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

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(Mult(sub_expr1, sub_expr2, sub_expr3), Mult(sub_expr2, Mult(sub_expr1, sub_expr3))), domain = _m_domain)

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 k} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) = \left(\frac{1}{2^{\frac{t}{2}}} \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)\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: 56 body: 5
4	ExprTuple	56
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	56, 14
12	Operation	operator: 50 operands: 15
13	Operation	operator: 50 operands: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	28, 19, 29
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 46 operands: 23
20	Operation	operator: 50 operands: 24
21	Literal
22	Operation	operator: 25 operands: 26
23	ExprTuple	39, 27
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	49, 30
27	Operation	operator: 52 operands: 31
28	Operation	operator: 52 operands: 32
29	Operation	operator: 52 operands: 33
30	Operation	operator: 42 operand: 39
31	ExprTuple	58, 35
32	ExprTuple	37, 36
33	ExprTuple	37, 38
34	ExprTuple	39
35	Operation	operator: 46 operands: 40
36	Operation	operator: 50 operands: 41
37	Literal
38	Operation	operator: 42 operand: 45
39	Literal
40	ExprTuple	59, 58
41	ExprTuple	58, 54, 55, 44, 56
42	Literal
43	ExprTuple	45
44	Literal
45	Operation	operator: 46 operands: 47
46	Literal
47	ExprTuple	48, 49
48	Operation	operator: 50 operands: 51
49	Operation	operator: 52 operands: 53
50	Literal
51	ExprTuple	58, 54, 55, 56, 57
52	Literal
53	ExprTuple	58, 59
54	Literal
55	Literal
56	Variable
57	Variable
58	Literal
59	Literal