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

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

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}^{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}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right.. = \left\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{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..\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: 49 body: 5
4	ExprTuple	49
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Conditional	value: 10 condition: 12
9	Conditional	value: 11 condition: 12
10	Operation	operator: 43 operands: 13
11	Operation	operator: 43 operands: 14
12	Operation	operator: 15 operands: 16
13	ExprTuple	18, 17
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	49, 19
17	Operation	operator: 45 operands: 20
18	Operation	operator: 45 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	25, 24
21	ExprTuple	25, 26
22	Literal
23	ExprTuple	27, 28
24	Operation	operator: 38 operand: 33
25	Literal
26	Operation	operator: 43 operands: 30
27	Literal
28	Operation	operator: 31 operands: 32
29	ExprTuple	33
30	ExprTuple	51, 47, 48, 34, 49
31	Literal
32	ExprTuple	41, 35
33	Operation	operator: 36 operands: 37
34	Literal
35	Operation	operator: 38 operand: 42
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	42
40	Operation	operator: 43 operands: 44
41	Operation	operator: 45 operands: 46
42	Literal
43	Literal
44	ExprTuple	51, 47, 48, 49, 50
45	Literal
46	ExprTuple	51, 52
47	Literal
48	Literal
49	Variable
50	Variable
51	Literal
52	Literal