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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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, Lambda, k, m
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
sub_expr4 = InSet(k, _m_domain)
expr = Equals(Lambda(k, Conditional(Mult(sub_expr1, sub_expr2, sub_expr3), sub_expr4)), Lambda(k, Conditional(Mult(sub_expr2, Mult(sub_expr1, sub_expr3)), sub_expr4))).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[k \mapsto \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\frac{1}{2^{\frac{t}{2}}} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\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
3Lambdaparameter: 55
body: 5
4Lambdaparameter: 55
body: 7
5Conditionalvalue: 8
condition: 10
6ExprTuple55
7Conditionalvalue: 9
condition: 10
8Operationoperator: 49
operands: 11
9Operationoperator: 49
operands: 12
10Operationoperator: 13
operands: 14
11ExprTuple23, 15, 24
12ExprTuple15, 16
13Literal
14ExprTuple55, 17
15Operationoperator: 45
operands: 18
16Operationoperator: 49
operands: 19
17Operationoperator: 20
operands: 21
18ExprTuple44, 22
19ExprTuple23, 24
20Literal
21ExprTuple25, 26
22Operationoperator: 51
operands: 27
23Operationoperator: 51
operands: 28
24Operationoperator: 51
operands: 29
25Literal
26Operationoperator: 30
operands: 31
27ExprTuple57, 32
28ExprTuple34, 33
29ExprTuple34, 35
30Literal
31ExprTuple48, 36
32Operationoperator: 45
operands: 37
33Operationoperator: 49
operands: 38
34Literal
35Operationoperator: 40
operand: 43
36Operationoperator: 40
operand: 44
37ExprTuple58, 57
38ExprTuple57, 53, 54, 42, 55
39ExprTuple43
40Literal
41ExprTuple44
42Literal
43Operationoperator: 45
operands: 46
44Literal
45Literal
46ExprTuple47, 48
47Operationoperator: 49
operands: 50
48Operationoperator: 51
operands: 52
49Literal
50ExprTuple57, 53, 54, 55, 56
51Literal
52ExprTuple57, 58
53Literal
54Literal
55Variable
56Variable
57Literal
58Literal