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

# build up the expression from sub-expressions
expr = Lambda(K, Forall(instance_param_or_params = [H], instance_expr = 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), 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())

K \mapsto \left[\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\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]\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: 23 body: 1
1	Operation	operator: 19 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 42 body: 5
4	ExprTuple	42
5	Conditional	value: 6 condition: 7
6	Operation	operator: 19 operand: 11
7	Operation	operator: 9 operands: 10
8	ExprTuple	11
9	Literal
10	ExprTuple	42, 12
11	Lambda	parameter: 47 body: 14
12	Operation	operator: 15 operand: 23
13	ExprTuple	47
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	23
17	Operation	operator: 19 operand: 22
18	Operation	operator: 38 operands: 21
19	Literal
20	ExprTuple	22
21	ExprTuple	47, 23
22	Lambda	parameters: 46 body: 24
23	Variable
24	Conditional	value: 25 condition: 26
25	Operation	operator: 27 operands: 28
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	31, 32
29	Literal
30	ExprTuple	33, 34
31	Operation	operator: 45 operands: 35
32	Operation	operator: 43 operands: 36
33	Operation	operator: 38 operands: 37
34	Operation	operator: 38 operands: 39
35	ExprTuple	40, 49
36	ExprTuple	47, 41
37	ExprTuple	48, 42
38	Literal
39	ExprTuple	49, 42
40	Operation	operator: 43 operands: 44
41	Operation	operator: 45 operands: 46
42	Variable
43	Literal
44	ExprTuple	47, 48
45	Literal
46	ExprTuple	48, 49
47	Variable
48	Variable
49	Variable