Expression of type Lambda

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, Lambda, 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 = Lambda([V, W], 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))))

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())

\left(V, W\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 28 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 48 operand: 6
3	Operation	operator: 46 operands: 5
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameter: 82 body: 10
7	Operation	operator: 12 operands: 11
8	Operation	operator: 12 operands: 13
9	ExprTuple	82
10	Operation	operator: 14 operands: 15
11	ExprTuple	75, 16
12	Literal
13	ExprTuple	51, 16
14	Literal
15	ExprTuple	17, 18
16	Operation	operator: 19 operand: 56
17	Operation	operator: 70 operands: 21
18	Operation	operator: 46 operands: 22
19	Literal
20	ExprTuple	56
21	ExprTuple	82, 23
22	ExprTuple	24, 25, 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 48 operand: 32
25	Operation	operator: 48 operand: 33
26	Operation	operator: 48 operand: 34
27	Literal
28	ExprTuple	75, 51
29	ExprTuple	32
30	ExprTuple	33
31	ExprTuple	34
32	Lambda	parameter: 85 body: 35
33	Lambda	parameters: 66 body: 36
34	Lambda	parameter: 84 body: 38
35	Conditional	value: 39 condition: 64
36	Conditional	value: 40 condition: 41
37	ExprTuple	84
38	Conditional	value: 42 condition: 43
39	Operation	operator: 70 operands: 44
40	Operation	operator: 68 operands: 45
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operand: 55
43	Operation	operator: 70 operands: 50
44	ExprTuple	79, 51
45	ExprTuple	52, 53
46	Literal
47	ExprTuple	64, 54
48	Literal
49	ExprTuple	55
50	ExprTuple	84, 56
51	Variable
52	Operation	operator: 82 operand: 61
53	Operation	operator: 65 operands: 58
54	Operation	operator: 70 operands: 59
55	Lambda	parameter: 85 body: 60
56	Variable
57	ExprTuple	61
58	ExprTuple	79, 62
59	ExprTuple	72, 75
60	Conditional	value: 63 condition: 64
61	Operation	operator: 65 operands: 66
62	Operation	operator: 82 operand: 72
63	Operation	operator: 68 operands: 69
64	Operation	operator: 70 operands: 71
65	Literal
66	ExprTuple	85, 72
67	ExprTuple	72
68	Literal
69	ExprTuple	73, 74
70	Literal
71	ExprTuple	85, 75
72	Variable
73	Operation	operator: 82 operand: 78
74	Operation	operator: 80 operands: 77
75	Variable
76	ExprTuple	78
77	ExprTuple	84, 79
78	Operation	operator: 80 operands: 81
79	Operation	operator: 82 operand: 85
80	Literal
81	ExprTuple	84, 85
82	Variable
83	ExprTuple	85
84	Variable
85	Variable