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 Conditional, Lambda, k, m
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Interval, one, subtract

# build up the expression from sub-expressions
sub_expr1 = Conditional(k, InSet(k, Interval(one, m)))
sub_expr2 = Conditional(k, InSet(k, Interval(one, subtract(Add(m, one), one))))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr2, sub_expr1)), Equals(Lambda(k, sub_expr2), Lambda(k, sub_expr1)).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{k}~\left(\left\{k \textrm{ if } k \in \{1~\ldotp \ldotp~\left(m + 1\right) - 1\}\right.. = \left\{k \textrm{ if } k \in \{1~\ldotp \ldotp~m\}\right..\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[k \mapsto \left\{k \textrm{ if } k \in \{1~\ldotp \ldotp~\left(m + 1\right) - 1\}\right..\right] =  \\ \left[k \mapsto \left\{k \textrm{ if } k \in \{1~\ldotp \ldotp~m\}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(2)	('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: 5 operand: 8
4	Operation	operator: 13 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 23 body: 11
9	Lambda	parameter: 23 body: 15
10	Lambda	parameter: 23 body: 16
11	Operation	operator: 13 operands: 14
12	ExprTuple	23
13	Literal
14	ExprTuple	15, 16
15	Conditional	value: 23 condition: 17
16	Conditional	value: 23 condition: 18
17	Operation	operator: 20 operands: 19
18	Operation	operator: 20 operands: 21
19	ExprTuple	23, 22
20	Literal
21	ExprTuple	23, 24
22	Operation	operator: 26 operands: 25
23	Variable
24	Operation	operator: 26 operands: 27
25	ExprTuple	37, 28
26	Literal
27	ExprTuple	37, 36
28	Operation	operator: 32 operands: 29
29	ExprTuple	30, 31
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operand: 37
32	Literal
33	ExprTuple	36, 37
34	Literal
35	ExprTuple	37
36	Variable
37	Literal