Expression of type Forall ¶

from the theory of proveit.logic.sets.comprehension ¶

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 Q, R, f, n
from proveit.core_expr_types import Q__x_1_to_n, R__x_1_to_n, S_1_to_n, f__x_1_to_n, x_1_to_n
from proveit.logic import Forall, Implies, SetOfAll, SubsetEq
from proveit.logic.sets import general_comprehension_fyn
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
sub_expr2 = [S_1_to_n]
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [f, Q, R, S_1_to_n], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Q__x_1_to_n, domains = sub_expr2, condition = R__x_1_to_n), SubsetEq(SetOfAll(instance_param_or_params = sub_expr1, instance_element = f__x_1_to_n, domains = sub_expr2, condition = R__x_1_to_n), general_comprehension_fyn).with_wrapping_at(2)).with_wrapping_at(2)), domain = NaturalPos)

# 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_{f, Q, R, S_{1}, S_{2}, \ldots, S_{n}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{\left(x_{1} \in S_{1}\right), \left(x_{2} \in S_{2}\right), \ldots, \left(x_{n} \in S_{n}\right)~|~R\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left\{f\left(x_{1}, x_{2}, \ldots, x_{n}\right)~|~R\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right\}_{\left(x_{1} \in S_{1}\right), \left(x_{2} \in S_{2}\right), \ldots, \left(x_{n} \in S_{n}\right)} \subseteq  \\ \left\{f\left(y_{1}, y_{2}, \ldots, y_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right\}_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)} \end{array} \end{array}\right) \end{array} \end{array}\right)\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)

	core type	sub-expressions
0	Operation	operator: 19 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 63 body: 4
3	ExprTuple	63
4	Conditional	value: 5 condition: 6
5	Operation	operator: 19 operand: 9
6	Operation	operator: 59 operands: 8
7	ExprTuple	9
8	ExprTuple	63, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	40, 51, 48, 13
12	Operation	operator: 14 operands: 15
13	ExprRange	lambda_map: 16 start_index: 62 end_index: 63
14	Literal
15	ExprTuple	17, 18
16	Lambda	parameter: 71 body: 65
17	Operation	operator: 19 operand: 23
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23
21	Literal
22	ExprTuple	24, 25
23	Lambda	parameters: 49 body: 26
24	Operation	operator: 28 operand: 31
25	Operation	operator: 28 operand: 32
26	Conditional	value: 30 condition: 36
27	ExprTuple	31
28	Literal
29	ExprTuple	32
30	Operation	operator: 51 operands: 49
31	Lambda	parameters: 49 body: 33
32	Lambda	parameters: 52 body: 34
33	Conditional	value: 35 condition: 36
34	Conditional	value: 37 condition: 38
35	Operation	operator: 40 operands: 49
36	Operation	operator: 41 operands: 39
37	Operation	operator: 40 operands: 52
38	Operation	operator: 41 operands: 42
39	ExprTuple	43, 44
40	Variable
41	Literal
42	ExprTuple	45, 46
43	ExprRange	lambda_map: 47 start_index: 62 end_index: 63
44	Operation	operator: 48 operands: 49
45	ExprRange	lambda_map: 50 start_index: 62 end_index: 63
46	Operation	operator: 51 operands: 52
47	Lambda	parameter: 71 body: 53
48	Variable
49	ExprTuple	54
50	Lambda	parameter: 71 body: 55
51	Variable
52	ExprTuple	56
53	Operation	operator: 59 operands: 57
54	ExprRange	lambda_map: 58 start_index: 62 end_index: 63
55	Operation	operator: 59 operands: 60
56	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
57	ExprTuple	64, 65
58	Lambda	parameter: 71 body: 64
59	Literal
60	ExprTuple	66, 65
61	Lambda	parameter: 71 body: 66
62	Literal
63	Variable
64	IndexedVar	variable: 67 index: 71
65	IndexedVar	variable: 68 index: 71
66	IndexedVar	variable: 69 index: 71
67	Variable
68	Variable
69	Variable
70	ExprTuple	71
71	Variable

Expression of type Forall¶

from the theory of proveit.logic.sets.comprehension¶

Expression of type Forall ¶

from the theory of proveit.logic.sets.comprehension ¶