Expression of type Lambda

from the theory of proveit.linear_algebra.tensors

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, U, V, Variable, W, a, b, c, d, e, i, k
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(U, sub_expr1)
sub_expr3 = IndexedVar(W, sub_expr1)
expr = Lambda([a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k], Conditional(Forall(instance_param_or_params = [b], instance_expr = Implies(Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, e_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, b, e_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), sub_expr2), one, i), ExprRange(sub_expr1, InSet(IndexedVar(c, sub_expr1), sub_expr3), one, k), ExprRange(sub_expr1, InSet(IndexedVar(d, sub_expr1), sub_expr2), one, i), ExprRange(sub_expr1, InSet(IndexedVar(e, sub_expr1), sub_expr3), 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}, d_{1}, d_{2}, \ldots, d_{i}, e_{1}, e_{2}, \ldots, e_{k}\right) \mapsto \left\{\forall_{b \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i} {\otimes} b{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right) \textrm{ if } \left(a_{1} \in U_{1}\right) ,  \left(a_{2} \in U_{2}\right) ,  \ldots ,  \left(a_{i} \in U_{i}\right), \left(c_{1} \in W_{1}\right) ,  \left(c_{2} \in W_{2}\right) ,  \ldots ,  \left(c_{k} \in W_{k}\right), \left(d_{1} \in U_{1}\right) ,  \left(d_{2} \in U_{2}\right) ,  \ldots ,  \left(d_{i} \in U_{i}\right), \left(e_{1} \in W_{1}\right) ,  \left(e_{2} \in W_{2}\right) ,  \ldots ,  \left(e_{k} \in W_{k}\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	53, 54, 55, 57
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11, 12, 13
9	Lambda	parameter: 56 body: 15
10	ExprRange	lambda_map: 16 start_index: 63 end_index: 61
11	ExprRange	lambda_map: 17 start_index: 63 end_index: 64
12	ExprRange	lambda_map: 18 start_index: 63 end_index: 61
13	ExprRange	lambda_map: 19 start_index: 63 end_index: 64
14	ExprTuple	56
15	Conditional	value: 20 condition: 21
16	Lambda	parameter: 74 body: 22
17	Lambda	parameter: 74 body: 23
18	Lambda	parameter: 74 body: 24
19	Lambda	parameter: 74 body: 25
20	Operation	operator: 26 operands: 27
21	Operation	operator: 32 operands: 28
22	Operation	operator: 32 operands: 29
23	Operation	operator: 32 operands: 30
24	Operation	operator: 32 operands: 31
25	Operation	operator: 32 operands: 33
26	Literal
27	ExprTuple	34, 35
28	ExprTuple	56, 36
29	ExprTuple	65, 37
30	ExprTuple	66, 38
31	ExprTuple	67, 37
32	Literal
33	ExprTuple	68, 38
34	Operation	operator: 40 operands: 39
35	Operation	operator: 40 operands: 41
36	Variable
37	IndexedVar	variable: 42 index: 74
38	IndexedVar	variable: 43 index: 74
39	ExprTuple	44, 45
40	Literal
41	ExprTuple	46, 47
42	Variable
43	Variable
44	Operation	operator: 51 operands: 48
45	Operation	operator: 51 operands: 49
46	Operation	operator: 51 operands: 50
47	Operation	operator: 51 operands: 52
48	ExprTuple	53, 54
49	ExprTuple	55, 57
50	ExprTuple	53, 56, 54
51	Literal
52	ExprTuple	55, 56, 57
53	ExprRange	lambda_map: 58 start_index: 63 end_index: 61
54	ExprRange	lambda_map: 59 start_index: 63 end_index: 64
55	ExprRange	lambda_map: 60 start_index: 63 end_index: 61
56	Variable
57	ExprRange	lambda_map: 62 start_index: 63 end_index: 64
58	Lambda	parameter: 74 body: 65
59	Lambda	parameter: 74 body: 66
60	Lambda	parameter: 74 body: 67
61	Variable
62	Lambda	parameter: 74 body: 68
63	Literal
64	Variable
65	IndexedVar	variable: 69 index: 74
66	IndexedVar	variable: 70 index: 74
67	IndexedVar	variable: 71 index: 74
68	IndexedVar	variable: 72 index: 74
69	Variable
70	Variable
71	Variable
72	Variable
73	ExprTuple	74
74	Variable