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Expression of type Implies

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 k, m
from proveit.logic import Equals, Implies, 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)))
expr = Implies(InSet(k, _m_domain), Equals(Mult(sub_expr1, Mult(sub_expr2, sub_expr3)), Mult(sub_expr1, sub_expr2, sub_expr3)))
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(k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right) \Rightarrow \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right) = \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}}}\right)\right)
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.()()('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
operands: 6
4Operationoperator: 7
operands: 8
5Literal
6ExprTuple53, 9
7Literal
8ExprTuple10, 11
9Operationoperator: 12
operands: 13
10Operationoperator: 47
operands: 14
11Operationoperator: 47
operands: 15
12Literal
13ExprTuple16, 17
14ExprTuple19, 18
15ExprTuple19, 25, 26
16Literal
17Operationoperator: 20
operands: 21
18Operationoperator: 47
operands: 22
19Operationoperator: 49
operands: 23
20Literal
21ExprTuple46, 24
22ExprTuple25, 26
23ExprTuple34, 27
24Operationoperator: 38
operand: 32
25Operationoperator: 43
operands: 29
26Operationoperator: 49
operands: 30
27Operationoperator: 47
operands: 31
28ExprTuple32
29ExprTuple32, 33
30ExprTuple34, 35
31ExprTuple55, 51, 52, 36, 53
32Literal
33Operationoperator: 49
operands: 37
34Literal
35Operationoperator: 38
operand: 41
36Literal
37ExprTuple55, 40
38Literal
39ExprTuple41
40Operationoperator: 43
operands: 42
41Operationoperator: 43
operands: 44
42ExprTuple56, 55
43Literal
44ExprTuple45, 46
45Operationoperator: 47
operands: 48
46Operationoperator: 49
operands: 50
47Literal
48ExprTuple55, 51, 52, 53, 54
49Literal
50ExprTuple55, 56
51Literal
52Literal
53Variable
54Variable
55Literal
56Literal