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 k, t
from proveit.linear_algebra import ScalarMult, TensorProd
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(k, two_pow_t)
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), NumKet(sub_expr1, Add(t, one))), ScalarMult(Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Mult(two, pi, i, _phase, two_pow_t))), TensorProd(ket1, NumKet(k, t)))), domain = Interval(zero, subtract(two_pow_t, 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 \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\right) = \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\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: 58 body: 5
4	ExprTuple	58
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	58, 14
12	Operation	operator: 16 operands: 15
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operands: 19
15	ExprTuple	20, 21
16	Literal
17	ExprTuple	22, 23
18	Literal
19	ExprTuple	24, 25
20	Operation	operator: 63 operands: 26
21	Operation	operator: 44 operands: 27
22	Operation	operator: 56 operands: 28
23	Operation	operator: 29 operands: 30
24	Literal
25	Operation	operator: 53 operands: 31
26	ExprTuple	50, 32
27	ExprTuple	48, 33
28	ExprTuple	34, 35
29	Literal
30	ExprTuple	36, 37
31	ExprTuple	62, 38
32	Operation	operator: 56 operands: 39
33	Operation	operator: 53 operands: 40
34	Operation	operator: 63 operands: 41
35	Operation	operator: 63 operands: 42
36	Operation	operator: 43 operand: 52
37	Operation	operator: 44 operands: 45
38	Operation	operator: 46 operand: 52
39	ExprTuple	65, 59, 60, 61, 48
40	ExprTuple	66, 52
41	ExprTuple	50, 49
42	ExprTuple	50, 51
43	Literal
44	Literal
45	ExprTuple	58, 66
46	Literal
47	ExprTuple	52
48	Operation	operator: 53 operands: 54
49	Operation	operator: 56 operands: 55
50	Literal
51	Operation	operator: 56 operands: 57
52	Literal
53	Literal
54	ExprTuple	58, 62
55	ExprTuple	65, 59, 60, 61, 58
56	Literal
57	ExprTuple	65, 59, 60, 61, 62
58	Variable
59	Literal
60	Literal
61	Literal
62	Operation	operator: 63 operands: 64
63	Literal
64	ExprTuple	65, 66
65	Literal
66	Variable