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
from proveit.logic import And, Equals, Forall, Iff, InSet

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(P, 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))

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())

P \mapsto \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)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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