Expression of type Lambda

from the theory of proveit.numbers.addition

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, Lambda, b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import And, Forall, InSet
from proveit.numbers import Add, LessEq, Natural, Real, RealNonPos

# build up the expression from sub-expressions
expr = Lambda([i, j], Conditional(Forall(instance_param_or_params = [a_1_to_i, c_1_to_j], instance_expr = Forall(instance_param_or_params = [b], instance_expr = LessEq(Add(a_1_to_i, b, c_1_to_j), b), domain = Real), domain = RealNonPos), And(InSet(i, Natural), InSet(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())

\left(i, j\right) \mapsto \left\{\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{R}^{\le 0}}~\left[\forall_{b \in \mathbb{R}}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) \leq b\right)\right] \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	47, 50
2	Conditional	value: 3 condition: 4
3	Operation	operator: 17 operand: 7
4	Operation	operator: 19 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 36 operands: 12
9	Operation	operator: 36 operands: 13
10	ExprTuple	43, 45
11	Conditional	value: 14 condition: 15
12	ExprTuple	47, 16
13	ExprTuple	50, 16
14	Operation	operator: 17 operand: 21
15	Operation	operator: 19 operands: 20
16	Literal
17	Literal
18	ExprTuple	21
19	Literal
20	ExprTuple	22, 23
21	Lambda	parameter: 44 body: 25
22	ExprRange	lambda_map: 26 start_index: 49 end_index: 47
23	ExprRange	lambda_map: 27 start_index: 49 end_index: 50
24	ExprTuple	44
25	Conditional	value: 28 condition: 29
26	Lambda	parameter: 56 body: 30
27	Lambda	parameter: 56 body: 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 36 operands: 34
30	Operation	operator: 36 operands: 35
31	Operation	operator: 36 operands: 37
32	Literal
33	ExprTuple	38, 44
34	ExprTuple	44, 39
35	ExprTuple	51, 40
36	Literal
37	ExprTuple	52, 40
38	Operation	operator: 41 operands: 42
39	Literal
40	Literal
41	Literal
42	ExprTuple	43, 44, 45
43	ExprRange	lambda_map: 46 start_index: 49 end_index: 47
44	Variable
45	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
46	Lambda	parameter: 56 body: 51
47	Variable
48	Lambda	parameter: 56 body: 52
49	Literal
50	Variable
51	IndexedVar	variable: 53 index: 56
52	IndexedVar	variable: 54 index: 56
53	Variable
54	Variable
55	ExprTuple	56
56	Variable