Expression of type Forall

from the theory of proveit.linear_algebra.inner_products

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 H, K, alpha, x
from proveit.linear_algebra import InnerProdSpaces, Norm, ScalarMult
from proveit.logic import Equals, Forall
from proveit.numbers import Abs, Mult

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [H], instance_expr = Forall(instance_param_or_params = [alpha, x], instance_expr = Equals(Norm(ScalarMult(alpha, x)), Mult(Abs(alpha), Norm(x))), domains = [K, H]), domain = InnerProdSpaces(K))

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_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\left[\forall_{\alpha \in K, x \in H}~\left(\left \|\alpha \cdot x\right \| = \left(\left|\alpha\right| \cdot \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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 36 body: 4
3	ExprTuple	36
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	36, 12
11	Lambda	parameters: 38 body: 13
12	Operation	operator: 14 operand: 35
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	35
16	Operation	operator: 18 operands: 19
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	24, 25
22	Operation	operator: 41 operand: 32
23	Operation	operator: 27 operands: 28
24	Operation	operator: 30 operands: 29
25	Operation	operator: 30 operands: 31
26	ExprTuple	32
27	Literal
28	ExprTuple	33, 34
29	ExprTuple	43, 35
30	Literal
31	ExprTuple	44, 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 39 operand: 43
34	Operation	operator: 41 operand: 44
35	Variable
36	Variable
37	Literal
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	43
41	Literal
42	ExprTuple	44
43	Variable
44	Variable