Expression of type Forall

from the theory of proveit.linear_algebra.addition

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 K, V, i, j, k, v
from proveit.linear_algebra import ScalarMult, VecSpaces, VecSum
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Integer, Interval, LessEq, one, subtract

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [j, k], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [v], instance_expr = Equals(VecSum(index_or_indices = [i], summand = v, domain = Interval(j, k)), ScalarMult(Add(subtract(k, j), one), v)), domain = V), domain = VecSpaces(K)), domain = Integer, condition = LessEq(j, 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_{j, k \in \mathbb{Z}~|~j \leq k}~\left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{v \in V}~\left(\left(\sum_{i=j}^{k} v\right) = \left(\left(\left(k - j\right) + 1\right) \cdot 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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 61 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 21 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11, 12
9	Lambda	parameter: 39 body: 14
10	Operation	operator: 53 operands: 15
11	Operation	operator: 53 operands: 16
12	Operation	operator: 17 operands: 61
13	ExprTuple	39
14	Conditional	value: 18 condition: 19
15	ExprTuple	65, 20
16	ExprTuple	64, 20
17	Literal
18	Operation	operator: 21 operand: 25
19	Operation	operator: 23 operands: 24
20	Literal
21	Literal
22	ExprTuple	25
23	Literal
24	ExprTuple	39, 26
25	Lambda	parameter: 49 body: 28
26	Operation	operator: 29 operand: 33
27	ExprTuple	49
28	Conditional	value: 31 condition: 32
29	Literal
30	ExprTuple	33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 53 operands: 36
33	Variable
34	Literal
35	ExprTuple	37, 38
36	ExprTuple	49, 39
37	Operation	operator: 40 operand: 44
38	Operation	operator: 42 operands: 43
39	Variable
40	Literal
41	ExprTuple	44
42	Literal
43	ExprTuple	45, 49
44	Lambda	parameter: 57 body: 47
45	Operation	operator: 55 operands: 48
46	ExprTuple	57
47	Conditional	value: 49 condition: 50
48	ExprTuple	51, 52
49	Variable
50	Operation	operator: 53 operands: 54
51	Operation	operator: 55 operands: 56
52	Literal
53	Literal
54	ExprTuple	57, 58
55	Literal
56	ExprTuple	64, 59
57	Variable
58	Operation	operator: 60 operands: 61
59	Operation	operator: 62 operand: 65
60	Literal
61	ExprTuple	65, 64
62	Literal
63	ExprTuple	65
64	Variable
65	Variable