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 A, B, X, Y
from proveit.linear_algebra import HilbertSpaces, Hspace, LinMap
from proveit.logic import Exists, Forall, InClass, InSet, Set
from proveit.numbers import Complex
from proveit.physics.quantum import Qmult, QmultCodomain

# build up the expression from sub-expressions
sub_expr1 = LinMap(Hspace, X)
expr = Forall(instance_param_or_params = [A, B], instance_expr = Exists(instance_param_or_params = [Hspace, X, Y], instance_expr = InSet([Qmult(A), Qmult(B)], Set([sub_expr1, Hspace], [X, LinMap(Hspace, Complex)], [LinMap(X, Y), sub_expr1], [Complex, QmultCodomain], [QmultCodomain, Complex])), domain = HilbertSpaces), condition = InClass(Qmult(A, B), QmultCodomain))

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_{A, B~|~\left(A \thinspace B\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}}~\left[\exists_{\mathcal{H}, X, Y \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}}~\left(\left(\left[A\right], \left[B\right]\right) \in \left\{\left(\mathcal{L}\left(\mathcal{H}, X\right), \mathcal{H}\right), \left(X, \mathcal{L}\left(\mathcal{H}, \mathbb{C}\right)\right), \left(\mathcal{L}\left(X, Y\right), \mathcal{L}\left(\mathcal{H}, X\right)\right), \left(\mathbb{C}, \mathcal{Q^*}\right), \left(\mathcal{Q^*}, \mathbb{C}\right)\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	parameters: 14 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 32 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 48
10	Lambda	parameters: 12 body: 13
11	Operation	operator: 35 operands: 14
12	ExprTuple	55, 56, 54
13	Conditional	value: 15 condition: 16
14	ExprTuple	43, 44
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24, 25
21	ExprTuple	26, 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 32 operands: 30
24	Operation	operator: 32 operands: 31
25	Operation	operator: 32 operands: 33
26	Operation	operator: 35 operand: 43
27	Operation	operator: 35 operand: 44
28	Literal
29	ExprTuple	37, 38, 39, 40, 41
30	ExprTuple	55, 42
31	ExprTuple	56, 42
32	Literal
33	ExprTuple	54, 42
34	ExprTuple	43
35	Literal
36	ExprTuple	44
37	ExprTuple	47, 55
38	ExprTuple	56, 45
39	ExprTuple	46, 47
40	ExprTuple	53, 48
41	ExprTuple	48, 53
42	Literal
43	Variable
44	Variable
45	Operation	operator: 51 operands: 49
46	Operation	operator: 51 operands: 50
47	Operation	operator: 51 operands: 52
48	Literal
49	ExprTuple	55, 53
50	ExprTuple	56, 54
51	Literal
52	ExprTuple	55, 56
53	Literal
54	Variable
55	Variable
56	Variable