Expression of type Lambda

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 Conditional, ExprRange, Function, Lambda, P, Pk, Variable, i, j, k
from proveit.logic import And, Forall, Implies, InSet
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 = Lambda([i, j], Conditional(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))), And(InSet(i, Integer), InSet(j, 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())

\left(i, j\right) \mapsto \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) \textrm{ if } i \in \mathbb{Z} ,  j \in \mathbb{Z}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 46 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 23 operand: 6
3	Operation	operator: 25 operands: 5
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameter: 40 body: 10
7	Operation	operator: 38 operands: 11
8	Operation	operator: 38 operands: 12
9	ExprTuple	40
10	Conditional	value: 13 condition: 14
11	ExprTuple	47, 15
12	ExprTuple	48, 15
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operands: 19
15	Literal
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	47, 22
20	Operation	operator: 23 operand: 29
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	Literal
24	ExprTuple	29
25	Literal
26	ExprTuple	30
27	Literal
28	ExprTuple	48, 31
29	Lambda	parameter: 42 body: 32
30	ExprRange	lambda_map: 33 start_index: 47 end_index: 48
31	Literal
32	Conditional	value: 34 condition: 35
33	Lambda	parameter: 44 body: 36
34	Operation	operator: 40 operand: 42
35	Operation	operator: 38 operands: 39
36	Operation	operator: 40 operand: 44
37	ExprTuple	42
38	Literal
39	ExprTuple	42, 43
40	Variable
41	ExprTuple	44
42	Variable
43	Operation	operator: 45 operands: 46
44	Variable
45	Literal
46	ExprTuple	47, 48
47	Variable
48	Variable