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 = [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_{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)

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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 51 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operands: 7
5	Operation	operator: 8 operands: 9
6	Literal
7	ExprTuple	10, 11
8	Literal
9	ExprTuple	12, 13
10	Operation	operator: 14 operand: 19
11	Operation	operator: 31 operands: 16
12	Operation	operator: 45 operands: 17
13	Operation	operator: 45 operands: 18
14	Literal
15	ExprTuple	19
16	ExprTuple	20, 21
17	ExprTuple	52, 22
18	ExprTuple	53, 22
19	Lambda	parameter: 47 body: 23
20	Operation	operator: 25 operand: 29
21	Operation	operator: 25 operand: 30
22	Literal
23	Conditional	value: 27 condition: 28
24	ExprTuple	29
25	Literal
26	ExprTuple	30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 45 operands: 33
29	Lambda	parameter: 48 body: 34
30	Lambda	parameter: 48 body: 35
31	Literal
32	ExprTuple	36, 37
33	ExprTuple	47, 49
34	Conditional	value: 38 condition: 40
35	Conditional	value: 39 condition: 40
36	Operation	operator: 42 operand: 47
37	Operation	operator: 43 operand: 47
38	Operation	operator: 42 operand: 48
39	Operation	operator: 43 operand: 48
40	Operation	operator: 45 operands: 46
41	ExprTuple	47
42	Variable
43	Variable
44	ExprTuple	48
45	Literal
46	ExprTuple	48, 49
47	Variable
48	Variable
49	Operation	operator: 50 operands: 51
50	Literal
51	ExprTuple	52, 53
52	Variable
53	Variable