Expression of type Forall

from the theory of proveit.linear_algebra.scalar_multiplication

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 ExprRange, IndexedVar, K, V, Variable, a, i, v
from proveit.core_expr_types import a_1_to_i
from proveit.linear_algebra import ScalarMult, VecAdd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [i], instance_expr = Forall(instance_param_or_params = [a_1_to_i, v], instance_expr = Equals(VecAdd(ExprRange(sub_expr1, ScalarMult(IndexedVar(a, sub_expr1), v), one, i)), ScalarMult(VecAdd(a_1_to_i), v)).with_wrapping_at(2), domains = [Complex, V]), domain = Natural), 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 \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{i \in \mathbb{N}}~\left[\forall_{\left(a_{1} \in \mathbb{C}\right), \left(a_{2} \in \mathbb{C}\right), \ldots, \left(a_{i} \in \mathbb{C}\right), v \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} \cdot v\right) +  \left(a_{2} \cdot v\right) +  \ldots +  \left(a_{i} \cdot v\right)\right) =  \\ \left(\left(a_{1} +  a_{2} +  \ldots +  a_{i}\right) \cdot v\right) \end{array} \end{array}\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: 19 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 43 body: 4
3	ExprTuple	43
4	Conditional	value: 5 condition: 6
5	Operation	operator: 19 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	43, 11
10	Lambda	parameter: 56 body: 13
11	Operation	operator: 14 operand: 18
12	ExprTuple	56
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 19 operand: 22
17	Operation	operator: 47 operands: 21
18	Variable
19	Literal
20	ExprTuple	22
21	ExprTuple	56, 23
22	Lambda	parameters: 24 body: 25
23	Literal
24	ExprTuple	50, 57
25	Conditional	value: 26 condition: 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 45 operands: 36
33	Operation	operator: 52 operands: 37
34	ExprRange	lambda_map: 38 start_index: 55 end_index: 56
35	Operation	operator: 47 operands: 39
36	ExprTuple	40
37	ExprTuple	41, 57
38	Lambda	parameter: 61 body: 42
39	ExprTuple	57, 43
40	ExprRange	lambda_map: 44 start_index: 55 end_index: 56
41	Operation	operator: 45 operands: 46
42	Operation	operator: 47 operands: 48
43	Variable
44	Lambda	parameter: 61 body: 49
45	Literal
46	ExprTuple	50
47	Literal
48	ExprTuple	58, 51
49	Operation	operator: 52 operands: 53
50	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
51	Literal
52	Literal
53	ExprTuple	58, 57
54	Lambda	parameter: 61 body: 58
55	Literal
56	Variable
57	Variable
58	IndexedVar	variable: 59 index: 61
59	Variable
60	ExprTuple	61
61	Variable