Expression of type Forall

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, 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 = 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))

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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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