Expression of type Forall

from the theory of proveit.logic.booleans.disjunction

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 ExprRange, Function, P, Pk, Variable, i, j, k
from proveit.logic import Exists, Forall, Implies, Or
from proveit.numbers import Add, Interval, LessEq, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [P], instance_expr = Implies(Exists(instance_param_or_params = [k], instance_expr = Pk, domain = Interval(i, j)), Or(ExprRange(sub_expr1, Function(P, [sub_expr1]), i, j))), condition = LessEq(i, Add(j, one)))

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_{P~|~i \leq \left(j + 1\right)}~\left(\left[\exists_{k \in \{i~\ldotp \ldotp~j\}}~P\left(k\right)\right] \Rightarrow \left(P\left(i\right) \lor  P\left(i + 1\right) \lor  \ldots \lor  P\left(j\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 32 body: 5
4	ExprTuple	32
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	39, 14
12	Operation	operator: 15 operand: 21
13	Operation	operator: 17 operands: 18
14	Operation	operator: 19 operands: 20
15	Literal
16	ExprTuple	21
17	Literal
18	ExprTuple	22
19	Literal
20	ExprTuple	40, 23
21	Lambda	parameter: 34 body: 24
22	ExprRange	lambda_map: 25 start_index: 39 end_index: 40
23	Literal
24	Conditional	value: 26 condition: 27
25	Lambda	parameter: 36 body: 28
26	Operation	operator: 32 operand: 34
27	Operation	operator: 30 operands: 31
28	Operation	operator: 32 operand: 36
29	ExprTuple	34
30	Literal
31	ExprTuple	34, 35
32	Variable
33	ExprTuple	36
34	Variable
35	Operation	operator: 37 operands: 38
36	Variable
37	Literal
38	ExprTuple	39, 40
39	Variable
40	Variable