Expression of type Forall

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

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
expr = Forall(instance_param_or_params = [n], instance_expr = 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)), domain = 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())

\forall_{n \in \mathbb{N}^+}~\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]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 46 body: 4
3	ExprTuple	46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 38 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	46, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	42, 37
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operand: 20
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20
18	Literal
19	ExprTuple	37, 21
20	Lambda	parameters: 41 body: 22
21	Operation	operator: 23 operand: 42
22	Conditional	value: 25 condition: 26
23	Literal
24	ExprTuple	42
25	Operation	operator: 27 operands: 41
26	Operation	operator: 28 operands: 29
27	Literal
28	Literal
29	ExprTuple	30, 31
30	ExprRange	lambda_map: 32 start_index: 45 end_index: 46
31	Operation	operator: 33 operands: 34
32	Lambda	parameter: 50 body: 35
33	Literal
34	ExprTuple	36, 37
35	Operation	operator: 38 operands: 39
36	Operation	operator: 40 operands: 41
37	Variable
38	Literal
39	ExprTuple	47, 42
40	Literal
41	ExprTuple	43
42	Variable
43	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
44	Lambda	parameter: 50 body: 47
45	Literal
46	Variable
47	IndexedVar	variable: 48 index: 50
48	Variable
49	ExprTuple	50
50	Variable