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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
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import CartExp, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = CartExp(Complex, _two_pow_t)
sub_expr3 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr3, sub_expr2), domain = _m_domain), InSet(VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = _m_domain), sub_expr2)).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_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right) \in \mathbb{C}^{2^{t}}\right)\right] \Rightarrow  \\ \left(\left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) \in \mathbb{C}^{2^{t}}\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: 8
4Operationoperator: 25
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 18
8Lambdaparameter: 47
body: 10
9Operationoperator: 11
operand: 14
10Conditionalvalue: 13
condition: 20
11Literal
12ExprTuple14
13Operationoperator: 25
operands: 15
14Lambdaparameter: 47
body: 17
15ExprTuple19, 18
16ExprTuple47
17Conditionalvalue: 19
condition: 20
18Operationoperator: 21
operands: 22
19Operationoperator: 23
operands: 24
20Operationoperator: 25
operands: 26
21Literal
22ExprTuple27, 48
23Literal
24ExprTuple28, 29
25Literal
26ExprTuple47, 30
27Literal
28Operationoperator: 50
operands: 31
29Operationoperator: 32
operands: 33
30Operationoperator: 34
operands: 35
31ExprTuple36, 37
32Literal
33ExprTuple47, 55
34Literal
35ExprTuple38, 39
36Literal
37Operationoperator: 40
operands: 41
38Literal
39Operationoperator: 42
operands: 43
40Literal
41ExprTuple54, 44, 45, 46, 47
42Literal
43ExprTuple48, 49
44Literal
45Literal
46Literal
47Variable
48Operationoperator: 50
operands: 51
49Operationoperator: 52
operand: 56
50Literal
51ExprTuple54, 55
52Literal
53ExprTuple56
54Literal
55Literal
56Literal