Expression of type Lambda

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 Conditional, Lambda, i, j, x, y
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import And, Forall, InSet
from proveit.numbers import LessEq, Mult, Natural, RealNonNeg

# build up the expression from sub-expressions
expr = Lambda([i, j], Conditional(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))), And(InSet(i, Natural), InSet(j, 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())

\left(i, j\right) \mapsto \left\{\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] \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	43, 46
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 22 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 35 body: 10
8	Operation	operator: 48 operands: 11
9	Operation	operator: 48 operands: 12
10	Operation	operator: 13 operand: 16
11	ExprTuple	43, 15
12	ExprTuple	46, 15
13	Literal
14	ExprTuple	16
15	Literal
16	Lambda	parameters: 17 body: 18
17	ExprTuple	36, 37
18	Conditional	value: 19 condition: 20
19	Operation	operator: 34 operands: 21
20	Operation	operator: 22 operands: 23
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26, 27, 28
24	Operation	operator: 30 operands: 29
25	Operation	operator: 30 operands: 31
26	ExprRange	lambda_map: 32 start_index: 45 end_index: 43
27	ExprRange	lambda_map: 33 start_index: 45 end_index: 46
28	Operation	operator: 34 operands: 35
29	ExprTuple	36, 40, 37
30	Literal
31	ExprTuple	36, 41, 37
32	Lambda	parameter: 56 body: 38
33	Lambda	parameter: 56 body: 39
34	Literal
35	ExprTuple	40, 41
36	ExprRange	lambda_map: 42 start_index: 45 end_index: 43
37	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
38	Operation	operator: 48 operands: 47
39	Operation	operator: 48 operands: 49
40	Variable
41	Variable
42	Lambda	parameter: 56 body: 50
43	Variable
44	Lambda	parameter: 56 body: 51
45	Literal
46	Variable
47	ExprTuple	50, 52
48	Literal
49	ExprTuple	51, 52
50	IndexedVar	variable: 53 index: 56
51	IndexedVar	variable: 54 index: 56
52	Literal
53	Variable
54	Variable
55	ExprTuple	56
56	Variable