Expression of type Forall

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, k
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, Forall, 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))
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(Conditional(sub_expr1, InSet(k, Interval(Add(zero, one), one))), Conditional(sub_expr1, InSet(k, Interval(one, one)))))

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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 41 body: 4
4	Operation	operator: 5 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Conditional	value: 10 condition: 9
8	Conditional	value: 10 condition: 11
9	Operation	operator: 15 operands: 12
10	Operation	operator: 13 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	41, 17
13	Literal
14	ExprTuple	18, 19
15	Literal
16	ExprTuple	41, 20
17	Operation	operator: 26 operands: 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 24 operand: 41
20	Operation	operator: 26 operands: 27
21	ExprTuple	28, 36
22	Literal
23	ExprTuple	29, 30
24	Literal
25	ExprTuple	41
26	Literal
27	ExprTuple	36, 36
28	Operation	operator: 31 operands: 32
29	Literal
30	Operation	operator: 33 operands: 34
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37, 38, 39, 40, 41
35	Literal
36	Literal
37	Literal
38	Literal
39	Literal
40	Literal
41	Variable