Expression of type Conditional

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 Conditional, H, K, a, x, y
from proveit.linear_algebra import InnerProd, InnerProdSpaces, ScalarMult
from proveit.logic import Equals, Forall, InClass

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [a], instance_expr = Forall(instance_param_or_params = [x, y], instance_expr = Equals(InnerProd(ScalarMult(a, x), y), ScalarMult(a, InnerProd(x, y))), domain = H), domain = K), InClass(H, 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())

\left\{\forall_{a \in K}~\left[\forall_{x, y \in H}~\left(\left \langle a \cdot x, y\right \rangle = \left(a \cdot \left \langle x, y\right \rangle\right)\right)\right] \textrm{ if } H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 14 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	37, 7
6	Lambda	parameter: 42 body: 9
7	Operation	operator: 10 operand: 18
8	ExprTuple	42
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	18
12	Operation	operator: 14 operand: 17
13	Operation	operator: 33 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	42, 18
17	Lambda	parameters: 41 body: 19
18	Variable
19	Conditional	value: 20 condition: 21
20	Operation	operator: 22 operands: 23
21	Operation	operator: 24 operands: 25
22	Literal
23	ExprTuple	26, 27
24	Literal
25	ExprTuple	28, 29
26	Operation	operator: 40 operands: 30
27	Operation	operator: 38 operands: 31
28	Operation	operator: 33 operands: 32
29	Operation	operator: 33 operands: 34
30	ExprTuple	35, 44
31	ExprTuple	42, 36
32	ExprTuple	43, 37
33	Literal
34	ExprTuple	44, 37
35	Operation	operator: 38 operands: 39
36	Operation	operator: 40 operands: 41
37	Variable
38	Literal
39	ExprTuple	42, 43
40	Literal
41	ExprTuple	43, 44
42	Variable
43	Variable
44	Variable