Expression of type Lambda

from the theory of proveit.numbers.multiplication

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, IndexedVar, Lambda, Variable, b, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Abs, Add, Complex, Mult, Natural, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
expr = Lambda([i, j, k], Conditional(Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Forall(instance_param_or_params = [b_1_to_j], instance_expr = Equals(Mult(a_1_to_i, Abs(Add(b_1_to_j)), c_1_to_k), Abs(Add(ExprRange(sub_expr1, Mult(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j)))).with_wrapping_at(2), domain = Complex).with_wrapping(), domain = RealNonNeg).with_wrapping(), And(InSet(i, Natural), InSet(j, Natural), InSet(k, 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, k\right) \mapsto \left\{\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{R}^{\ge 0}}~\\
\left[\begin{array}{l}\forall_{b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left|b_{1} +  b_{2} +  \ldots +  b_{j}\right|\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left|\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{1}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{2}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \ldots +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right| \end{array} \end{array}\right)\end{array}\right]\end{array} \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  k \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	70, 62, 75
2	Conditional	value: 3 condition: 4
3	Operation	operator: 19 operand: 7
4	Operation	operator: 34 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9, 10
7	Lambda	parameters: 11 body: 12
8	Operation	operator: 51 operands: 13
9	Operation	operator: 51 operands: 14
10	Operation	operator: 51 operands: 15
11	ExprTuple	66, 68
12	Conditional	value: 16 condition: 17
13	ExprTuple	70, 18
14	ExprTuple	62, 18
15	ExprTuple	75, 18
16	Operation	operator: 19 operand: 22
17	Operation	operator: 34 operands: 21
18	Literal
19	Literal
20	ExprTuple	22
21	ExprTuple	23, 24
22	Lambda	parameters: 57 body: 25
23	ExprRange	lambda_map: 26 start_index: 74 end_index: 70
24	ExprRange	lambda_map: 27 start_index: 74 end_index: 75
25	Conditional	value: 28 condition: 29
26	Lambda	parameter: 82 body: 30
27	Lambda	parameter: 82 body: 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 51 operands: 36
31	Operation	operator: 51 operands: 37
32	Literal
33	ExprTuple	38, 39
34	Literal
35	ExprTuple	40
36	ExprTuple	76, 41
37	ExprTuple	78, 41
38	Operation	operator: 63 operands: 42
39	Operation	operator: 48 operand: 46
40	ExprRange	lambda_map: 44 start_index: 74 end_index: 62
41	Literal
42	ExprTuple	66, 45, 68
43	ExprTuple	46
44	Lambda	parameter: 82 body: 47
45	Operation	operator: 48 operand: 53
46	Operation	operator: 56 operands: 50
47	Operation	operator: 51 operands: 52
48	Literal
49	ExprTuple	53
50	ExprTuple	54
51	Literal
52	ExprTuple	65, 55
53	Operation	operator: 56 operands: 57
54	ExprRange	lambda_map: 58 start_index: 74 end_index: 62
55	Literal
56	Literal
57	ExprTuple	59
58	Lambda	parameter: 77 body: 60
59	ExprRange	lambda_map: 61 start_index: 74 end_index: 62
60	Operation	operator: 63 operands: 64
61	Lambda	parameter: 82 body: 65
62	Variable
63	Literal
64	ExprTuple	66, 67, 68
65	IndexedVar	variable: 71 index: 82
66	ExprRange	lambda_map: 69 start_index: 74 end_index: 70
67	IndexedVar	variable: 71 index: 77
68	ExprRange	lambda_map: 73 start_index: 74 end_index: 75
69	Lambda	parameter: 82 body: 76
70	Variable
71	Variable
72	ExprTuple	77
73	Lambda	parameter: 82 body: 78
74	Literal
75	Variable
76	IndexedVar	variable: 79 index: 82
77	Variable
78	IndexedVar	variable: 80 index: 82
79	Variable
80	Variable
81	ExprTuple	82
82	Variable