Expression of type Forall

from the theory of proveit.physics.quantum.algebra

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, k
from proveit.core_expr_types import a_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals, Forall
from proveit.numbers import BinarySequence, Bit, NaturalPos, one
from proveit.physics.quantum import Ket, NumKet

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [k], instance_expr = Forall(instance_param_or_params = [a_1_to_k], instance_expr = Equals(NumKet(BinarySequence(a_1_to_k), k), TensorProd(ExprRange(sub_expr1, Ket(IndexedVar(a, sub_expr1)), one, k))), domain = Bit), 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_{k \in \mathbb{N}^+}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{k} \in \mathbb{N}^{\leq 1}}~\left(\lvert \texttt{\#}\texttt{b}(a_{1}, a_{2}, \ldots, a_{k}) \rangle_{k} = \left(\lvert a_{1} \rangle {\otimes}  \lvert a_{2} \rangle {\otimes}  \ldots {\otimes}  \lvert a_{k} \rangle\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 40 body: 4
3	ExprTuple	40
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 33 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	40, 11
10	Lambda	parameters: 31 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	ExprRange	lambda_map: 26 start_index: 39 end_index: 40
22	Literal
23	ExprTuple	27, 40
24	Literal
25	ExprTuple	28
26	Lambda	parameter: 46 body: 29
27	Operation	operator: 30 operands: 31
28	ExprRange	lambda_map: 32 start_index: 39 end_index: 40
29	Operation	operator: 33 operands: 34
30	Literal
31	ExprTuple	35
32	Lambda	parameter: 46 body: 36
33	Literal
34	ExprTuple	43, 37
35	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
36	Operation	operator: 41 operand: 43
37	Literal
38	Lambda	parameter: 46 body: 43
39	Literal
40	Variable
41	Literal
42	ExprTuple	43
43	IndexedVar	variable: 44 index: 46
44	Variable
45	ExprTuple	46
46	Variable