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, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import ScalarMult, VecAdd
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 = [n], instance_expr = Forall(instance_param_or_params = [a], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(ScalarMult(a, VecAdd(x_1_to_n)), VecAdd(ExprRange(sub_expr1, ScalarMult(a, IndexedVar(x, sub_expr1)), one, n))).with_wrapping_at(2), domain = V), domain = K), domain = Natural)

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_{n \in \mathbb{N}}~\left[\forall_{a \in K}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in V}~\left(\begin{array}{c} \begin{array}{l} \left(a \cdot \left(x_{1} +  x_{2} +  \ldots +  x_{n}\right)\right) =  \\ \left(\left(a \cdot x_{1}\right) +  \left(a \cdot x_{2}\right) +  \ldots +  \left(a \cdot x_{n}\right)\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: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 46 body: 4
3	ExprTuple	46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 15 operand: 9
6	Operation	operator: 39 operands: 8
7	ExprTuple	9
8	ExprTuple	46, 10
9	Lambda	parameter: 49 body: 12
10	Literal
11	ExprTuple	49
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 18
14	Operation	operator: 39 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	49, 19
18	Lambda	parameters: 37 body: 20
19	Variable
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operands: 24
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29
27	Operation	operator: 47 operands: 30
28	Operation	operator: 36 operands: 31
29	ExprRange	lambda_map: 32 start_index: 45 end_index: 46
30	ExprTuple	49, 33
31	ExprTuple	34
32	Lambda	parameter: 53 body: 35
33	Operation	operator: 36 operands: 37
34	ExprRange	lambda_map: 38 start_index: 45 end_index: 46
35	Operation	operator: 39 operands: 40
36	Literal
37	ExprTuple	41
38	Lambda	parameter: 53 body: 42
39	Literal
40	ExprTuple	50, 43
41	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
42	Operation	operator: 47 operands: 48
43	Variable
44	Lambda	parameter: 53 body: 50
45	Literal
46	Variable
47	Literal
48	ExprTuple	49, 50
49	Variable
50	IndexedVar	variable: 51 index: 53
51	Variable
52	ExprTuple	53
53	Variable