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 = [K], instance_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_{K}~\left[\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]\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: 23 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 29 body: 3
3	Operation	operator: 23 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 43 body: 6
6	Conditional	value: 7 condition: 8
7	Operation	operator: 23 operand: 12
8	Operation	operator: 10 operands: 11
9	ExprTuple	12
10	Literal
11	ExprTuple	13, 14
12	Lambda	parameter: 51 body: 16
13	Operation	operator: 18 operands: 17
14	Operation	operator: 18 operands: 19
15	ExprTuple	51
16	Conditional	value: 20 condition: 21
17	ExprTuple	53, 22
18	Literal
19	ExprTuple	46, 22
20	Operation	operator: 23 operand: 28
21	Operation	operator: 49 operands: 25
22	Operation	operator: 26 operand: 29
23	Literal
24	ExprTuple	28
25	ExprTuple	51, 29
26	Literal
27	ExprTuple	29
28	Lambda	parameter: 54 body: 31
29	Variable
30	ExprTuple	54
31	Conditional	value: 32 condition: 33
32	Operation	operator: 34 operands: 35
33	Operation	operator: 49 operands: 36
34	Literal
35	ExprTuple	37, 38
36	ExprTuple	54, 39
37	Operation	operator: 47 operands: 40
38	Lambda	parameter: 56 body: 41
39	Operation	operator: 42 operands: 43
40	ExprTuple	51, 54
41	Conditional	value: 44 condition: 45
42	Literal
43	ExprTuple	53, 46
44	Operation	operator: 47 operands: 48
45	Operation	operator: 49 operands: 50
46	Variable
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	56, 53
51	Variable
52	Operation	operator: 54 operand: 56
53	Variable
54	Variable
55	ExprTuple	56
56	Variable