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Expression of type Implies

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 fi, i, u, v, w, x, z
from proveit.linear_algebra import TensorProd, VecSum
from proveit.logic import CartExp, Equals, Forall, Implies, InSet
from proveit.numbers import Interval, Real, five, one, three
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = CartExp(Real, three)
sub_expr3 = Interval(one, five)
sub_expr4 = TensorProd(u, v, w, x, fi, z)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr4, TensorProd(sub_expr2, sub_expr2, sub_expr2, sub_expr2, sub_expr2, sub_expr2)), domain = sub_expr3), Equals(TensorProd(u, v, w, x, VecSum(index_or_indices = sub_expr1, summand = fi, domain = sub_expr3), z), VecSum(index_or_indices = sub_expr1, summand = sub_expr4, domain = sub_expr3)).with_wrapping_at(1)).with_wrapping_at(2)
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())
\begin{array}{c} \begin{array}{l} \left[\forall_{i \in \{1~\ldotp \ldotp~5\}}~\left(\left(u {\otimes} v {\otimes} w {\otimes} x {\otimes} f\left(i\right) {\otimes} z\right) \in \left(\mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(u {\otimes} v {\otimes} w {\otimes} x {\otimes} \left(\sum_{i=1}^{5} f\left(i\right)\right) {\otimes} z\right) \\  = \left(\sum_{i=1}^{5} \left(u {\otimes} v {\otimes} w {\otimes} x {\otimes} f\left(i\right) {\otimes} z\right)\right) \end{array} \end{array}\right) \end{array} \end{array}
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(2)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 5
operand: 9
4Operationoperator: 7
operands: 8
5Literal
6ExprTuple9
7Literal
8ExprTuple10, 11
9Lambdaparameter: 46
body: 12
10Operationoperator: 27
operands: 13
11Operationoperator: 19
operand: 17
12Conditionalvalue: 15
condition: 30
13ExprTuple31, 32, 33, 34, 16, 36
14ExprTuple17
15Operationoperator: 39
operands: 18
16Operationoperator: 19
operand: 23
17Lambdaparameter: 46
body: 21
18ExprTuple24, 22
19Literal
20ExprTuple23
21Conditionalvalue: 24
condition: 30
22Operationoperator: 27
operands: 25
23Lambdaparameter: 46
body: 26
24Operationoperator: 27
operands: 28
25ExprTuple29, 29, 29, 29, 29, 29
26Conditionalvalue: 35
condition: 30
27Literal
28ExprTuple31, 32, 33, 34, 35, 36
29Operationoperator: 37
operands: 38
30Operationoperator: 39
operands: 40
31Variable
32Variable
33Variable
34Variable
35Operationoperator: 41
operand: 46
36Variable
37Literal
38ExprTuple43, 44
39Literal
40ExprTuple46, 45
41Variable
42ExprTuple46
43Literal
44Literal
45Operationoperator: 47
operands: 48
46Variable
47Literal
48ExprTuple49, 50
49Literal
50Literal