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
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Interval, eight, nine, one, subtract, two

# build up the expression from sub-expressions
sub_expr1 = subtract(k, two)
sub_expr2 = Conditional(sub_expr1, InSet(k, Interval(nine, nine)))
sub_expr3 = Conditional(sub_expr1, InSet(k, Interval(Add(eight, one), nine)))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr3, sub_expr2)), Equals(Lambda(k, sub_expr3), Lambda(k, sub_expr2)).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 - 2 \textrm{ if } k \in \{8 + 1~\ldotp \ldotp~9\}\right.. = \left\{k - 2 \textrm{ if } k \in \{9~\ldotp \ldotp~9\}\right..\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[k \mapsto \left\{k - 2 \textrm{ if } k \in \{8 + 1~\ldotp \ldotp~9\}\right..\right] =  \\ \left[k \mapsto \left\{k - 2 \textrm{ if } k \in \{9~\ldotp \ldotp~9\}\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: 26 body: 11
9	Lambda	parameter: 26 body: 15
10	Lambda	parameter: 26 body: 16
11	Operation	operator: 13 operands: 14
12	ExprTuple	26
13	Literal
14	ExprTuple	15, 16
15	Conditional	value: 18 condition: 17
16	Conditional	value: 18 condition: 19
17	Operation	operator: 22 operands: 20
18	Operation	operator: 36 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	26, 24
21	ExprTuple	26, 25
22	Literal
23	ExprTuple	26, 27
24	Operation	operator: 31 operands: 28
25	Operation	operator: 29 operand: 34
26	Variable
27	Operation	operator: 31 operands: 32
28	ExprTuple	33, 35
29	Literal
30	ExprTuple	34
31	Literal
32	ExprTuple	35, 35
33	Operation	operator: 36 operands: 37
34	Literal
35	Literal
36	Literal
37	ExprTuple	38, 39
38	Literal
39	Literal