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, 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 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), TensorProd(ket1, NumKet(k, t)))
sub_expr3 = TensorProd(QubitSpace, CartExp(Complex, two_pow_t))
sub_expr4 = Interval(zero, subtract(two_pow_t, one))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, sub_expr3), domain = sub_expr4), InSet(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = sub_expr4), sub_expr3)).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 k} \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\right)\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 \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\right)\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: 24 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 18
8	Lambda	parameter: 56 body: 10
9	Operation	operator: 11 operand: 14
10	Conditional	value: 13 condition: 20
11	Literal
12	ExprTuple	14
13	Operation	operator: 24 operands: 15
14	Lambda	parameter: 56 body: 17
15	ExprTuple	19, 18
16	ExprTuple	56
17	Conditional	value: 19 condition: 20
18	Operation	operator: 35 operands: 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	ExprTuple	26, 27
22	Literal
23	ExprTuple	28, 29
24	Literal
25	ExprTuple	56, 30
26	Operation	operator: 32 operands: 31
27	Operation	operator: 32 operands: 33
28	Operation	operator: 59 operands: 34
29	Operation	operator: 35 operands: 36
30	Operation	operator: 37 operands: 38
31	ExprTuple	39, 63
32	Literal
33	ExprTuple	39, 57
34	ExprTuple	40, 41
35	Literal
36	ExprTuple	42, 43
37	Literal
38	ExprTuple	44, 45
39	Literal
40	Literal
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operand: 65
43	Operation	operator: 49 operands: 50
44	Literal
45	Operation	operator: 51 operands: 52
46	Literal
47	ExprTuple	63, 53, 54, 55, 56
48	Literal
49	Literal
50	ExprTuple	56, 64
51	Literal
52	ExprTuple	57, 58
53	Literal
54	Literal
55	Literal
56	Variable
57	Operation	operator: 59 operands: 60
58	Operation	operator: 61 operand: 65
59	Literal
60	ExprTuple	63, 64
61	Literal
62	ExprTuple	65
63	Literal
64	Variable
65	Literal