Expression of type Forall

from the theory of proveit.core_expr_types.lambda_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, Lambda, Q, f, g, i
from proveit.core_expr_types import Q__a_1_to_i, Q__b_1_to_i, Q__c_1_to_i, a_1_to_i, b_1_to_i, c_1_to_i, f__a_1_to_i, f__b_1_to_i, g__a_1_to_i, g__c_1_to_i
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i], instance_expr = Forall(instance_param_or_params = [f, g, Q], instance_expr = Implies(Forall(instance_param_or_params = [a_1_to_i], instance_expr = Equals(f__a_1_to_i, g__a_1_to_i), condition = Q__a_1_to_i), Equals(Lambda([b_1_to_i], Conditional(f__b_1_to_i, Q__b_1_to_i)), Lambda([c_1_to_i], Conditional(g__c_1_to_i, Q__c_1_to_i))).with_wrapping_at(2)).with_wrapping_at(2)), domain = NaturalPos)

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