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 A, B, Conditional, Function, K, Lambda, V, W, x
from proveit.linear_algebra import LinMap, VecAdd, VecSpaces
from proveit.logic import Equals, Forall, InSet

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [V, W], instance_expr = Forall(instance_param_or_params = [A, B], instance_expr = Equals(VecAdd(A, B), Lambda(x, Conditional(VecAdd(Function(A, sub_expr1), Function(B, sub_expr1)), InSet(x, V)))), domain = LinMap(V, W)), 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_{K}~\left[\forall_{V, W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{A, B \in \mathcal{L}\left(V, W\right)}~\left(\left(A + B\right) = \left[x \mapsto \left\{A\left(x\right) + B\left(x\right) \textrm{ if } x \in V\right..\right]\right)\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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 32 body: 3
3	Operation	operator: 9 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 45 body: 6
6	Conditional	value: 7 condition: 8
7	Operation	operator: 9 operand: 12
8	Operation	operator: 24 operands: 11
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Lambda	parameters: 33 body: 15
13	Operation	operator: 17 operands: 16
14	Operation	operator: 17 operands: 18
15	Conditional	value: 19 condition: 20
16	ExprTuple	48, 21
17	Literal
18	ExprTuple	49, 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 26 operand: 32
22	Literal
23	ExprTuple	28, 29
24	Literal
25	ExprTuple	30, 31
26	Literal
27	ExprTuple	32
28	Operation	operator: 40 operands: 33
29	Lambda	parameter: 53 body: 34
30	Operation	operator: 42 operands: 35
31	Operation	operator: 42 operands: 36
32	Variable
33	ExprTuple	50, 51
34	Conditional	value: 37 condition: 38
35	ExprTuple	50, 39
36	ExprTuple	51, 39
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operands: 43
39	Operation	operator: 44 operands: 45
40	Literal
41	ExprTuple	46, 47
42	Literal
43	ExprTuple	53, 48
44	Literal
45	ExprTuple	48, 49
46	Operation	operator: 50 operand: 53
47	Operation	operator: 51 operand: 53
48	Variable
49	Variable
50	Variable
51	Variable
52	ExprTuple	53
53	Variable