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.logic import Equals, Forall, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = InSet(k, _m_domain)
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(Conditional(Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), sub_expr1), Conditional(one, sub_expr1)))

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}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. = \left\{1 \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 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: 42 body: 5
4	ExprTuple	42
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Conditional	value: 10 condition: 11
9	Conditional	value: 38 condition: 11
10	Operation	operator: 44 operands: 12
11	Operation	operator: 13 operands: 14
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	42, 17
15	Operation	operator: 48 operands: 18
16	Operation	operator: 48 operands: 19
17	Operation	operator: 20 operands: 21
18	ExprTuple	23, 22
19	ExprTuple	23, 24
20	Literal
21	ExprTuple	25, 26
22	Operation	operator: 35 operand: 31
23	Literal
24	Operation	operator: 44 operands: 28
25	Literal
26	Operation	operator: 29 operands: 30
27	ExprTuple	31
28	ExprTuple	50, 40, 41, 47, 42
29	Literal
30	ExprTuple	46, 32
31	Operation	operator: 33 operands: 34
32	Operation	operator: 35 operand: 38
33	Literal
34	ExprTuple	37, 46
35	Literal
36	ExprTuple	38
37	Operation	operator: 44 operands: 39
38	Literal
39	ExprTuple	50, 40, 41, 42, 43
40	Literal
41	Literal
42	Variable
43	Operation	operator: 44 operands: 45
44	Literal
45	ExprTuple	46, 47
46	Operation	operator: 48 operands: 49
47	Literal
48	Literal
49	ExprTuple	50, 51
50	Literal
51	Literal