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, VecSum
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Interval, Mult, NaturalPos, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, _psi_t_ket, two_pow_t

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [t], instance_expr = Equals(_psi_t_ket, ScalarMult(frac(one, Exp(two, frac(t, two))), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one))))), domain = NaturalPos)

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_{t \in \mathbb{N}^+}~\left(\lvert \psi_{t} \rangle = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\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: 62 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 33 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	62, 12
10	Operation	operator: 13 operand: 62
11	Operation	operator: 31 operands: 15
12	Literal
13	Literal
14	ExprTuple	62
15	ExprTuple	16, 17
16	Operation	operator: 29 operands: 18
17	Operation	operator: 19 operand: 22
18	ExprTuple	63, 21
19	Literal
20	ExprTuple	22
21	Operation	operator: 57 operands: 23
22	Lambda	parameter: 54 body: 25
23	ExprTuple	61, 26
24	ExprTuple	54
25	Conditional	value: 27 condition: 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	Literal
30	ExprTuple	62, 61
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	54, 37
35	Operation	operator: 57 operands: 38
36	Operation	operator: 39 operands: 40
37	Operation	operator: 41 operands: 42
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	54, 62
41	Literal
42	ExprTuple	45, 46
43	Literal
44	Operation	operator: 47 operands: 48
45	Literal
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	61, 51, 52, 53, 54
49	Literal
50	ExprTuple	55, 56
51	Literal
52	Literal
53	Literal
54	Variable
55	Operation	operator: 57 operands: 58
56	Operation	operator: 59 operand: 63
57	Literal
58	ExprTuple	61, 62
59	Literal
60	ExprTuple	63
61	Literal
62	Variable
63	Literal