Expression of type Conditional

from the theory of proveit.core_expr_types.tuples

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, f, i, j, m, n
from proveit.core_expr_types import Len
from proveit.core_expr_types.tuples import general_range_of_ranges
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Natural, one, zero

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [f, i, j], instance_expr = Equals(Len(operands = general_range_of_ranges), zero).with_wrapping_at(1)), And(InSet(m, Natural), InSet(n, Natural), Equals(Add(n, one), m)))

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\{\forall_{f, i, j}~\left(\begin{array}{c} \begin{array}{l} |\left(f\left(m, i\left(m\right)\right), f\left(m, i\left(m\right) + 1\right), \ldots, f\left(m, j\left(m\right)\right), f\left(m + 1, i\left(m + 1\right)\right), f\left(m + 1, i\left(m + 1\right) + 1\right), \ldots, f\left(m + 1, j\left(m + 1\right)\right), \ldots\ldots, f\left(n, i\left(n\right)\right), f\left(n, i\left(n\right) + 1\right), \ldots, f\left(n, j\left(n\right)\right)\right)| \\  = 0 \end{array} \end{array}\right) \textrm{ if } m \in \mathbb{N} ,  n \in \mathbb{N} ,  \left(n + 1\right) = m\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9, 10
7	Lambda	parameters: 11 body: 12
8	Operation	operator: 14 operands: 13
9	Operation	operator: 14 operands: 15
10	Operation	operator: 17 operands: 16
11	ExprTuple	41, 38, 39
12	Operation	operator: 17 operands: 18
13	ExprTuple	30, 19
14	Literal
15	ExprTuple	31, 19
16	ExprTuple	20, 30
17	Literal
18	ExprTuple	21, 22
19	Literal
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	Literal
23	Literal
24	ExprTuple	31, 27
25	Literal
26	ExprTuple	28
27	Literal
28	ExprRange	lambda_map: 29 start_index: 30 end_index: 31
29	Lambda	parameter: 43 body: 32
30	Variable
31	Variable
32	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
33	Lambda	parameter: 44 body: 37
34	Operation	operator: 38 operand: 43
35	Operation	operator: 39 operand: 43
36	ExprTuple	44
37	Operation	operator: 41 operands: 42
38	Variable
39	Variable
40	ExprTuple	43
41	Variable
42	ExprTuple	43, 44
43	Variable
44	Variable