Expression of type Forall

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, a, b, x, y, z
from proveit.linear_algebra import InnerProd, InnerProdSpaces, ScalarMult, VecAdd
from proveit.logic import Equals, Forall

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

\forall_{K}~\left[\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\left[\forall_{a, b \in K}~\left[\forall_{x, y, z \in H}~\left(\left \langle \left(a \cdot x\right) + \left(b \cdot y\right), z\right \rangle = \left(\left(a \cdot \left \langle x, z\right \rangle\right) + \left(b \cdot \left \langle y, z\right \rangle\right)\right)\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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 33 body: 3
3	Operation	operator: 21 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 52 body: 7
6	ExprTuple	52
7	Conditional	value: 8 condition: 9
8	Operation	operator: 21 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	52, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operand: 33
15	ExprTuple	67, 68
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	33
19	Operation	operator: 21 operand: 24
20	Operation	operator: 36 operands: 23
21	Literal
22	ExprTuple	24
23	ExprTuple	25, 26
24	Lambda	parameters: 27 body: 28
25	Operation	operator: 47 operands: 29
26	Operation	operator: 47 operands: 30
27	ExprTuple	69, 70, 71
28	Conditional	value: 31 condition: 32
29	ExprTuple	67, 33
30	ExprTuple	68, 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 37
33	Variable
34	Literal
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	40, 41, 42
38	Operation	operator: 65 operands: 43
39	Operation	operator: 53 operands: 44
40	Operation	operator: 47 operands: 45
41	Operation	operator: 47 operands: 46
42	Operation	operator: 47 operands: 48
43	ExprTuple	49, 71
44	ExprTuple	50, 51
45	ExprTuple	69, 52
46	ExprTuple	70, 52
47	Literal
48	ExprTuple	71, 52
49	Operation	operator: 53 operands: 54
50	Operation	operator: 62 operands: 55
51	Operation	operator: 62 operands: 56
52	Variable
53	Literal
54	ExprTuple	57, 58
55	ExprTuple	67, 59
56	ExprTuple	68, 60
57	Operation	operator: 62 operands: 61
58	Operation	operator: 62 operands: 63
59	Operation	operator: 65 operands: 64
60	Operation	operator: 65 operands: 66
61	ExprTuple	67, 69
62	Literal
63	ExprTuple	68, 70
64	ExprTuple	69, 71
65	Literal
66	ExprTuple	70, 71
67	Variable
68	Variable
69	Variable
70	Variable
71	Variable