Expression of type Forall

from the theory of proveit.numbers.multiplication

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 i, j, x, y
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import Forall
from proveit.numbers import Less, Mult, Natural, RealPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [x, y], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = Less(Mult(a_1_to_i, x, b_1_to_j), Mult(a_1_to_i, y, b_1_to_j)), domain = RealPos, condition = Less(x, y))), domain = Natural)

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