Expression of type Implies

from the theory of proveit.physics.quantum.QPE

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, t
from proveit.linear_algebra import ScalarMult, TensorProd
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(k, two_pow_t)
sub_expr2 = Interval(zero, subtract(two_pow_t, one))
sub_expr3 = InSet(k, sub_expr2)
sub_expr4 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), NumKet(sub_expr1, Add(t, one)))
sub_expr5 = ScalarMult(Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Mult(two, pi, i, _phase, two_pow_t))), TensorProd(ket1, NumKet(k, t)))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr4, sub_expr5), domain = sub_expr2), Equals(Lambda(k, Conditional(sub_expr4, sub_expr3)), Lambda(k, Conditional(sub_expr5, sub_expr3))).with_wrapping_at(2)).with_wrapping_at(2)

expr:

# 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

# 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 \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\right) = \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{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 \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(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')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 67 body: 11
9	Lambda	parameter: 67 body: 12
10	Lambda	parameter: 67 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	67
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	67, 23
21	Operation	operator: 25 operands: 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	ExprTuple	29, 30
25	Literal
26	ExprTuple	31, 32
27	Literal
28	ExprTuple	33, 34
29	Operation	operator: 72 operands: 35
30	Operation	operator: 53 operands: 36
31	Operation	operator: 65 operands: 37
32	Operation	operator: 38 operands: 39
33	Literal
34	Operation	operator: 62 operands: 40
35	ExprTuple	59, 41
36	ExprTuple	57, 42
37	ExprTuple	43, 44
38	Literal
39	ExprTuple	45, 46
40	ExprTuple	71, 47
41	Operation	operator: 65 operands: 48
42	Operation	operator: 62 operands: 49
43	Operation	operator: 72 operands: 50
44	Operation	operator: 72 operands: 51
45	Operation	operator: 52 operand: 61
46	Operation	operator: 53 operands: 54
47	Operation	operator: 55 operand: 61
48	ExprTuple	74, 68, 69, 70, 57
49	ExprTuple	75, 61
50	ExprTuple	59, 58
51	ExprTuple	59, 60
52	Literal
53	Literal
54	ExprTuple	67, 75
55	Literal
56	ExprTuple	61
57	Operation	operator: 62 operands: 63
58	Operation	operator: 65 operands: 64
59	Literal
60	Operation	operator: 65 operands: 66
61	Literal
62	Literal
63	ExprTuple	67, 71
64	ExprTuple	74, 68, 69, 70, 67
65	Literal
66	ExprTuple	74, 68, 69, 70, 71
67	Variable
68	Literal
69	Literal
70	Literal
71	Operation	operator: 72 operands: 73
72	Literal
73	ExprTuple	74, 75
74	Literal
75	Variable