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, fa, fx, x
from proveit.logic import Forall, 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 = 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)), Add(fa, 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_{f \in \textrm{MonDecFuncs}\left(S\right)~|~S \subseteq \mathbb{R}}~\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)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 57 body: 4
3	ExprTuple	57
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 21 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 64 body: 13
11	Operation	operator: 59 operands: 14
12	Operation	operator: 38 operands: 15
13	Conditional	value: 16 condition: 17
14	ExprTuple	57, 18
15	ExprTuple	44, 19
16	Operation	operator: 37 operands: 20
17	Operation	operator: 21 operands: 22
18	Operation	operator: 23 operand: 44
19	Literal
20	ExprTuple	25, 26
21	Literal
22	ExprTuple	27, 28, 29, 30
23	Literal
24	ExprTuple	44
25	Operation	operator: 31 operand: 40
26	Operation	operator: 33 operands: 34
27	Operation	operator: 59 operands: 35
28	Operation	operator: 59 operands: 36
29	Operation	operator: 37 operands: 64
30	Operation	operator: 38 operands: 39
31	Literal
32	ExprTuple	40
33	Literal
34	ExprTuple	41, 42
35	ExprTuple	65, 43
36	ExprTuple	66, 43
37	Literal
38	Literal
39	ExprTuple	62, 44
40	Lambda	parameter: 61 body: 45
41	Operation	operator: 57 operand: 65
42	Operation	operator: 47 operand: 50
43	Literal
44	Variable
45	Conditional	value: 54 condition: 49
46	ExprTuple	65
47	Literal
48	ExprTuple	50
49	Operation	operator: 59 operands: 51
50	Lambda	parameter: 61 body: 52
51	ExprTuple	61, 53
52	Conditional	value: 54 condition: 55
53	Operation	operator: 56 operands: 64
54	Operation	operator: 57 operand: 61
55	Operation	operator: 59 operands: 60
56	Literal
57	Variable
58	ExprTuple	61
59	Literal
60	ExprTuple	61, 62
61	Variable
62	Operation	operator: 63 operands: 64
63	Literal
64	ExprTuple	65, 66
65	Variable
66	Variable