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, f, i, k
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, f_1_to_k
from proveit.linear_algebra import TensorProd, VecZero
from proveit.logic import And, Equals, Forall, Implies, InSet, NotEquals
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, f_1_to_k], Conditional(Forall(instance_param_or_params = [b, e], instance_expr = Implies(Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, e, f_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, f_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V, conditions = [Equals(b, e), NotEquals(b, VecZero(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(f, 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}, f_{1}, f_{2}, \ldots, f_{k}\right) \mapsto \left\{\forall_{b, e \in V~|~b = e, b \neq \vec{0}\left(V\right)}~\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} 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} e{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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(f_{1} \in W_{1}\right) ,  \left(f_{2} \in W_{2}\right) ,  \ldots ,  \left(f_{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	65, 66, 67, 68
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 26 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10, 11, 12
8	Lambda	parameters: 46 body: 13
9	ExprRange	lambda_map: 14 start_index: 75 end_index: 73
10	ExprRange	lambda_map: 15 start_index: 75 end_index: 76
11	ExprRange	lambda_map: 16 start_index: 75 end_index: 73
12	ExprRange	lambda_map: 17 start_index: 75 end_index: 76
13	Conditional	value: 18 condition: 19
14	Lambda	parameter: 86 body: 20
15	Lambda	parameter: 86 body: 21
16	Lambda	parameter: 86 body: 22
17	Lambda	parameter: 86 body: 23
18	Operation	operator: 24 operands: 25
19	Operation	operator: 26 operands: 27
20	Operation	operator: 43 operands: 28
21	Operation	operator: 43 operands: 29
22	Operation	operator: 43 operands: 30
23	Operation	operator: 43 operands: 31
24	Literal
25	ExprTuple	32, 33
26	Literal
27	ExprTuple	34, 35, 36, 37
28	ExprTuple	77, 38
29	ExprTuple	78, 39
30	ExprTuple	79, 38
31	ExprTuple	80, 39
32	Operation	operator: 45 operands: 40
33	Operation	operator: 45 operands: 41
34	Operation	operator: 43 operands: 42
35	Operation	operator: 43 operands: 44
36	Operation	operator: 45 operands: 46
37	Operation	operator: 47 operands: 48
38	IndexedVar	variable: 49 index: 86
39	IndexedVar	variable: 50 index: 86
40	ExprTuple	51, 52
41	ExprTuple	53, 54
42	ExprTuple	63, 69
43	Literal
44	ExprTuple	64, 69
45	Literal
46	ExprTuple	63, 64
47	Literal
48	ExprTuple	63, 55
49	Variable
50	Variable
51	Operation	operator: 59 operands: 56
52	Operation	operator: 59 operands: 57
53	Operation	operator: 59 operands: 58
54	Operation	operator: 59 operands: 60
55	Operation	operator: 61 operand: 69
56	ExprTuple	65, 63, 66
57	ExprTuple	67, 64, 68
58	ExprTuple	65, 66
59	Literal
60	ExprTuple	67, 68
61	Literal
62	ExprTuple	69
63	Variable
64	Variable
65	ExprRange	lambda_map: 70 start_index: 75 end_index: 73
66	ExprRange	lambda_map: 71 start_index: 75 end_index: 76
67	ExprRange	lambda_map: 72 start_index: 75 end_index: 73
68	ExprRange	lambda_map: 74 start_index: 75 end_index: 76
69	Variable
70	Lambda	parameter: 86 body: 77
71	Lambda	parameter: 86 body: 78
72	Lambda	parameter: 86 body: 79
73	Variable
74	Lambda	parameter: 86 body: 80
75	Literal
76	Variable
77	IndexedVar	variable: 81 index: 86
78	IndexedVar	variable: 82 index: 86
79	IndexedVar	variable: 83 index: 86
80	IndexedVar	variable: 84 index: 86
81	Variable
82	Variable
83	Variable
84	Variable
85	ExprTuple	86
86	Variable