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

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
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))

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_{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]

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