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, Function, Lambda, Q, f, 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, Implies, InSet
from proveit.numbers import Complex, Mult, Natural, NaturalPos, Sum
from proveit.numbers.summation import summation_b1toj_fQ

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(f, sub_expr1)
sub_expr3 = Function(Q, sub_expr1)
expr = Lambda([i, j, k], Conditional(Forall(instance_param_or_params = [f, Q], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Complex), condition = sub_expr3), Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, summation_b1toj_fQ, c_1_to_k), Sum(index_or_indices = sub_expr1, summand = Mult(a_1_to_i, sub_expr2, c_1_to_k), condition = sub_expr3)).with_wrapping_at(2), domain = Complex).with_wrapping()).with_wrapping_at(2)), And(InSet(i, Natural), InSet(j, NaturalPos), 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\{\forall_{f, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}\right)\end{array}\right] \end{array} \end{array}\right) \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	68, 79, 72
2	Conditional	value: 3 condition: 4
3	Operation	operator: 23 operand: 7
4	Operation	operator: 36 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9, 10
7	Lambda	parameters: 11 body: 12
8	Operation	operator: 54 operands: 13
9	Operation	operator: 54 operands: 14
10	Operation	operator: 54 operands: 15
11	ExprTuple	69, 66
12	Operation	operator: 16 operands: 17
13	ExprTuple	68, 19
14	ExprTuple	79, 18
15	ExprTuple	72, 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	Literal
20	Operation	operator: 23 operand: 25
21	Operation	operator: 23 operand: 26
22	ExprTuple	25
23	Literal
24	ExprTuple	26
25	Lambda	parameters: 70 body: 27
26	Lambda	parameters: 28 body: 29
27	Conditional	value: 30 condition: 62
28	ExprTuple	63, 65
29	Conditional	value: 31 condition: 32
30	Operation	operator: 54 operands: 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 37
33	ExprTuple	64, 58
34	Literal
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	40, 41
38	Operation	operator: 60 operands: 42
39	Operation	operator: 50 operand: 47
40	ExprRange	lambda_map: 44 start_index: 78 end_index: 68
41	ExprRange	lambda_map: 45 start_index: 78 end_index: 72
42	ExprTuple	63, 46, 65
43	ExprTuple	47
44	Lambda	parameter: 84 body: 48
45	Lambda	parameter: 84 body: 49
46	Operation	operator: 50 operand: 56
47	Lambda	parameters: 70 body: 52
48	Operation	operator: 54 operands: 53
49	Operation	operator: 54 operands: 55
50	Literal
51	ExprTuple	56
52	Conditional	value: 57 condition: 62
53	ExprTuple	73, 58
54	Literal
55	ExprTuple	75, 58
56	Lambda	parameters: 70 body: 59
57	Operation	operator: 60 operands: 61
58	Literal
59	Conditional	value: 64 condition: 62
60	Literal
61	ExprTuple	63, 64, 65
62	Operation	operator: 66 operands: 70
63	ExprRange	lambda_map: 67 start_index: 78 end_index: 68
64	Operation	operator: 69 operands: 70
65	ExprRange	lambda_map: 71 start_index: 78 end_index: 72
66	Variable
67	Lambda	parameter: 84 body: 73
68	Variable
69	Variable
70	ExprTuple	74
71	Lambda	parameter: 84 body: 75
72	Variable
73	IndexedVar	variable: 76 index: 84
74	ExprRange	lambda_map: 77 start_index: 78 end_index: 79
75	IndexedVar	variable: 80 index: 84
76	Variable
77	Lambda	parameter: 84 body: 81
78	Literal
79	Variable
80	Variable
81	IndexedVar	variable: 82 index: 84
82	Variable
83	ExprTuple	84
84	Variable