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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, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
sub_expr3 = InSet(k, _m_domain)
sub_expr4 = Conditional(Mult(sub_expr1, sub_expr2), sub_expr3)
sub_expr5 = Conditional(Mult(sub_expr2, sub_expr1), sub_expr3)
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr4, sub_expr5)), Equals(Lambda(k, sub_expr4), Lambda(k, sub_expr5)).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}~\left(\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}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. = \left\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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 \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\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \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: 13
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Lambdaparameter: 56
body: 11
9Lambdaparameter: 56
body: 15
10Lambdaparameter: 56
body: 16
11Operationoperator: 13
operands: 14
12ExprTuple56
13Literal
14ExprTuple15, 16
15Conditionalvalue: 17
condition: 19
16Conditionalvalue: 18
condition: 19
17Operationoperator: 50
operands: 20
18Operationoperator: 50
operands: 21
19Operationoperator: 22
operands: 23
20ExprTuple25, 24
21ExprTuple24, 25
22Literal
23ExprTuple56, 26
24Operationoperator: 52
operands: 27
25Operationoperator: 52
operands: 28
26Operationoperator: 29
operands: 30
27ExprTuple32, 31
28ExprTuple32, 33
29Literal
30ExprTuple34, 35
31Operationoperator: 45
operand: 40
32Literal
33Operationoperator: 50
operands: 37
34Literal
35Operationoperator: 38
operands: 39
36ExprTuple40
37ExprTuple58, 54, 55, 41, 56
38Literal
39ExprTuple48, 42
40Operationoperator: 43
operands: 44
41Literal
42Operationoperator: 45
operand: 49
43Literal
44ExprTuple47, 48
45Literal
46ExprTuple49
47Operationoperator: 50
operands: 51
48Operationoperator: 52
operands: 53
49Literal
50Literal
51ExprTuple58, 54, 55, 56, 57
52Literal
53ExprTuple58, 59
54Literal
55Literal
56Variable
57Variable
58Literal
59Literal