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 k, m
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Interval, Mult, NaturalPos, Sum, frac, one, two

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Add(m, one)
expr = Forall(instance_param_or_params = [m], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = k, domain = Interval(one, sub_expr2)), frac(Mult(sub_expr2, Add(m, two)), two)), domain = NaturalPos, condition = Equals(Sum(index_or_indices = sub_expr1, summand = k, domain = Interval(one, m)), frac(Mult(m, sub_expr2), two)))

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