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

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = NumKet(k, t)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr4 = ScalarMult(sub_expr3, sub_expr2)
sub_expr5 = Interval(zero, subtract(two_pow_t, one))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(ket1, sub_expr4), TensorProd(QubitSpace, CartExp(Complex, two_pow_t))), domain = sub_expr5), Equals(TensorProd(ket1, VecSum(index_or_indices = sub_expr1, summand = sub_expr4, domain = sub_expr5)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, TensorProd(ket1, sub_expr2)), domain = sub_expr5)).with_wrapping_at(1)).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(\lvert 1 \rangle {\otimes} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\lvert 1 \rangle {\otimes} \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)\right) \\  = \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\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: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 67 body: 12
10	Operation	operator: 43 operands: 13
11	Operation	operator: 19 operand: 17
12	Conditional	value: 15 condition: 34
13	ExprTuple	48, 16
14	ExprTuple	17
15	Operation	operator: 41 operands: 18
16	Operation	operator: 19 operand: 24
17	Lambda	parameter: 67 body: 21
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	24
21	Conditional	value: 25 condition: 34
22	Operation	operator: 43 operands: 26
23	Operation	operator: 43 operands: 27
24	Lambda	parameter: 67 body: 29
25	Operation	operator: 39 operands: 30
26	ExprTuple	48, 33
27	ExprTuple	31, 32
28	ExprTuple	67
29	Conditional	value: 33 condition: 34
30	ExprTuple	46, 35
31	Operation	operator: 37 operands: 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 39 operands: 40
34	Operation	operator: 41 operands: 42
35	Operation	operator: 43 operands: 44
36	ExprTuple	45, 74
37	Literal
38	ExprTuple	45, 68
39	Literal
40	ExprTuple	46, 49
41	Literal
42	ExprTuple	67, 47
43	Literal
44	ExprTuple	48, 49
45	Literal
46	Operation	operator: 70 operands: 50
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operand: 76
49	Operation	operator: 54 operands: 55
50	ExprTuple	56, 57
51	Literal
52	ExprTuple	58, 59
53	Literal
54	Literal
55	ExprTuple	67, 75
56	Literal
57	Operation	operator: 60 operands: 61
58	Literal
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	74, 64, 65, 66, 67
62	Literal
63	ExprTuple	68, 69
64	Literal
65	Literal
66	Literal
67	Variable
68	Operation	operator: 70 operands: 71
69	Operation	operator: 72 operand: 76
70	Literal
71	ExprTuple	74, 75
72	Literal
73	ExprTuple	76
74	Literal
75	Variable
76	Literal