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, a, b, c, 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, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda([a_1_to_i, c_1_to_k], Conditional(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(), And(ExprRange(sub_expr2, InSet(IndexedVar(a, sub_expr2), RealNonNeg), one, i), ExprRange(sub_expr2, InSet(IndexedVar(c, sub_expr2), RealNonNeg), one, k))))

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(a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}\right) \mapsto \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} \textrm{ if } \left(a_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(a_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(a_{i} \in \mathbb{R}^{\ge 0}\right), \left(c_{1} \in \mathbb{R}^{\ge 0}\right) ,  \left(c_{2} \in \mathbb{R}^{\ge 0}\right) ,  \ldots ,  \left(c_{k} \in \mathbb{R}^{\ge 0}\right)\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	52, 54
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 20 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 43 body: 11
9	ExprRange	lambda_map: 12 start_index: 60 end_index: 56
10	ExprRange	lambda_map: 13 start_index: 60 end_index: 61
11	Conditional	value: 14 condition: 15
12	Lambda	parameter: 68 body: 16
13	Lambda	parameter: 68 body: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operands: 21
16	Operation	operator: 37 operands: 22
17	Operation	operator: 37 operands: 23
18	Literal
19	ExprTuple	24, 25
20	Literal
21	ExprTuple	26
22	ExprTuple	62, 27
23	ExprTuple	64, 27
24	Operation	operator: 49 operands: 28
25	Operation	operator: 34 operand: 32
26	ExprRange	lambda_map: 30 start_index: 60 end_index: 48
27	Literal
28	ExprTuple	52, 31, 54
29	ExprTuple	32
30	Lambda	parameter: 68 body: 33
31	Operation	operator: 34 operand: 39
32	Operation	operator: 42 operands: 36
33	Operation	operator: 37 operands: 38
34	Literal
35	ExprTuple	39
36	ExprTuple	40
37	Literal
38	ExprTuple	51, 41
39	Operation	operator: 42 operands: 43
40	ExprRange	lambda_map: 44 start_index: 60 end_index: 48
41	Literal
42	Literal
43	ExprTuple	45
44	Lambda	parameter: 63 body: 46
45	ExprRange	lambda_map: 47 start_index: 60 end_index: 48
46	Operation	operator: 49 operands: 50
47	Lambda	parameter: 68 body: 51
48	Variable
49	Literal
50	ExprTuple	52, 53, 54
51	IndexedVar	variable: 57 index: 68
52	ExprRange	lambda_map: 55 start_index: 60 end_index: 56
53	IndexedVar	variable: 57 index: 63
54	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
55	Lambda	parameter: 68 body: 62
56	Variable
57	Variable
58	ExprTuple	63
59	Lambda	parameter: 68 body: 64
60	Literal
61	Variable
62	IndexedVar	variable: 65 index: 68
63	Variable
64	IndexedVar	variable: 66 index: 68
65	Variable
66	Variable
67	ExprTuple	68
68	Variable