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
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import CartExp, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = CartExp(Complex, _two_pow_t)
sub_expr3 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr3, sub_expr2), domain = _m_domain), InSet(VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = _m_domain), sub_expr2)).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 \lvert k \rangle_{t}\right) \in \mathbb{C}^{2^{t}}\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 \lvert k \rangle_{t}\right)\right) \in \mathbb{C}^{2^{t}}\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: 25 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 18
8	Lambda	parameter: 47 body: 10
9	Operation	operator: 11 operand: 14
10	Conditional	value: 13 condition: 20
11	Literal
12	ExprTuple	14
13	Operation	operator: 25 operands: 15
14	Lambda	parameter: 47 body: 17
15	ExprTuple	19, 18
16	ExprTuple	47
17	Conditional	value: 19 condition: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	27, 48
23	Literal
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	47, 30
27	Literal
28	Operation	operator: 50 operands: 31
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operands: 35
31	ExprTuple	36, 37
32	Literal
33	ExprTuple	47, 55
34	Literal
35	ExprTuple	38, 39
36	Literal
37	Operation	operator: 40 operands: 41
38	Literal
39	Operation	operator: 42 operands: 43
40	Literal
41	ExprTuple	54, 44, 45, 46, 47
42	Literal
43	ExprTuple	48, 49
44	Literal
45	Literal
46	Literal
47	Variable
48	Operation	operator: 50 operands: 51
49	Operation	operator: 52 operand: 56
50	Literal
51	ExprTuple	54, 55
52	Literal
53	ExprTuple	56
54	Literal
55	Literal
56	Literal