34, 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	Variable
36	Literal
37	ExprTuple	45, 40
38	Literal
39	ExprTuple	41
40	Variable
41	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
42	Lambda	parameter: 48 body: 45
43	Literal
44	Variable
45	IndexedVar	variable: 46 index: 48
46	Variable
47	ExprTuple	48
48	Variable