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 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 LessEq, Mult, RealNonNeg

# build up the expression from sub-expressions
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 = LessEq(Mult(a_1_to_i, x, b_1_to_j), Mult(a_1_to_i, y, b_1_to_j)), domain = RealNonNeg, condition = LessEq(x, y)))

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_{x, y}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{R}^{\ge 0}~|~x \leq 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) \leq \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]

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