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
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = InSet(k, _m_domain)
sub_expr2 = Conditional(one, sub_expr1)
sub_expr3 = Conditional(Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), sub_expr1)
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr3, sub_expr2)), Equals(Lambda(k, sub_expr3), Lambda(k, sub_expr2)).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}~\left(\left\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{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.. = \left\{1 \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}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{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] =  \\ \left[k \mapsto \left\{1 \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: 13 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 49 body: 11
9	Lambda	parameter: 49 body: 15
10	Lambda	parameter: 49 body: 16
11	Operation	operator: 13 operands: 14
12	ExprTuple	49
13	Literal
14	ExprTuple	15, 16
15	Conditional	value: 17 condition: 18
16	Conditional	value: 45 condition: 18
17	Operation	operator: 51 operands: 19
18	Operation	operator: 20 operands: 21
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	49, 24
22	Operation	operator: 55 operands: 25
23	Operation	operator: 55 operands: 26
24	Operation	operator: 27 operands: 28
25	ExprTuple	30, 29
26	ExprTuple	30, 31
27	Literal
28	ExprTuple	32, 33
29	Operation	operator: 42 operand: 38
30	Literal
31	Operation	operator: 51 operands: 35
32	Literal
33	Operation	operator: 36 operands: 37
34	ExprTuple	38
35	ExprTuple	57, 47, 48, 54, 49
36	Literal
37	ExprTuple	53, 39
38	Operation	operator: 40 operands: 41
39	Operation	operator: 42 operand: 45
40	Literal
41	ExprTuple	44, 53
42	Literal
43	ExprTuple	45
44	Operation	operator: 51 operands: 46
45	Literal
46	ExprTuple	57, 47, 48, 49, 50
47	Literal
48	Literal
49	Variable
50	Operation	operator: 51 operands: 52
51	Literal
52	ExprTuple	53, 54
53	Operation	operator: 55 operands: 56
54	Literal
55	Literal
56	ExprTuple	57, 58
57	Literal
58	Literal