Expression of type Implies

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, n
from proveit.logic import And, Equals, Forall, Implies, 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 = Implies(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)))), Forall(instance_param_or_params = [n], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = k, domain = Interval(one, n)), frac(Mult(n, Add(n, one)), two)), domain = NaturalPos))

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())

\left(\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]\right) \Rightarrow \left[\forall_{n \in \mathbb{N}^+}~\left(\left(\sum_{k = 1}^{n} k\right) = \frac{n \cdot \left(n + 1\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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 26 operands: 5
4	Operation	operator: 10 operand: 9
5	ExprTuple	7, 8
6	ExprTuple	9
7	Literal
8	Operation	operator: 10 operand: 14
9	Lambda	parameter: 73 body: 13
10	Literal
11	ExprTuple	14
12	ExprTuple	73
13	Conditional	value: 15 condition: 16
14	Lambda	parameter: 85 body: 18
15	Operation	operator: 39 operands: 19
16	Operation	operator: 75 operands: 20
17	ExprTuple	85
18	Conditional	value: 21 condition: 22
19	ExprTuple	23, 24
20	ExprTuple	73, 45
21	Operation	operator: 39 operands: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 52 operand: 34
24	Operation	operator: 54 operands: 29
25	ExprTuple	30, 31
26	Literal
27	ExprTuple	32, 33
28	ExprTuple	34
29	ExprTuple	35, 71
30	Operation	operator: 52 operand: 43
31	Operation	operator: 54 operands: 37
32	Operation	operator: 75 operands: 38
33	Operation	operator: 39 operands: 40
34	Lambda	parameter: 78 body: 41
35	Operation	operator: 67 operands: 42
36	ExprTuple	43
37	ExprTuple	44, 71
38	ExprTuple	85, 45
39	Literal
40	ExprTuple	46, 47
41	Conditional	value: 78 condition: 48
42	ExprTuple	73, 49
43	Lambda	parameter: 78 body: 50
44	Operation	operator: 67 operands: 51
45	Literal
46	Operation	operator: 52 operand: 60
47	Operation	operator: 54 operands: 55
48	Operation	operator: 75 operands: 56
49	Operation	operator: 80 operands: 57
50	Conditional	value: 78 condition: 58
51	ExprTuple	77, 59
52	Literal
53	ExprTuple	60
54	Literal
55	ExprTuple	61, 71
56	ExprTuple	78, 62
57	ExprTuple	73, 84
58	Operation	operator: 75 operands: 63
59	Operation	operator: 80 operands: 64
60	Lambda	parameter: 78 body: 66
61	Operation	operator: 67 operands: 68
62	Operation	operator: 82 operands: 69
63	ExprTuple	78, 70
64	ExprTuple	85, 71
65	ExprTuple	78
66	Conditional	value: 78 condition: 72
67	Literal
68	ExprTuple	85, 77
69	ExprTuple	84, 73
70	Operation	operator: 82 operands: 74
71	Literal
72	Operation	operator: 75 operands: 76
73	Variable
74	ExprTuple	84, 77
75	Literal
76	ExprTuple	78, 79
77	Operation	operator: 80 operands: 81
78	Variable
79	Operation	operator: 82 operands: 83
80	Literal
81	ExprTuple	85, 84
82	Literal
83	ExprTuple	84, 85
84	Literal
85	Variable