Expression of type Conditional ¶

from the theory of proveit.linear_algebra.linear_maps ¶

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, Function, K, P, Px, Py, V, W, c, x, y
from proveit.linear_algebra import LinMap, ScalarMult, VecAdd, VecSpaces
from proveit.logic import And, Equals, Forall, Iff, InClass, InSet

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = VecSpaces(K)
expr = Conditional(Forall(instance_param_or_params = [P], instance_expr = Iff(InSet(P, LinMap(V, W)), And(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(Px, W), domain = V), Forall(instance_param_or_params = [x, y], instance_expr = Equals(Function(P, [VecAdd(x, y)]), VecAdd(Px, Py)), domain = V), Forall(instance_param_or_params = [c], instance_expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Function(P, [ScalarMult(c, x)]), ScalarMult(c, Px)), domain = V), domain = K)).with_wrapping_at(2,4)).with_wrapping_at(2)), And(InClass(V, sub_expr2), InClass(W, sub_expr2)))

# 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}~\left(\begin{array}{c} \begin{array}{l} \left(P \in \mathcal{L}\left(V, W\right)\right) \Leftrightarrow  \\ \left(\begin{array}{c} \left[\forall_{x \in V}~\left(P\left(x\right) \in W\right)\right] \land  \\ \left[\forall_{x, y \in V}~\left(P\left(x + y\right) = \left(P\left(x\right) + P\left(y\right)\right)\right)\right] \land  \\ \left[\forall_{c \in K}~\left[\forall_{x \in V}~\left(P\left(c \cdot x\right) = \left(c \cdot P\left(x\right)\right)\right)\right]\right] \end{array}\right) \end{array} \end{array}\right) \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right) ,  W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\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: 47 operand: 5
2	Operation	operator: 45 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameter: 81 body: 9
6	Operation	operator: 11 operands: 10
7	Operation	operator: 11 operands: 12
8	ExprTuple	81
9	Operation	operator: 13 operands: 14
10	ExprTuple	74, 15
11	Literal
12	ExprTuple	50, 15
13	Literal
14	ExprTuple	16, 17
15	Operation	operator: 18 operand: 55
16	Operation	operator: 69 operands: 20
17	Operation	operator: 45 operands: 21
18	Literal
19	ExprTuple	55
20	ExprTuple	81, 22
21	ExprTuple	23, 24, 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 47 operand: 31
24	Operation	operator: 47 operand: 32
25	Operation	operator: 47 operand: 33
26	Literal
27	ExprTuple	74, 50
28	ExprTuple	31
29	ExprTuple	32
30	ExprTuple	33
31	Lambda	parameter: 84 body: 34
32	Lambda	parameters: 65 body: 35
33	Lambda	parameter: 83 body: 37
34	Conditional	value: 38 condition: 63
35	Conditional	value: 39 condition: 40
36	ExprTuple	83
37	Conditional	value: 41 condition: 42
38	Operation	operator: 69 operands: 43
39	Operation	operator: 67 operands: 44
40	Operation	operator: 45 operands: 46
41	Operation	operator: 47 operand: 54
42	Operation	operator: 69 operands: 49
43	ExprTuple	78, 50
44	ExprTuple	51, 52
45	Literal
46	ExprTuple	63, 53
47	Literal
48	ExprTuple	54
49	ExprTuple	83, 55
50	Variable
51	Operation	operator: 81 operand: 60
52	Operation	operator: 64 operands: 57
53	Operation	operator: 69 operands: 58
54	Lambda	parameter: 84 body: 59
55	Variable
56	ExprTuple	60
57	ExprTuple	78, 61
58	ExprTuple	71, 74
59	Conditional	value: 62 condition: 63
60	Operation	operator: 64 operands: 65
61	Operation	operator: 81 operand: 71
62	Operation	operator: 67 operands: 68
63	Operation	operator: 69 operands: 70
64	Literal
65	ExprTuple	84, 71
66	ExprTuple	71
67	Literal
68	ExprTuple	72, 73
69	Literal
70	ExprTuple	84, 74
71	Variable
72	Operation	operator: 81 operand: 77
73	Operation	operator: 79 operands: 76
74	Variable
75	ExprTuple	77
76	ExprTuple	83, 78
77	Operation	operator: 79 operands: 80
78	Operation	operator: 81 operand: 84
79	Literal
80	ExprTuple	83, 84
81	Variable
82	ExprTuple	84
83	Variable
84	Variable

Expression of type Conditional¶

from the theory of proveit.linear_algebra.linear_maps¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.linear_maps ¶