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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 Conditional, Lambda, k, m
from proveit.logic import Equals, Forall, 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)))
sub_expr4 = InSet(k, _m_domain)
sub_expr5 = Mult(sub_expr1, sub_expr2, sub_expr3)
sub_expr6 = Mult(sub_expr2, Mult(sub_expr1, sub_expr3))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr5, sub_expr6), domain = _m_domain), Equals(Lambda(k, Conditional(sub_expr5, sub_expr4)), Lambda(k, Conditional(sub_expr6, sub_expr4))).with_wrapping_at(2)).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 \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) = \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)\right)\right)\right] \Rightarrow  \\ \left(\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}\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: 17
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Lambdaparameter: 65
body: 11
9Lambdaparameter: 65
body: 12
10Lambdaparameter: 65
body: 14
11Conditionalvalue: 15
condition: 16
12Conditionalvalue: 21
condition: 16
13ExprTuple65
14Conditionalvalue: 22
condition: 16
15Operationoperator: 17
operands: 18
16Operationoperator: 19
operands: 20
17Literal
18ExprTuple21, 22
19Literal
20ExprTuple65, 23
21Operationoperator: 59
operands: 24
22Operationoperator: 59
operands: 25
23Operationoperator: 26
operands: 27
24ExprTuple37, 28, 38
25ExprTuple28, 29
26Literal
27ExprTuple30, 31
28Operationoperator: 55
operands: 32
29Operationoperator: 59
operands: 33
30Literal
31Operationoperator: 34
operands: 35
32ExprTuple48, 36
33ExprTuple37, 38
34Literal
35ExprTuple58, 39
36Operationoperator: 61
operands: 40
37Operationoperator: 61
operands: 41
38Operationoperator: 61
operands: 42
39Operationoperator: 51
operand: 48
40ExprTuple67, 44
41ExprTuple46, 45
42ExprTuple46, 47
43ExprTuple48
44Operationoperator: 55
operands: 49
45Operationoperator: 59
operands: 50
46Literal
47Operationoperator: 51
operand: 54
48Literal
49ExprTuple68, 67
50ExprTuple67, 63, 64, 53, 65
51Literal
52ExprTuple54
53Literal
54Operationoperator: 55
operands: 56
55Literal
56ExprTuple57, 58
57Operationoperator: 59
operands: 60
58Operationoperator: 61
operands: 62
59Literal
60ExprTuple67, 63, 64, 65, 66
61Literal
62ExprTuple67, 68
63Literal
64Literal
65Variable
66Variable
67Literal
68Literal