Expression of type Forall

from the theory of proveit.linear_algebra.addition

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 K, V, a, b
from proveit.linear_algebra import InnerProd, Norm, VecAdd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Exp, sqrt, two, zero

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(Norm(VecAdd(a, b)), sqrt(Add(Exp(Norm(a), two), Exp(Norm(b), two)))), domain = V, condition = Equals(InnerProd(a, b), zero)), domain = VecSpaces(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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{a, b \in V~|~\left \langle a, b\right \rangle = 0}~\left(\left \|a + b\right \| = \sqrt{\left(\left \|a\right \|^{2} + \left \|b\right \|^{2}\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: 37 body: 4
3	ExprTuple	37
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	37, 12
11	Lambda	parameters: 46 body: 13
12	Operation	operator: 14 operand: 18
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 32 operands: 19
17	Operation	operator: 20 operands: 21
18	Variable
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	24, 25, 26
22	Operation	operator: 57 operand: 34
23	Operation	operator: 51 operands: 28
24	Operation	operator: 30 operands: 29
25	Operation	operator: 30 operands: 31
26	Operation	operator: 32 operands: 33
27	ExprTuple	34
28	ExprTuple	35, 36
29	ExprTuple	59, 37
30	Literal
31	ExprTuple	60, 37
32	Literal
33	ExprTuple	38, 39
34	Operation	operator: 40 operands: 46
35	Operation	operator: 41 operands: 42
36	Operation	operator: 43 operands: 44
37	Variable
38	Operation	operator: 45 operands: 46
39	Literal
40	Literal
41	Literal
42	ExprTuple	47, 48
43	Literal
44	ExprTuple	49, 55
45	Literal
46	ExprTuple	59, 60
47	Operation	operator: 51 operands: 50
48	Operation	operator: 51 operands: 52
49	Literal
50	ExprTuple	53, 55
51	Literal
52	ExprTuple	54, 55
53	Operation	operator: 57 operand: 59
54	Operation	operator: 57 operand: 60
55	Literal
56	ExprTuple	59
57	Literal
58	ExprTuple	60
59	Variable
60	Variable