Expression of type Forall

from the theory of proveit.numbers.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 b, i, j
from proveit.core_expr_types import a_1_to_i, c_1_to_j
from proveit.logic import Forall
from proveit.numbers import Add, Less, Natural, Real, RealNeg, greater, zero

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, c_1_to_j], instance_expr = Forall(instance_param_or_params = [b], instance_expr = Less(Add(a_1_to_i, b, c_1_to_j), b), domain = Real), domain = RealNeg), domain = Natural, condition = greater(Add(i, j), zero))

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_{i, j \in \mathbb{N}~|~\left(i + j\right) > 0}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{j} \in \mathbb{R}^-}~\left[\forall_{b \in \mathbb{R}}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + b+ c_{1} +  c_{2} +  \ldots +  c_{j}\right) < b\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: 22 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 26 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 22 operand: 8
5	Operation	operator: 24 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10, 11
8	Lambda	parameters: 12 body: 13
9	Operation	operator: 42 operands: 14
10	Operation	operator: 42 operands: 15
11	Operation	operator: 38 operands: 16
12	ExprTuple	49, 51
13	Conditional	value: 17 condition: 18
14	ExprTuple	53, 19
15	ExprTuple	56, 19
16	ExprTuple	20, 21
17	Operation	operator: 22 operand: 27
18	Operation	operator: 24 operands: 25
19	Literal
20	Literal
21	Operation	operator: 47 operands: 26
22	Literal
23	ExprTuple	27
24	Literal
25	ExprTuple	28, 29
26	ExprTuple	53, 56
27	Lambda	parameter: 50 body: 31
28	ExprRange	lambda_map: 32 start_index: 55 end_index: 53
29	ExprRange	lambda_map: 33 start_index: 55 end_index: 56
30	ExprTuple	50
31	Conditional	value: 34 condition: 35
32	Lambda	parameter: 62 body: 36
33	Lambda	parameter: 62 body: 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 42 operands: 40
36	Operation	operator: 42 operands: 41
37	Operation	operator: 42 operands: 43
38	Literal
39	ExprTuple	44, 50
40	ExprTuple	50, 45
41	ExprTuple	57, 46
42	Literal
43	ExprTuple	58, 46
44	Operation	operator: 47 operands: 48
45	Literal
46	Literal
47	Literal
48	ExprTuple	49, 50, 51
49	ExprRange	lambda_map: 52 start_index: 55 end_index: 53
50	Variable
51	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
52	Lambda	parameter: 62 body: 57
53	Variable
54	Lambda	parameter: 62 body: 58
55	Literal
56	Variable
57	IndexedVar	variable: 59 index: 62
58	IndexedVar	variable: 60 index: 62
59	Variable
60	Variable
61	ExprTuple	62
62	Variable