Expression of type Forall

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 S, a, b, f, fx, x
from proveit.logic import Forall, 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 = Forall(instance_param_or_params = [S], instance_expr = Forall(instance_param_or_params = [f], instance_expr = 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)]), domain = MonDecFuncs(S), condition = SubsetEq(S, Real)))

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())

\forall_{S}~\left[\forall_{f \in \textrm{MonDecFuncs}\left(S\right)~|~S \subseteq \mathbb{R}}~\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)\right]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 46 body: 3
3	Operation	operator: 10 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 53 body: 7
6	ExprTuple	53
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 13
9	Operation	operator: 24 operands: 12
10	Literal
11	ExprTuple	13
12	ExprTuple	14, 15
13	Lambda	parameters: 61 body: 16
14	Operation	operator: 55 operands: 17
15	Operation	operator: 41 operands: 18
16	Conditional	value: 19 condition: 20
17	ExprTuple	53, 21
18	ExprTuple	46, 22
19	Operation	operator: 40 operands: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 26 operand: 46
22	Literal
23	ExprTuple	28, 29
24	Literal
25	ExprTuple	30, 31, 32, 33
26	Literal
27	ExprTuple	46
28	Operation	operator: 34 operand: 43
29	Operation	operator: 36 operand: 44
30	Operation	operator: 55 operands: 38
31	Operation	operator: 55 operands: 39
32	Operation	operator: 40 operands: 61
33	Operation	operator: 41 operands: 42
34	Literal
35	ExprTuple	43
36	Literal
37	ExprTuple	44
38	ExprTuple	68, 45
39	ExprTuple	65, 45
40	Literal
41	Literal
42	ExprTuple	59, 46
43	Lambda	parameter: 58 body: 47
44	Lambda	parameter: 58 body: 48
45	Literal
46	Variable
47	Conditional	value: 50 condition: 49
48	Conditional	value: 50 condition: 51
49	Operation	operator: 55 operands: 52
50	Operation	operator: 53 operand: 58
51	Operation	operator: 55 operands: 56
52	ExprTuple	58, 57
53	Variable
54	ExprTuple	58
55	Literal
56	ExprTuple	58, 59
57	Operation	operator: 60 operands: 61
58	Variable
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	68, 65
62	Literal
63	ExprTuple	64, 65
64	Operation	operator: 66 operands: 67
65	Variable
66	Literal
67	ExprTuple	68, 69
68	Variable
69	Operation	operator: 70 operand: 72
70	Literal
71	ExprTuple	72
72	Literal