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, 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(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)))

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[\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)

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