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, VecSpaces
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 = [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())

\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]

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