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 a, b, f, fb, fx, x
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Integer, Interval, Less, Sum, one, subtract

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, b)), Add(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, subtract(b, one))), fb)), domain = Integer, condition = Less(a, b)))

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}~\left[\forall_{a, b \in \mathbb{Z}~|~a < b}~\left(\left(\sum_{x = a}^{b} f\left(x\right)\right) = \left(\left(\sum_{x = a}^{b - 1} f\left(x\right)\right) + f\left(b\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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 41 body: 4
3	ExprTuple	41
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameters: 40 body: 8
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operands: 12
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	17, 18, 19
15	Operation	operator: 30 operand: 25
16	Operation	operator: 51 operands: 21
17	Operation	operator: 43 operands: 22
18	Operation	operator: 43 operands: 23
19	Operation	operator: 24 operands: 40
20	ExprTuple	25
21	ExprTuple	26, 27
22	ExprTuple	49, 28
23	ExprTuple	53, 28
24	Literal
25	Lambda	parameter: 45 body: 29
26	Operation	operator: 30 operand: 34
27	Operation	operator: 41 operand: 53
28	Literal
29	Conditional	value: 38 condition: 33
30	Literal
31	ExprTuple	34
32	ExprTuple	53
33	Operation	operator: 43 operands: 35
34	Lambda	parameter: 45 body: 36
35	ExprTuple	45, 37
36	Conditional	value: 38 condition: 39
37	Operation	operator: 47 operands: 40
38	Operation	operator: 41 operand: 45
39	Operation	operator: 43 operands: 44
40	ExprTuple	49, 53
41	Variable
42	ExprTuple	45
43	Literal
44	ExprTuple	45, 46
45	Variable
46	Operation	operator: 47 operands: 48
47	Literal
48	ExprTuple	49, 50
49	Variable
50	Operation	operator: 51 operands: 52
51	Literal
52	ExprTuple	53, 54
53	Variable
54	Operation	operator: 55 operand: 57
55	Literal
56	ExprTuple	57
57	Literal