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Expression of type Lambda

from the theory of proveit.linear_algebra.tensors

In [1]:
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, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import And, Equals, InSet
from proveit.numbers import one
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda([a_1_to_i, b_1_to_j, c_1_to_k], Conditional(Equals(TensorProd(a_1_to_i, b_1_to_j, c_1_to_k), TensorProd(a_1_to_i, TensorProd(b_1_to_j), c_1_to_k)).with_wrapping_at(2), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), IndexedVar(U, sub_expr1)), one, i), ExprRange(sub_expr1, InSet(IndexedVar(b, sub_expr1), IndexedVar(V, sub_expr1)), one, j), ExprRange(sub_expr1, InSet(IndexedVar(c, sub_expr1), IndexedVar(W, sub_expr1)), one, k))))
expr:
In [3]:
# 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
In [4]:
# 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}, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k}\right) \mapsto \left\{\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} b_{1} {\otimes}  b_{2} {\otimes}  \ldots {\otimes}  b_{j}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} {\otimes}  b_{2} {\otimes}  \ldots {\otimes}  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array} \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(b_{1} \in V_{1}\right) ,  \left(b_{2} \in V_{2}\right) ,  \ldots ,  \left(b_{j} \in V_{j}\right), \left(c_{1} \in W_{1}\right) ,  \left(c_{2} \in W_{2}\right) ,  \ldots ,  \left(c_{k} \in W_{k}\right)\right..
In [5]:
stored_expr.style_options()
no style options
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Lambdaparameters: 13
body: 1
1Conditionalvalue: 2
condition: 3
2Operationoperator: 4
operands: 5
3Operationoperator: 6
operands: 7
4Literal
5ExprTuple8, 9
6Literal
7ExprTuple10, 11, 12
8Operationoperator: 26
operands: 13
9Operationoperator: 26
operands: 14
10ExprRangelambda_map: 15
start_index: 41
end_index: 25
11ExprRangelambda_map: 16
start_index: 41
end_index: 42
12ExprRangelambda_map: 17
start_index: 41
end_index: 29
13ExprTuple18, 34, 20
14ExprTuple18, 19, 20
15Lambdaparameter: 51
body: 21
16Lambdaparameter: 51
body: 22
17Lambdaparameter: 51
body: 23
18ExprRangelambda_map: 24
start_index: 41
end_index: 25
19Operationoperator: 26
operands: 27
20ExprRangelambda_map: 28
start_index: 41
end_index: 29
21Operationoperator: 32
operands: 30
22Operationoperator: 32
operands: 31
23Operationoperator: 32
operands: 33
24Lambdaparameter: 51
body: 35
25Variable
26Literal
27ExprTuple34
28Lambdaparameter: 51
body: 38
29Variable
30ExprTuple35, 36
31ExprTuple48, 37
32Literal
33ExprTuple38, 39
34ExprRangelambda_map: 40
start_index: 41
end_index: 42
35IndexedVarvariable: 43
index: 51
36IndexedVarvariable: 44
index: 51
37IndexedVarvariable: 45
index: 51
38IndexedVarvariable: 46
index: 51
39IndexedVarvariable: 47
index: 51
40Lambdaparameter: 51
body: 48
41Literal
42Variable
43Variable
44Variable
45Variable
46Variable
47Variable
48IndexedVarvariable: 49
index: 51
49Variable
50ExprTuple51
51Variable