Expression of type Forall

from the theory of proveit.linear_algebra.linear_maps

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, K, Lambda, P, Px, V, W, c, x
from proveit.linear_algebra import LinMap, ScalarMult, VecSpaces
from proveit.logic import Equals, Forall, InSet

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V, W], instance_expr = Forall(instance_param_or_params = [c], instance_expr = Forall(instance_param_or_params = [P], instance_expr = Equals(ScalarMult(c, P), Lambda(x, Conditional(ScalarMult(c, Px), InSet(x, V)))), domain = LinMap(V, W)), domain = K), 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_{V, W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{c \in K}~\left[\forall_{P \in \mathcal{L}\left(V, W\right)}~\left(\left(c \cdot P\right) = \left[x \mapsto \left\{c \cdot P\left(x\right) \textrm{ if } x \in V\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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 40 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 20 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 48 body: 13
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	48
13	Conditional	value: 17 condition: 18
14	ExprTuple	50, 19
15	Literal
16	ExprTuple	43, 19
17	Operation	operator: 20 operand: 25
18	Operation	operator: 46 operands: 22
19	Operation	operator: 23 operand: 26
20	Literal
21	ExprTuple	25
22	ExprTuple	48, 26
23	Literal
24	ExprTuple	26
25	Lambda	parameter: 51 body: 28
26	Variable
27	ExprTuple	51
28	Conditional	value: 29 condition: 30
29	Operation	operator: 31 operands: 32
30	Operation	operator: 46 operands: 33
31	Literal
32	ExprTuple	34, 35
33	ExprTuple	51, 36
34	Operation	operator: 44 operands: 37
35	Lambda	parameter: 53 body: 38
36	Operation	operator: 39 operands: 40
37	ExprTuple	48, 51
38	Conditional	value: 41 condition: 42
39	Literal
40	ExprTuple	50, 43
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operands: 47
43	Variable
44	Literal
45	ExprTuple	48, 49
46	Literal
47	ExprTuple	53, 50
48	Variable
49	Operation	operator: 51 operand: 53
50	Variable
51	Variable
52	ExprTuple	53
53	Variable