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 M, m, n
from proveit.linear_algebra import LinMap, MatrixSpace
from proveit.logic import CartExp, Forall, InSet
from proveit.numbers import Complex, NaturalPos
from proveit.physics.quantum import Qmult

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [M], instance_expr = InSet(Qmult(M), LinMap(CartExp(Complex, n), CartExp(Complex, m))), domain = MatrixSpace(field = Complex, rows = m, columns = n)), 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_{m, n \in \mathbb{N}^+}~\left[\forall_{M \in \mathbb{C}^{m \times n}}~\left(\left[M\right] \in \mathcal{L}\left(\mathbb{C}^{n}, \mathbb{C}^{m}\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	parameters: 3 body: 4
3	ExprTuple	40, 38
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13
11	Lambda	parameter: 32 body: 14
12	Operation	operator: 21 operands: 15
13	Operation	operator: 21 operands: 16
14	Conditional	value: 17 condition: 18
15	ExprTuple	40, 19
16	ExprTuple	38, 19
17	Operation	operator: 21 operands: 20
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	32, 25
23	Operation	operator: 26 operand: 32
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	32
28	Literal
29	ExprTuple	33, 34
30	Literal
31	NamedExprs	field: 39 rows: 40 columns: 38
32	Variable
33	Operation	operator: 36 operands: 35
34	Operation	operator: 36 operands: 37
35	ExprTuple	39, 38
36	Literal
37	ExprTuple	39, 40
38	Variable
39	Literal
40	Variable