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, fx, x
from proveit.logic import And, Forall, InSet, SubsetEq
from proveit.numbers import Integer, Integrate, Interval, IntervalCC, LessEq, MonDecFuncs, Real, Sum, one, subtract

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(subtract(a, one), 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)), 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 - 1,b\right] \subseteq S}~\left(\left(\sum_{x = a}^{b} f\left(x\right)\right) \leq \left(\int_{a - 1}^{b} f\left(x\right)\,dx\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: 54 body: 9
7	Operation	operator: 48 operands: 10
8	Operation	operator: 34 operands: 11
9	Conditional	value: 12 condition: 13
10	ExprTuple	46, 14
11	ExprTuple	39, 15
12	Operation	operator: 33 operands: 16
13	Operation	operator: 17 operands: 18
14	Operation	operator: 19 operand: 39
15	Literal
16	ExprTuple	21, 22
17	Literal
18	ExprTuple	23, 24, 25, 26
19	Literal
20	ExprTuple	39
21	Operation	operator: 27 operand: 36
22	Operation	operator: 29 operand: 37
23	Operation	operator: 48 operands: 31
24	Operation	operator: 48 operands: 32
25	Operation	operator: 33 operands: 54
26	Operation	operator: 34 operands: 35
27	Literal
28	ExprTuple	36
29	Literal
30	ExprTuple	37
31	ExprTuple	61, 38
32	ExprTuple	58, 38
33	Literal
34	Literal
35	ExprTuple	52, 39
36	Lambda	parameter: 51 body: 40
37	Lambda	parameter: 51 body: 41
38	Literal
39	Variable
40	Conditional	value: 43 condition: 42
41	Conditional	value: 43 condition: 44
42	Operation	operator: 48 operands: 45
43	Operation	operator: 46 operand: 51
44	Operation	operator: 48 operands: 49
45	ExprTuple	51, 50
46	Variable
47	ExprTuple	51
48	Literal
49	ExprTuple	51, 52
50	Operation	operator: 53 operands: 54
51	Variable
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	61, 58
55	Literal
56	ExprTuple	57, 58
57	Operation	operator: 59 operands: 60
58	Variable
59	Literal
60	ExprTuple	61, 62
61	Variable
62	Operation	operator: 63 operand: 65
63	Literal
64	ExprTuple	65
65	Literal

Expression of type Conditional¶

from the theory of proveit.numbers.summation¶

Expression of type Conditional ¶

from the theory of proveit.numbers.summation ¶