Expression of type Conditional ¶

from the theory of proveit.linear_algebra.inner_products ¶

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, P, Px, Py, X, x, y
from proveit.linear_algebra import HilbertSpaces, Hspace, InnerProd, LinMap, OrthoProj, VecZero
from proveit.logic import And, Equals, Forall, Iff, Implies, InClass, SubsetEq
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Conditional(Forall(instance_param_or_params = [P], instance_expr = Iff(Equals(P, OrthoProj(Hspace, X)), And(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Px, x), domain = X), Forall(instance_param_or_params = [y], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), domain = Hspace)).with_wrapping_at(2)).with_wrapping_at(2), domain = LinMap(Hspace, Hspace)), And(InClass(Hspace, HilbertSpaces), InClass(X, HilbertSpaces), SubsetEq(X, Hspace)))

# 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())

\left\{\forall_{P \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(P = \textrm{OrthoProj}\left(\mathcal{H}, X\right)\right) \Leftrightarrow  \\ \left(\begin{array}{c} \left[\forall_{x \in X}~\left(P\left(x\right) = x\right)\right] \land  \\ \left[\forall_{y \in \mathcal{H}}~\left(\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right)\right)\right] \end{array}\right) \end{array} \end{array}\right) \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  X \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  X \subseteq \mathcal{H}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 51 operand: 5
2	Operation	operator: 26 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameter: 59 body: 10
6	Operation	operator: 12 operands: 11
7	Operation	operator: 12 operands: 13
8	Operation	operator: 14 operands: 15
9	ExprTuple	59
10	Conditional	value: 16 condition: 17
11	ExprTuple	65, 18
12	Literal
13	ExprTuple	72, 18
14	Literal
15	ExprTuple	72, 65
16	Operation	operator: 19 operands: 20
17	Operation	operator: 68 operands: 21
18	Literal
19	Literal
20	ExprTuple	22, 23
21	ExprTuple	59, 24
22	Operation	operator: 66 operands: 25
23	Operation	operator: 26 operands: 27
24	Operation	operator: 28 operands: 29
25	ExprTuple	59, 30
26	Literal
27	ExprTuple	31, 32
28	Literal
29	ExprTuple	65, 65
30	Operation	operator: 33 operands: 34
31	Operation	operator: 51 operand: 37
32	Operation	operator: 51 operand: 38
33	Literal
34	ExprTuple	65, 72
35	ExprTuple	37
36	ExprTuple	38
37	Lambda	parameter: 75 body: 39
38	Lambda	parameter: 76 body: 40
39	Conditional	value: 41 condition: 64
40	Conditional	value: 42 condition: 43
41	Operation	operator: 66 operands: 44
42	Operation	operator: 45 operands: 46
43	Operation	operator: 68 operands: 47
44	ExprTuple	48, 75
45	Literal
46	ExprTuple	49, 50
47	ExprTuple	76, 65
48	Operation	operator: 59 operand: 75
49	Operation	operator: 51 operand: 54
50	Operation	operator: 66 operands: 53
51	Literal
52	ExprTuple	54
53	ExprTuple	55, 56
54	Lambda	parameter: 75 body: 58
55	Operation	operator: 59 operand: 76
56	Operation	operator: 61 operand: 65
57	ExprTuple	75
58	Conditional	value: 63 condition: 64
59	Variable
60	ExprTuple	76
61	Literal
62	ExprTuple	65
63	Operation	operator: 66 operands: 67
64	Operation	operator: 68 operands: 69
65	Variable
66	Literal
67	ExprTuple	70, 71
68	Literal
69	ExprTuple	75, 72
70	Operation	operator: 73 operands: 74
71	Literal
72	Variable
73	Literal
74	ExprTuple	75, 76
75	Variable
76	Variable

Expression of type Conditional¶

from the theory of proveit.linear_algebra.inner_products¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.inner_products ¶