Theorem existential_implication of type Forall

from the theory of proveit.logic.booleans.quantification.existence

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.logic.booleans.quantification.existence import existential_implication

# check that the built expression is the same as the stored expression
assert existential_implication.expr == stored_expr
assert existential_implication.expr._style_id == stored_expr._style_id
print("Passed sanity check: existential_implication matches stored_expr")

Passed sanity check: existential_implication matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{n \in \mathbb{N}^+}~\left[\forall_{P, Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~\left(P\left(x_{1}, x_{2}, \ldots, x_{n}\right) \Rightarrow R\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right] \Rightarrow  \\ \left[\exists_{z_{1}, z_{2}, \ldots, z_{n}~|~Q\left(z_{1}, z_{2}, \ldots, z_{n}\right)}~R\left(z_{1}, z_{2}, \ldots, z_{n}\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\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: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 54 body: 4
3	ExprTuple	54
4	Conditional	value: 5 condition: 6
5	Operation	operator: 17 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	54, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	42, 45, 44
13	Operation	operator: 31 operands: 14
14	ExprTuple	15, 16
15	Operation	operator: 17 operand: 20
16	Operation	operator: 31 operands: 19
17	Literal
18	ExprTuple	20
19	ExprTuple	21, 22
20	Lambda	parameters: 41 body: 23
21	Operation	operator: 25 operand: 29
22	Operation	operator: 25 operand: 30
23	Conditional	value: 27 condition: 28
24	ExprTuple	29
25	Literal
26	ExprTuple	30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 45 operands: 41
29	Lambda	parameters: 43 body: 33
30	Lambda	parameters: 46 body: 34
31	Literal
32	ExprTuple	35, 36
33	Conditional	value: 37 condition: 38
34	Conditional	value: 39 condition: 40
35	Operation	operator: 42 operands: 41
36	Operation	operator: 44 operands: 41
37	Operation	operator: 42 operands: 43
38	Operation	operator: 45 operands: 43
39	Operation	operator: 44 operands: 46
40	Operation	operator: 45 operands: 46
41	ExprTuple	47
42	Variable
43	ExprTuple	48
44	Variable
45	Variable
46	ExprTuple	49
47	ExprRange	lambda_map: 50 start_index: 53 end_index: 54
48	ExprRange	lambda_map: 51 start_index: 53 end_index: 54
49	ExprRange	lambda_map: 52 start_index: 53 end_index: 54
50	Lambda	parameter: 62 body: 55
51	Lambda	parameter: 62 body: 56
52	Lambda	parameter: 62 body: 57
53	Literal
54	Variable
55	IndexedVar	variable: 58 index: 62
56	IndexedVar	variable: 59 index: 62
57	IndexedVar	variable: 60 index: 62
58	Variable
59	Variable
60	Variable
61	ExprTuple	62
62	Variable