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 S, f, fx, g, gx, x
from proveit.logic import Forall, Implies
from proveit.numbers import LessEq, Sum

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [S, f, g], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = LessEq(fx, gx), domain = S), LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = S), Sum(index_or_indices = sub_expr1, summand = gx, domain = S))))

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_{S, f, g}~\left(\left[\forall_{x \in S}~\left(f\left(x\right) \leq g\left(x\right)\right)\right] \Rightarrow \left(\left[\sum_{x \in S}~f\left(x\right)\right] \leq \left[\sum_{x \in S}~g\left(x\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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	35, 29, 30
4	Operation	operator: 5 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Operation	operator: 9 operand: 12
8	Operation	operator: 22 operands: 11
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Lambda	parameter: 34 body: 15
13	Operation	operator: 17 operand: 20
14	Operation	operator: 17 operand: 21
15	Conditional	value: 19 condition: 28
16	ExprTuple	20
17	Literal
18	ExprTuple	21
19	Operation	operator: 22 operands: 23
20	Lambda	parameter: 34 body: 24
21	Lambda	parameter: 34 body: 25
22	Literal
23	ExprTuple	26, 27
24	Conditional	value: 26 condition: 28
25	Conditional	value: 27 condition: 28
26	Operation	operator: 29 operand: 34
27	Operation	operator: 30 operand: 34
28	Operation	operator: 32 operands: 33
29	Variable
30	Variable
31	ExprTuple	34
32	Literal
33	ExprTuple	34, 35
34	Variable
35	Variable