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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 gamma, i, x, y
from proveit.linear_algebra import ScalarMult, TensorProd, VecSum
from proveit.logic import CartExp, Equals, Forall, Implies, InSet
from proveit.numbers import Interval, Real, four, three, two
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = ScalarMult(gamma, i)
sub_expr3 = CartExp(Real, three)
sub_expr4 = Interval(two, four)
sub_expr5 = ScalarMult(sub_expr2, y)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(x, sub_expr5), TensorProd(sub_expr3, sub_expr3)), domain = sub_expr4), Equals(TensorProd(x, VecSum(index_or_indices = sub_expr1, summand = sub_expr5, domain = sub_expr4)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, TensorProd(x, y)), domain = sub_expr4)).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 \{2~\ldotp \ldotp~4\}}~\left(\left(x {\otimes} \left(\left(\gamma \cdot i\right) \cdot y\right)\right) \in \left(\mathbb{R}^{3} {\otimes} \mathbb{R}^{3}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(x {\otimes} \left(\sum_{i=2}^{4} \left(\left(\gamma \cdot i\right) \cdot y\right)\right)\right) \\  = \left(\sum_{i=2}^{4} \left(\left(\gamma \cdot i\right) \cdot \left(x {\otimes} y\right)\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: 53
body: 12
10Operationoperator: 40
operands: 13
11Operationoperator: 19
operand: 17
12Conditionalvalue: 15
condition: 33
13ExprTuple46, 16
14ExprTuple17
15Operationoperator: 38
operands: 18
16Operationoperator: 19
operand: 24
17Lambdaparameter: 53
body: 21
18ExprTuple22, 23
19Literal
20ExprTuple24
21Conditionalvalue: 25
condition: 33
22Operationoperator: 40
operands: 26
23Operationoperator: 40
operands: 27
24Lambdaparameter: 53
body: 29
25Operationoperator: 48
operands: 30
26ExprTuple46, 32
27ExprTuple31, 31
28ExprTuple53
29Conditionalvalue: 32
condition: 33
30ExprTuple44, 34
31Operationoperator: 35
operands: 36
32Operationoperator: 48
operands: 37
33Operationoperator: 38
operands: 39
34Operationoperator: 40
operands: 41
35Literal
36ExprTuple42, 43
37ExprTuple44, 47
38Literal
39ExprTuple53, 45
40Literal
41ExprTuple46, 47
42Literal
43Literal
44Operationoperator: 48
operands: 49
45Operationoperator: 50
operands: 51
46Variable
47Variable
48Literal
49ExprTuple52, 53
50Literal
51ExprTuple54, 55
52Variable
53Variable
54Literal
55Literal