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, m
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, pi, subtract, 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 = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k)))
sub_expr3 = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(m, _two_pow_t)))), k)
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr2, sub_expr3), domain = _m_domain), Equals(Lambda(k, Conditional(sub_expr2, sub_expr1)), Lambda(k, Conditional(sub_expr3, sub_expr1))).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}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right) = (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k}\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 m}{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\{(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k} \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: 61 body: 11
9	Lambda	parameter: 61 body: 12
10	Lambda	parameter: 61 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	61
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	61, 23
21	Operation	operator: 55 operands: 24
22	Operation	operator: 67 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	28, 29
25	ExprTuple	30, 61
26	Literal
27	ExprTuple	31, 32
28	Operation	operator: 67 operands: 33
29	Operation	operator: 67 operands: 34
30	Operation	operator: 67 operands: 35
31	Literal
32	Operation	operator: 50 operands: 36
33	ExprTuple	39, 37
34	ExprTuple	39, 38
35	ExprTuple	39, 40
36	ExprTuple	66, 41
37	Operation	operator: 57 operand: 46
38	Operation	operator: 55 operands: 43
39	Literal
40	Operation	operator: 55 operands: 44
41	Operation	operator: 57 operand: 48
42	ExprTuple	46
43	ExprTuple	69, 59, 60, 53, 61
44	ExprTuple	69, 59, 60, 47
45	ExprTuple	48
46	Operation	operator: 63 operands: 49
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	52, 66
50	Literal
51	ExprTuple	53, 54
52	Operation	operator: 55 operands: 56
53	Literal
54	Operation	operator: 57 operand: 62
55	Literal
56	ExprTuple	69, 59, 60, 61, 65
57	Literal
58	ExprTuple	62
59	Literal
60	Literal
61	Variable
62	Operation	operator: 63 operands: 64
63	Literal
64	ExprTuple	65, 66
65	Variable
66	Operation	operator: 67 operands: 68
67	Literal
68	ExprTuple	69, 70
69	Literal
70	Literal