Expression of type Conditional

from the theory of proveit.numbers.multiplication

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, i, j, x, y
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import And, InSet
from proveit.numbers import LessEq, Mult, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(LessEq(Mult(a_1_to_i, x, b_1_to_j), Mult(a_1_to_i, y, b_1_to_j)), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), RealNonNeg), one, i), ExprRange(sub_expr1, InSet(IndexedVar(b, sub_expr1), RealNonNeg), one, j), LessEq(x, y)))

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\{\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot x\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right) \leq \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot y\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\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(b_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(b_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(b_{j} \in \mathbb{R}^{\ge 0}\right) ,  x \leq y\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: 16 operands: 3
2	Operation	operator: 4 operands: 5
3	ExprTuple	6, 7
4	Literal
5	ExprTuple	8, 9, 10
6	Operation	operator: 12 operands: 11
7	Operation	operator: 12 operands: 13
8	ExprRange	lambda_map: 14 start_index: 27 end_index: 25
9	ExprRange	lambda_map: 15 start_index: 27 end_index: 28
10	Operation	operator: 16 operands: 17
11	ExprTuple	18, 22, 19
12	Literal
13	ExprTuple	18, 23, 19
14	Lambda	parameter: 38 body: 20
15	Lambda	parameter: 38 body: 21
16	Literal
17	ExprTuple	22, 23
18	ExprRange	lambda_map: 24 start_index: 27 end_index: 25
19	ExprRange	lambda_map: 26 start_index: 27 end_index: 28
20	Operation	operator: 30 operands: 29
21	Operation	operator: 30 operands: 31
22	Variable
23	Variable
24	Lambda	parameter: 38 body: 32
25	Variable
26	Lambda	parameter: 38 body: 33
27	Literal
28	Variable
29	ExprTuple	32, 34
30	Literal
31	ExprTuple	33, 34
32	IndexedVar	variable: 35 index: 38
33	IndexedVar	variable: 36 index: 38
34	Literal
35	Variable
36	Variable
37	ExprTuple	38
38	Variable