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 = [K], instance_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_{K}~\left[\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]\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: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 21 body: 3
3	Operation	operator: 10 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 40 body: 7
6	ExprTuple	40
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 14
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14
12	Literal
13	ExprTuple	40, 15
14	Lambda	parameters: 49 body: 16
15	Operation	operator: 17 operand: 21
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 35 operands: 22
20	Operation	operator: 23 operands: 24
21	Variable
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28, 29
25	Operation	operator: 60 operand: 37
26	Operation	operator: 54 operands: 31
27	Operation	operator: 33 operands: 32
28	Operation	operator: 33 operands: 34
29	Operation	operator: 35 operands: 36
30	ExprTuple	37
31	ExprTuple	38, 39
32	ExprTuple	62, 40
33	Literal
34	ExprTuple	63, 40
35	Literal
36	ExprTuple	41, 42
37	Operation	operator: 43 operands: 49
38	Operation	operator: 44 operands: 45
39	Operation	operator: 46 operands: 47
40	Variable
41	Operation	operator: 48 operands: 49
42	Literal
43	Literal
44	Literal
45	ExprTuple	50, 51
46	Literal
47	ExprTuple	52, 58
48	Literal
49	ExprTuple	62, 63
50	Operation	operator: 54 operands: 53
51	Operation	operator: 54 operands: 55
52	Literal
53	ExprTuple	56, 58
54	Literal
55	ExprTuple	57, 58
56	Operation	operator: 60 operand: 62
57	Operation	operator: 60 operand: 63
58	Literal
59	ExprTuple	62
60	Literal
61	ExprTuple	63
62	Variable
63	Variable