Expression of type Lambda

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, Lambda, 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 = Lambda(H, 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())

H \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 39 body: 2
1	ExprTuple	39
2	Conditional	value: 3 condition: 4
3	Operation	operator: 16 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	39, 9
8	Lambda	parameter: 44 body: 11
9	Operation	operator: 12 operand: 20
10	ExprTuple	44
11	Conditional	value: 14 condition: 15
12	Literal
13	ExprTuple	20
14	Operation	operator: 16 operand: 19
15	Operation	operator: 35 operands: 18
16	Literal
17	ExprTuple	19
18	ExprTuple	44, 20
19	Lambda	parameters: 43 body: 21
20	Variable
21	Conditional	value: 22 condition: 23
22	Operation	operator: 24 operands: 25
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 42 operands: 32
29	Operation	operator: 40 operands: 33
30	Operation	operator: 35 operands: 34
31	Operation	operator: 35 operands: 36
32	ExprTuple	37, 46
33	ExprTuple	44, 38
34	ExprTuple	45, 39
35	Literal
36	ExprTuple	46, 39
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operands: 43
39	Variable
40	Literal
41	ExprTuple	44, 45
42	Literal
43	ExprTuple	45, 46
44	Variable
45	Variable
46	Variable