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 ExprRange, IndexedVar, Variable, a, c, k, m
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Exp, Mult, Neg, Sum, e, frac, i, one, pi, two, zero
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [k]
sub_expr3 = IndexedVar(a, one)
sub_expr4 = ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, zero)
sub_expr5 = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))
expr = Forall(instance_param_or_params = [sub_expr3, sub_expr4], instance_expr = Equals(Mult(sub_expr3, Sum(index_or_indices = sub_expr2, summand = sub_expr5, domain = _m_domain), sub_expr4), Sum(index_or_indices = sub_expr2, summand = Mult(sub_expr3, sub_expr5, sub_expr4), domain = _m_domain)).with_wrapping_at(2), domain = Complex).with_wrapping()

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}{l}\forall_{a_{1}, c_{1}, c_{2}, \ldots, c_{0} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right) =  \\ \left(\sum_{k = 0}^{2^{t} - 1} \left(a_{1} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right)\right) \end{array} \end{array}\right)\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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	parameters: 4 body: 5
4	ExprTuple	34, 36
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	14, 15
12	Operation	operator: 72 operands: 16
13	Operation	operator: 23 operand: 21
14	Operation	operator: 37 operands: 18
15	ExprRange	lambda_map: 19 start_index: 69 end_index: 52
16	ExprTuple	34, 20, 36
17	ExprTuple	21
18	ExprTuple	34, 29
19	Lambda	parameter: 57 body: 22
20	Operation	operator: 23 operand: 27
21	Lambda	parameter: 78 body: 25
22	Operation	operator: 37 operands: 26
23	Literal
24	ExprTuple	27
25	Conditional	value: 28 condition: 33
26	ExprTuple	45, 29
27	Lambda	parameter: 78 body: 31
28	Operation	operator: 72 operands: 32
29	Literal
30	ExprTuple	78
31	Conditional	value: 35 condition: 33
32	ExprTuple	34, 35, 36
33	Operation	operator: 37 operands: 38
34	IndexedVar	variable: 39 index: 69
35	Operation	operator: 72 operands: 40
36	ExprRange	lambda_map: 41 start_index: 69 end_index: 52
37	Literal
38	ExprTuple	78, 42
39	Variable
40	ExprTuple	43, 44
41	Lambda	parameter: 57 body: 45
42	Operation	operator: 46 operands: 47
43	Operation	operator: 74 operands: 48
44	Operation	operator: 74 operands: 49
45	IndexedVar	variable: 50 index: 57
46	Literal
47	ExprTuple	52, 53
48	ExprTuple	55, 54
49	ExprTuple	55, 56
50	Variable
51	ExprTuple	57
52	Literal
53	Operation	operator: 58 operands: 59
54	Operation	operator: 72 operands: 60
55	Literal
56	Operation	operator: 65 operand: 64
57	Variable
58	Literal
59	ExprTuple	71, 62
60	ExprTuple	80, 76, 77, 63, 78
61	ExprTuple	64
62	Operation	operator: 65 operand: 69
63	Literal
64	Operation	operator: 67 operands: 68
65	Literal
66	ExprTuple	69
67	Literal
68	ExprTuple	70, 71
69	Literal
70	Operation	operator: 72 operands: 73
71	Operation	operator: 74 operands: 75
72	Literal
73	ExprTuple	80, 76, 77, 78, 79
74	Literal
75	ExprTuple	80, 81
76	Literal
77	Literal
78	Variable
79	Variable
80	Literal
81	Literal