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, ExprRange, IndexedVar, Variable, a, b, c, 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, Real, RealNonNeg, greater_eq, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(Forall(instance_param_or_params = [b], instance_expr = greater_eq(Add(a_1_to_i, b, c_1_to_j), b), domain = Real), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), RealNonNeg), one, i), ExprRange(sub_expr1, InSet(IndexedVar(c, sub_expr1), RealNonNeg), one, j)))

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_{b \in \mathbb{R}}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) \geq b\right) \textrm{ if } \left(a_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(a_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(a_{i} \in \mathbb{R}^{\ge 0}\right), \left(c_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(c_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(c_{j} \in \mathbb{R}^{\ge 0}\right)\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: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9
7	Lambda	parameter: 30 body: 11
8	ExprRange	lambda_map: 12 start_index: 35 end_index: 33
9	ExprRange	lambda_map: 13 start_index: 35 end_index: 36
10	ExprTuple	30
11	Conditional	value: 14 condition: 15
12	Lambda	parameter: 42 body: 16
13	Lambda	parameter: 42 body: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 22 operands: 20
16	Operation	operator: 22 operands: 21
17	Operation	operator: 22 operands: 23
18	Literal
19	ExprTuple	30, 24
20	ExprTuple	30, 25
21	ExprTuple	37, 26
22	Literal
23	ExprTuple	38, 26
24	Operation	operator: 27 operands: 28
25	Literal
26	Literal
27	Literal
28	ExprTuple	29, 30, 31
29	ExprRange	lambda_map: 32 start_index: 35 end_index: 33
30	Variable
31	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
32	Lambda	parameter: 42 body: 37
33	Variable
34	Lambda	parameter: 42 body: 38
35	Literal
36	Variable
37	IndexedVar	variable: 39 index: 42
38	IndexedVar	variable: 40 index: 42
39	Variable
40	Variable
41	ExprTuple	42
42	Variable