Expression of type Forall

from the theory of proveit.numbers.addition

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 b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import Forall, InSet
from proveit.numbers import Add, Less, Natural, RationalNeg, RationalNonPos, zero

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_j], instance_expr = InSet(Add(a_1_to_i, b, c_1_to_j), RationalNeg), domain = RationalNonPos, condition = Less(b, zero)), domain = Natural)

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_{i, j \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{Q}^{\leq 0}~|~b < 0}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) \in \mathbb{Q}^-\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	parameters: 3 body: 4
3	ExprTuple	42, 45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 20 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 29 body: 13
11	Operation	operator: 47 operands: 14
12	Operation	operator: 47 operands: 15
13	Conditional	value: 16 condition: 17
14	ExprTuple	42, 18
15	ExprTuple	45, 18
16	Operation	operator: 47 operands: 19
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	24, 25, 26, 27
22	Operation	operator: 28 operands: 29
23	Literal
24	ExprRange	lambda_map: 30 start_index: 44 end_index: 42
25	Operation	operator: 47 operands: 31
26	ExprRange	lambda_map: 32 start_index: 44 end_index: 45
27	Operation	operator: 33 operands: 34
28	Literal
29	ExprTuple	35, 39, 36
30	Lambda	parameter: 55 body: 37
31	ExprTuple	39, 51
32	Lambda	parameter: 55 body: 38
33	Literal
34	ExprTuple	39, 40
35	ExprRange	lambda_map: 41 start_index: 44 end_index: 42
36	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
37	Operation	operator: 47 operands: 46
38	Operation	operator: 47 operands: 48
39	Variable
40	Literal
41	Lambda	parameter: 55 body: 49
42	Variable
43	Lambda	parameter: 55 body: 50
44	Literal
45	Variable
46	ExprTuple	49, 51
47	Literal
48	ExprTuple	50, 51
49	IndexedVar	variable: 52 index: 55
50	IndexedVar	variable: 53 index: 55
51	Literal
52	Variable
53	Variable
54	ExprTuple	55
55	Variable