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, H, K, n, v, w
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, InnerProd, InnerProdSpaces, OrthoNormBases
from proveit.logic import And, Equals, Forall, InClass, InSet, Set
from proveit.numbers import KroneckerDelta, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Conditional(Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(InSet(sub_expr1, OrthoNormBases(H)), And(InSet(sub_expr1, Bases(H)), Forall(instance_param_or_params = [v, w], instance_expr = Equals(InnerProd(v, w), KroneckerDelta(v, w)), domain = sub_expr1))).with_wrapping_at(2), domain = H), domain = NaturalPos), InClass(H, InnerProdSpaces(K)))

# 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_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in H}~\left(\begin{array}{c} \begin{array}{l} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(H\right)\right) =  \\ \left(\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{Bases}\left(H\right)\right) \land \left[\forall_{v, w \in \left\{x_{1}, x_{2}, \ldots, x_{n}\right\}}~\left(\left \langle v, w\right \rangle = \delta_{v, w}\right)\right]\right) \end{array} \end{array}\right)\right] \textrm{ if } H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\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: 36 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	44, 7
6	Lambda	parameter: 69 body: 9
7	Operation	operator: 10 operand: 14
8	ExprTuple	69
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	14
12	Operation	operator: 36 operand: 17
13	Operation	operator: 59 operands: 16
14	Variable
15	ExprTuple	17
16	ExprTuple	69, 18
17	Lambda	parameters: 65 body: 19
18	Literal
19	Conditional	value: 20 condition: 21
20	Operation	operator: 47 operands: 22
21	Operation	operator: 49 operands: 23
22	ExprTuple	24, 25
23	ExprTuple	26
24	Operation	operator: 59 operands: 27
25	Operation	operator: 49 operands: 28
26	ExprRange	lambda_map: 29 start_index: 68 end_index: 69
27	ExprTuple	63, 30
28	ExprTuple	31, 32
29	Lambda	parameter: 73 body: 33
30	Operation	operator: 34 operand: 44
31	Operation	operator: 59 operands: 35
32	Operation	operator: 36 operand: 40
33	Operation	operator: 59 operands: 38
34	Literal
35	ExprTuple	63, 39
36	Literal
37	ExprTuple	40
38	ExprTuple	70, 44
39	Operation	operator: 41 operand: 44
40	Lambda	parameters: 57 body: 43
41	Literal
42	ExprTuple	44
43	Conditional	value: 45 condition: 46
44	Variable
45	Operation	operator: 47 operands: 48
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53, 54
51	Operation	operator: 55 operands: 57
52	Operation	operator: 56 operands: 57
53	Operation	operator: 59 operands: 58
54	Operation	operator: 59 operands: 60
55	Literal
56	Literal
57	ExprTuple	61, 62
58	ExprTuple	61, 63
59	Literal
60	ExprTuple	62, 63
61	Variable
62	Variable
63	Operation	operator: 64 operands: 65
64	Literal
65	ExprTuple	66
66	ExprRange	lambda_map: 67 start_index: 68 end_index: 69
67	Lambda	parameter: 73 body: 70
68	Literal
69	Variable
70	IndexedVar	variable: 71 index: 73
71	Variable
72	ExprTuple	73
73	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 ¶