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, k
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 Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [i], instance_expr = Forall(instance_param_or_params = [k, a_1_to_i], instance_expr = Equals(VecAdd(ExprRange(sub_expr1, ScalarMult(k, IndexedVar(a, sub_expr1)), one, i)), ScalarMult(k, VecAdd(a_1_to_i))).with_wrapping_at(2), domains = [K, 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_{K}~\left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{i \in \mathbb{N}}~\left[\forall_{k \in K,\left(a_{1} \in V\right), \left(a_{2} \in V\right), \ldots, \left(a_{i} \in V\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(k \cdot a_{1}\right) +  \left(k \cdot a_{2}\right) +  \ldots +  \left(k \cdot a_{i}\right)\right) =  \\ \left(k \cdot \left(a_{1} +  a_{2} +  \ldots +  a_{i}\right)\right) \end{array} \end{array}\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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 44 body: 3
3	Operation	operator: 21 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 53 body: 7
6	ExprTuple	53
7	Conditional	value: 8 condition: 9
8	Operation	operator: 21 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	53, 14
13	Lambda	parameter: 58 body: 16
14	Operation	operator: 17 operand: 44
15	ExprTuple	58
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	44
19	Operation	operator: 21 operand: 24
20	Operation	operator: 49 operands: 23
21	Literal
22	ExprTuple	24
23	ExprTuple	58, 25
24	Lambda	parameters: 26 body: 27
25	Literal
26	ExprTuple	59, 52
27	Conditional	value: 28 condition: 29
28	Operation	operator: 30 operands: 31
29	Operation	operator: 32 operands: 33
30	Literal
31	ExprTuple	34, 35
32	Literal
33	ExprTuple	36, 37
34	Operation	operator: 47 operands: 38
35	Operation	operator: 54 operands: 39
36	Operation	operator: 49 operands: 40
37	ExprRange	lambda_map: 41 start_index: 57 end_index: 58
38	ExprTuple	42
39	ExprTuple	59, 43
40	ExprTuple	59, 44
41	Lambda	parameter: 63 body: 45
42	ExprRange	lambda_map: 46 start_index: 57 end_index: 58
43	Operation	operator: 47 operands: 48
44	Variable
45	Operation	operator: 49 operands: 50
46	Lambda	parameter: 63 body: 51
47	Literal
48	ExprTuple	52
49	Literal
50	ExprTuple	60, 53
51	Operation	operator: 54 operands: 55
52	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
53	Variable
54	Literal
55	ExprTuple	59, 60
56	Lambda	parameter: 63 body: 60
57	Literal
58	Variable
59	Variable
60	IndexedVar	variable: 61 index: 63
61	Variable
62	ExprTuple	63
63	Variable