Expression of type Conditional

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 Conditional, b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import And, Forall, InSet
from proveit.numbers import Add, Less, Natural, Real, RealNeg, greater, zero

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [a_1_to_i, c_1_to_j], instance_expr = Forall(instance_param_or_params = [b], instance_expr = Less(Add(a_1_to_i, b, c_1_to_j), b), domain = Real), domain = RealNeg), And(InSet(i, Natural), InSet(j, Natural), greater(Add(i, j), zero)))

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())

\left\{\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{R}^-}~\left[\forall_{b \in \mathbb{R}}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) < b\right)\right] \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  \left(i + j\right) > 0\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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