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 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, InSet

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(K, Forall(instance_param_or_params = [V, W], instance_expr = 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)), domain = VecSpaces(K)))

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())

K \mapsto \left[\forall_{V, W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\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)\right]\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 59 body: 1
1	Operation	operator: 51 operand: 3
2	ExprTuple	3
3	Lambda	parameters: 31 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 51 operand: 9
6	Operation	operator: 49 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameter: 85 body: 13
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	85
13	Operation	operator: 17 operands: 18
14	ExprTuple	78, 19
15	Literal
16	ExprTuple	54, 19
17	Literal
18	ExprTuple	20, 21
19	Operation	operator: 22 operand: 59
20	Operation	operator: 73 operands: 24
21	Operation	operator: 49 operands: 25
22	Literal
23	ExprTuple	59
24	ExprTuple	85, 26
25	ExprTuple	27, 28, 29
26	Operation	operator: 30 operands: 31
27	Operation	operator: 51 operand: 35
28	Operation	operator: 51 operand: 36
29	Operation	operator: 51 operand: 37
30	Literal
31	ExprTuple	78, 54
32	ExprTuple	35
33	ExprTuple	36
34	ExprTuple	37
35	Lambda	parameter: 88 body: 38
36	Lambda	parameters: 69 body: 39
37	Lambda	parameter: 87 body: 41
38	Conditional	value: 42 condition: 67
39	Conditional	value: 43 condition: 44
40	ExprTuple	87
41	Conditional	value: 45 condition: 46
42	Operation	operator: 73 operands: 47
43	Operation	operator: 71 operands: 48
44	Operation	operator: 49 operands: 50
45	Operation	operator: 51 operand: 58
46	Operation	operator: 73 operands: 53
47	ExprTuple	82, 54
48	ExprTuple	55, 56
49	Literal
50	ExprTuple	67, 57
51	Literal
52	ExprTuple	58
53	ExprTuple	87, 59
54	Variable
55	Operation	operator: 85 operand: 64
56	Operation	operator: 68 operands: 61
57	Operation	operator: 73 operands: 62
58	Lambda	parameter: 88 body: 63
59	Variable
60	ExprTuple	64
61	ExprTuple	82, 65
62	ExprTuple	75, 78
63	Conditional	value: 66 condition: 67
64	Operation	operator: 68 operands: 69
65	Operation	operator: 85 operand: 75
66	Operation	operator: 71 operands: 72
67	Operation	operator: 73 operands: 74
68	Literal
69	ExprTuple	88, 75
70	ExprTuple	75
71	Literal
72	ExprTuple	76, 77
73	Literal
74	ExprTuple	88, 78
75	Variable
76	Operation	operator: 85 operand: 81
77	Operation	operator: 83 operands: 80
78	Variable
79	ExprTuple	81
80	ExprTuple	87, 82
81	Operation	operator: 83 operands: 84
82	Operation	operator: 85 operand: 88
83	Literal
84	ExprTuple	87, 88
85	Variable
86	ExprTuple	88
87	Variable
88	Variable