Expression of type Forall

from the theory of proveit.core_expr_types.tuples

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 i, j, k
from proveit.core_expr_types import f_i_to_j, fk, g_i_to_j, gk
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import Integer, Interval

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(fk, gk), domain = Interval(i, j)), Equals([f_i_to_j], [g_i_to_j])), domain = Integer)

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_{i, j \in \mathbb{Z}}~\left(\left[\forall_{k \in \{i~\ldotp \ldotp~j\}}~\left(f\left(k\right) = g\left(k\right)\right)\right] \Rightarrow \left(\left(f\left(i\right), f\left(i + 1\right), \ldots, f\left(j\right)\right) = \left(g\left(i\right), g\left(i + 1\right), \ldots, g\left(j\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 44 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operands: 7
5	Operation	operator: 8 operands: 9
6	Literal
7	ExprTuple	10, 11
8	Literal
9	ExprTuple	12, 13
10	Operation	operator: 14 operand: 19
11	Operation	operator: 30 operands: 16
12	Operation	operator: 32 operands: 17
13	Operation	operator: 32 operands: 18
14	Literal
15	ExprTuple	19
16	ExprTuple	20, 21
17	ExprTuple	47, 22
18	ExprTuple	48, 22
19	Lambda	parameter: 46 body: 23
20	ExprTuple	24
21	ExprTuple	25
22	Literal
23	Conditional	value: 26 condition: 27
24	ExprRange	lambda_map: 28 start_index: 47 end_index: 48
25	ExprRange	lambda_map: 29 start_index: 47 end_index: 48
26	Operation	operator: 30 operands: 31
27	Operation	operator: 32 operands: 33
28	Lambda	parameter: 45 body: 34
29	Lambda	parameter: 45 body: 35
30	Literal
31	ExprTuple	36, 37
32	Literal
33	ExprTuple	46, 38
34	Operation	operator: 40 operand: 45
35	Operation	operator: 41 operand: 45
36	Operation	operator: 40 operand: 46
37	Operation	operator: 41 operand: 46
38	Operation	operator: 43 operands: 44
39	ExprTuple	45
40	Variable
41	Variable
42	ExprTuple	46
43	Literal
44	ExprTuple	47, 48
45	Variable
46	Variable
47	Variable
48	Variable