Expression of type And

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 And, Equals, Forall, TRUE
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 = And(TRUE, 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())

\top \land \left[\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)\right]

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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