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 Function, f, g, k, l, m, n
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import Integer, Interval, Sum

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = [l]
sub_expr3 = Interval(m, n)
expr = Forall(instance_param_or_params = [f, g], instance_expr = Forall(instance_param_or_params = [m, n], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Function(f, sub_expr1), Function(g, sub_expr1)), domain = sub_expr3), Equals(Sum(index_or_indices = sub_expr2, summand = Function(f, sub_expr2), domain = sub_expr3), Sum(index_or_indices = sub_expr2, summand = Function(g, sub_expr2), domain = sub_expr3))), domain = Integer))

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_{f, g}~\left[\forall_{m, n \in \mathbb{Z}}~\left(\left[\forall_{k \in \{m~\ldotp \ldotp~n\}}~\left(f\left(k\right) = g\left(k\right)\right)\right] \Rightarrow \left(\left(\sum_{l = m}^{n} f\left(l\right)\right) = \left(\sum_{l = m}^{n} g\left(l\right)\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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	46, 47
4	Operation	operator: 18 operand: 6
5	ExprTuple	6
6	Lambda	parameters: 55 body: 7
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14, 15
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 18 operand: 23
15	Operation	operator: 35 operands: 20
16	Operation	operator: 49 operands: 21
17	Operation	operator: 49 operands: 22
18	Literal
19	ExprTuple	23
20	ExprTuple	24, 25
21	ExprTuple	56, 26
22	ExprTuple	57, 26
23	Lambda	parameter: 51 body: 27
24	Operation	operator: 29 operand: 33
25	Operation	operator: 29 operand: 34
26	Literal
27	Conditional	value: 31 condition: 32
28	ExprTuple	33
29	Literal
30	ExprTuple	34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 49 operands: 37
33	Lambda	parameter: 52 body: 38
34	Lambda	parameter: 52 body: 39
35	Literal
36	ExprTuple	40, 41
37	ExprTuple	51, 53
38	Conditional	value: 42 condition: 44
39	Conditional	value: 43 condition: 44
40	Operation	operator: 46 operand: 51
41	Operation	operator: 47 operand: 51
42	Operation	operator: 46 operand: 52
43	Operation	operator: 47 operand: 52
44	Operation	operator: 49 operands: 50
45	ExprTuple	51
46	Variable
47	Variable
48	ExprTuple	52
49	Literal
50	ExprTuple	52, 53
51	Variable
52	Variable
53	Operation	operator: 54 operands: 55
54	Literal
55	ExprTuple	56, 57
56	Variable
57	Variable