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.linear_algebra import ScalarMult
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, two, zero
from proveit.physics.quantum import Ket
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
sub_expr1 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), Ket(k))
sub_expr2 = Conditional(sub_expr1, InSet(k, Interval(Add(zero, one), one)))
sub_expr3 = Conditional(sub_expr1, InSet(k, Interval(one, one)))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr2, sub_expr3)), Equals(Lambda(k, sub_expr2), Lambda(k, sub_expr3)).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}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle \textrm{ if } k \in \{0 + 1~\ldotp \ldotp~1\}\right.. = \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle \textrm{ if } k \in \{1~\ldotp \ldotp~1\}\right..\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[k \mapsto \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle \textrm{ if } k \in \{0 + 1~\ldotp \ldotp~1\}\right..\right] =  \\ \left[k \mapsto \left\{\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle \textrm{ if } k \in \{1~\ldotp \ldotp~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: 12 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 48 body: 11
9	Lambda	parameter: 48 body: 14
10	Lambda	parameter: 48 body: 15
11	Operation	operator: 12 operands: 13
12	Literal
13	ExprTuple	14, 15
14	Conditional	value: 17 condition: 16
15	Conditional	value: 17 condition: 18
16	Operation	operator: 22 operands: 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	ExprTuple	48, 24
20	Literal
21	ExprTuple	25, 26
22	Literal
23	ExprTuple	48, 27
24	Operation	operator: 33 operands: 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operand: 48
27	Operation	operator: 33 operands: 34
28	ExprTuple	35, 43
29	Literal
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	48
33	Literal
34	ExprTuple	43, 43
35	Operation	operator: 38 operands: 39
36	Literal
37	Operation	operator: 40 operands: 41
38	Literal
39	ExprTuple	42, 43
40	Literal
41	ExprTuple	44, 45, 46, 47, 48
42	Literal
43	Literal
44	Literal
45	Literal
46	Literal
47	Literal
48	Variable