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 = [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_{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]

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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 42 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 21 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameters: 30 body: 12
10	Operation	operator: 14 operands: 13
11	Operation	operator: 14 operands: 15
12	Conditional	value: 16 condition: 17
13	ExprTuple	45, 18
14	Literal
15	ExprTuple	46, 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	Operation	operator: 23 operand: 29
19	Literal
20	ExprTuple	25, 26
21	Literal
22	ExprTuple	27, 28
23	Literal
24	ExprTuple	29
25	Operation	operator: 37 operands: 30
26	Lambda	parameter: 50 body: 31
27	Operation	operator: 39 operands: 32
28	Operation	operator: 39 operands: 33
29	Variable
30	ExprTuple	47, 48
31	Conditional	value: 34 condition: 35
32	ExprTuple	47, 36
33	ExprTuple	48, 36
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operands: 42
37	Literal
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	50, 45
41	Literal
42	ExprTuple	45, 46
43	Operation	operator: 47 operand: 50
44	Operation	operator: 48 operand: 50
45	Variable
46	Variable
47	Variable
48	Variable
49	ExprTuple	50
50	Variable