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 Hspace, InnerProd, LinMap, OrthoProj, VecZero
from proveit.logic import And, Equals, Forall, Iff, Implies, InSet
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Conditional(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), InSet(P, LinMap(Hspace, 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\{\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} \textrm{ if } P \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\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: 3 operands: 4
2	Operation	operator: 52 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	43, 8
6	Operation	operator: 50 operands: 9
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	ExprTuple	43, 14
10	Literal
11	ExprTuple	15, 16
12	Literal
13	ExprTuple	49, 49
14	Operation	operator: 17 operands: 18
15	Operation	operator: 35 operand: 21
16	Operation	operator: 35 operand: 22
17	Literal
18	ExprTuple	49, 56
19	ExprTuple	21
20	ExprTuple	22
21	Lambda	parameter: 59 body: 23
22	Lambda	parameter: 60 body: 24
23	Conditional	value: 25 condition: 48
24	Conditional	value: 26 condition: 27
25	Operation	operator: 50 operands: 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 52 operands: 31
28	ExprTuple	32, 59
29	Literal
30	ExprTuple	33, 34
31	ExprTuple	60, 49
32	Operation	operator: 43 operand: 59
33	Operation	operator: 35 operand: 38
34	Operation	operator: 50 operands: 37
35	Literal
36	ExprTuple	38
37	ExprTuple	39, 40
38	Lambda	parameter: 59 body: 42
39	Operation	operator: 43 operand: 60
40	Operation	operator: 45 operand: 49
41	ExprTuple	59
42	Conditional	value: 47 condition: 48
43	Variable
44	ExprTuple	60
45	Literal
46	ExprTuple	49
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operands: 53
49	Variable
50	Literal
51	ExprTuple	54, 55
52	Literal
53	ExprTuple	59, 56
54	Operation	operator: 57 operands: 58
55	Literal
56	Variable
57	Literal
58	ExprTuple	59, 60
59	Variable
60	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 ¶