Expression of type Forall

from the theory of proveit.logic.booleans.conjunction

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 And, Forall, Implies
from proveit.numbers import Add, Integer, Interval, LessEq, one

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

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_{i, j \in \mathbb{Z}}~\left[\forall_{P~|~i \leq \left(j + 1\right)}~\left(\left[\forall_{k \in \{i~\ldotp \ldotp~j\}}~P\left(k\right)\right] \Rightarrow \left(P\left(i\right) \land  P\left(i + 1\right) \land  \ldots \land  P\left(j\right)\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 48 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 25 operand: 8
5	Operation	operator: 27 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 42 body: 12
9	Operation	operator: 40 operands: 13
10	Operation	operator: 40 operands: 14
11	ExprTuple	42
12	Conditional	value: 15 condition: 16
13	ExprTuple	49, 17
14	ExprTuple	50, 17
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	Literal
18	Literal
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	49, 24
22	Operation	operator: 25 operand: 31
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	Literal
26	ExprTuple	31
27	Literal
28	ExprTuple	32
29	Literal
30	ExprTuple	50, 33
31	Lambda	parameter: 44 body: 34
32	ExprRange	lambda_map: 35 start_index: 49 end_index: 50
33	Literal
34	Conditional	value: 36 condition: 37
35	Lambda	parameter: 46 body: 38
36	Operation	operator: 42 operand: 44
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operand: 46
39	ExprTuple	44
40	Literal
41	ExprTuple	44, 45
42	Variable
43	ExprTuple	46
44	Variable
45	Operation	operator: 47 operands: 48
46	Variable
47	Literal
48	ExprTuple	49, 50
49	Variable
50	Variable