Expression of type Forall

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 f, i, j, x
from proveit.core_expr_types import Len, f_i_to_j
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Add, Natural, Neg, two

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [f, x], instance_expr = Forall(instance_param_or_params = [i, j], instance_expr = Equals(Len(operands = [f_i_to_j, x]), Add(j, Neg(i), two)), condition = InSet(Len(operands = [f_i_to_j]), Natural)))

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())

\forall_{f, x}~\left[\forall_{i, j~|~|\left(f\left(i\right), f\left(i + 1\right), \ldots, f\left(j\right)\right)| \in \mathbb{N}}~\left(|\left(f\left(i\right), f\left(i + 1\right), \ldots, f\left(j\right), x\right)| = \left(j - i + 2\right)\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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