Expression of type Conditional ¶

from the theory of proveit.numbers.summation ¶

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, S, a, b, f, fa, fx, x
from proveit.logic import And, Forall, InSet, SubsetEq
from proveit.numbers import Add, Integer, Integrate, Interval, IntervalCC, LessEq, MonDecFuncs, Real, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(a, b)
expr = Conditional(Forall(instance_param_or_params = [a, b], instance_expr = LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, b)), Add(fa, Integrate(index_or_indices = sub_expr1, integrand = fx, domain = sub_expr2))), domain = Integer, conditions = [LessEq(a, b), SubsetEq(sub_expr2, S)]), And(InSet(f, MonDecFuncs(S)), SubsetEq(S, Real)))

# 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, b \in \mathbb{Z}~|~a \leq b, \left[a,b\right] \subseteq S}~\left(\left(\sum_{x = a}^{b} f\left(x\right)\right) \leq \left(f\left(a\right) + \left(\int_{a}^{b} f\left(x\right)\,dx\right)\right)\right) \textrm{ if } f \in \textrm{MonDecFuncs}\left(S\right) ,  S \subseteq \mathbb{R}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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

Expression of type Conditional¶

from the theory of proveit.numbers.summation¶

Expression of type Conditional ¶

from the theory of proveit.numbers.summation ¶