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, fa, fx, x
from proveit.logic import Forall, SubsetEq
from proveit.numbers import Add, Integer, Integrate, Interval, IntervalCC, LessEq, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = IntervalCC(a, b)
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)])

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_{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)

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 49 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 22 operands: 7
6	Operation	operator: 8 operands: 9
7	ExprTuple	10, 11
8	Literal
9	ExprTuple	12, 13, 14, 15
10	Operation	operator: 16 operand: 25
11	Operation	operator: 18 operands: 19
12	Operation	operator: 44 operands: 20
13	Operation	operator: 44 operands: 21
14	Operation	operator: 22 operands: 49
15	Operation	operator: 23 operands: 24
16	Literal
17	ExprTuple	25
18	Literal
19	ExprTuple	26, 27
20	ExprTuple	50, 28
21	ExprTuple	51, 28
22	Literal
23	Literal
24	ExprTuple	47, 29
25	Lambda	parameter: 46 body: 30
26	Operation	operator: 42 operand: 50
27	Operation	operator: 32 operand: 35
28	Literal
29	Variable
30	Conditional	value: 39 condition: 34
31	ExprTuple	50
32	Literal
33	ExprTuple	35
34	Operation	operator: 44 operands: 36
35	Lambda	parameter: 46 body: 37
36	ExprTuple	46, 38
37	Conditional	value: 39 condition: 40
38	Operation	operator: 41 operands: 49
39	Operation	operator: 42 operand: 46
40	Operation	operator: 44 operands: 45
41	Literal
42	Variable
43	ExprTuple	46
44	Literal
45	ExprTuple	46, 47
46	Variable
47	Operation	operator: 48 operands: 49
48	Literal
49	ExprTuple	50, 51
50	Variable
51	Variable