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Expression of type Forall

from the theory of proveit.linear_algebra.matrices.exponentiation

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 A, m, rho, x
from proveit.linear_algebra import MatrixExp, MatrixMult, ScalarMult
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Mult, Real, e, i, pi, two
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [rho], instance_expr = Forall(instance_param_or_params = [A, x], instance_expr = Equals(MatrixMult(MatrixExp(A, m), x), ScalarMult(Exp(e, Mult(two, pi, i, Mult(m, rho))), x)), condition = Equals(MatrixMult(A, x), ScalarMult(Exp(e, Mult(two, pi, i, rho)), x))), domain = Real)
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())
\forall_{\rho \in \mathbb{R}}~\left[\forall_{A, x~|~\left(A \thinspace x\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \rho} \cdot x\right)}~\left(\left(A^{m} \thinspace x\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(m \cdot \rho\right)} \cdot x\right)\right)\right]
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 51
body: 4
3ExprTuple51
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple51, 12
11Lambdaparameters: 26
body: 13
12Literal
13Conditionalvalue: 14
condition: 15
14Operationoperator: 17
operands: 16
15Operationoperator: 17
operands: 18
16ExprTuple19, 20
17Literal
18ExprTuple21, 22
19Operationoperator: 25
operands: 23
20Operationoperator: 27
operands: 24
21Operationoperator: 25
operands: 26
22Operationoperator: 27
operands: 28
23ExprTuple29, 32
24ExprTuple30, 32
25Literal
26ExprTuple38, 32
27Literal
28ExprTuple31, 32
29Operationoperator: 33
operands: 34
30Operationoperator: 36
operands: 35
31Operationoperator: 36
operands: 37
32Variable
33Literal
34ExprTuple38, 50
35ExprTuple40, 39
36Literal
37ExprTuple40, 41
38Variable
39Operationoperator: 48
operands: 42
40Literal
41Operationoperator: 48
operands: 43
42ExprTuple45, 46, 47, 44
43ExprTuple45, 46, 47, 51
44Operationoperator: 48
operands: 49
45Literal
46Literal
47Literal
48Literal
49ExprTuple50, 51
50Variable
51Variable