Expression of type Forall

from the theory of proveit.numbers.summation

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, alpha, b, c, k
from proveit.core_expr_types import alpha_k
from proveit.logic import And, Equals, Forall
from proveit.numbers import Add, Integer, Interval, LessEq, Sum, one

# build up the expression from sub-expressions
sub_expr1 = [k]
expr = Forall(instance_param_or_params = [a, b, c], instance_expr = Forall(instance_param_or_params = [alpha], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = alpha_k, domain = Interval(a, c)), Add(Sum(index_or_indices = sub_expr1, summand = alpha_k, domain = Interval(a, b)), Sum(index_or_indices = sub_expr1, summand = alpha_k, domain = Interval(Add(b, one), c))))), domain = Integer, condition = And(LessEq(a, b), LessEq(b, c)).with_total_ordering_style())

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_{a, b, c \in \mathbb{Z}~|~a \leq b \leq c}~\left[\forall_{\alpha}~\left(\left(\sum_{k = a}^{c} \alpha_{k}\right) = \left(\left(\sum_{k = a}^{b} \alpha_{k}\right) + \left(\sum_{k = b + 1}^{c} \alpha_{k}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	62, 67, 64
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 20 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13, 14
10	Lambda	parameter: 52 body: 16
11	Operation	operator: 54 operands: 17
12	Operation	operator: 54 operands: 18
13	Operation	operator: 54 operands: 19
14	Operation	operator: 20 operands: 21
15	ExprTuple	52
16	Operation	operator: 22 operands: 23
17	ExprTuple	62, 24
18	ExprTuple	67, 24
19	ExprTuple	64, 24
20	Literal
21	ExprTuple	25, 26
22	Literal
23	ExprTuple	27, 28
24	Literal
25	Operation	operator: 29 operands: 59
26	Operation	operator: 29 operands: 30
27	Operation	operator: 38 operand: 33
28	Operation	operator: 65 operands: 32
29	Literal
30	ExprTuple	67, 64
31	ExprTuple	33
32	ExprTuple	34, 35
33	Lambda	parameter: 57 body: 36
34	Operation	operator: 38 operand: 41
35	Operation	operator: 38 operand: 42
36	Conditional	value: 48 condition: 40
37	ExprTuple	41
38	Literal
39	ExprTuple	42
40	Operation	operator: 54 operands: 43
41	Lambda	parameter: 57 body: 44
42	Lambda	parameter: 57 body: 45
43	ExprTuple	57, 46
44	Conditional	value: 48 condition: 47
45	Conditional	value: 48 condition: 49
46	Operation	operator: 60 operands: 50
47	Operation	operator: 54 operands: 51
48	IndexedVar	variable: 52 index: 57
49	Operation	operator: 54 operands: 55
50	ExprTuple	62, 64
51	ExprTuple	57, 56
52	Variable
53	ExprTuple	57
54	Literal
55	ExprTuple	57, 58
56	Operation	operator: 60 operands: 59
57	Variable
58	Operation	operator: 60 operands: 61
59	ExprTuple	62, 67
60	Literal
61	ExprTuple	63, 64
62	Variable
63	Operation	operator: 65 operands: 66
64	Variable
65	Literal
66	ExprTuple	67, 68
67	Variable
68	Literal