You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


Inductive type

In type theory, a system has inductive types if it has facilities for creating a new type along with constants and functions that create terms of that type.  The feature serves a role similar to data structures in a programming language and allows a type theory to add concepts like numbers, relations, and trees.  As the name suggests, inductive types can be self-referential, but usually only in a way that permits structural recursion.

The standard example is encoding the natural numbers using Peano's encoding.
 Inductive nat : Type :=
   | 0 : nat
   | S : nat -> nat.
Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number.  "S" is the successor function which represents adding 1 to a number.  Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on.

Since their introduction, inductive types have been extended to encode more and more structures, while still being predicative and supporting structural recursion.

Elimination 

Inductive types usually come with a function to prove properties about them.  Thus, "nat" may come with:
 nat_elim : (forall P : nat -> Prop, (P 0) -> (forall n, P n -> P (S n)) -> (forall n, P n)).
This is the expected function for structural recursion for the type "nat".

Implementations

W- and M-types 

W-types are well-founded types in intuitionistic type theory (ITT). They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let  be a universe of types. Given a type  :  and a dependent family  :  → , one can form a W-type . The type  may be thought of as "labels" for the (potentially infinitely many) constructors of the inductive type being defined, whereas  indicates the (potentially infinite) arity of each constructor. W-types (resp. M-types) may also be understood as well-founded (resp. non-well-founded) trees with nodes labeled by elements  :  and where the node labeled by  has ()-many subtrees. Each W-type is isomorphic to the initial algebra of a so-called polynomial functor.

Let 0, 1, 2, etc. be finite types with inhabitants 11 : 1, 12, 22:2, etc. One may define the natural numbers as the W-type

with  : 2 →  is defined by (12) = 0 (representing the constructor for zero, which takes no arguments), and (22) = 1 (representing the successor function, which takes one argument).

One may define lists over a type  :  as  where

and 11 is the sole inhabitant of 1. The value of  corresponds to the constructor for the empty list, whereas the value of  corresponds to the constructor that appends  to the beginning of another list.

The constructor for elements of a generic W-type  has type

We can also write this rule in the style of a natural deduction proof,

The elimination rule for W-types works similarly to structural induction on trees. If, whenever a property (under the propositions-as-types interpretation)  :  →  holds for all subtrees of a given tree it also holds for that tree, then it holds for all trees.

In extensional type theories, W-types (resp. M-types) can be defined up to isomorphism as initial algebras (resp. final coalgebras) for polynomial functors. In this case, the property of initiality (res. finality) corresponds directly to the appropriate induction principle. In intensional type theories with the univalence axiom, this correspondence holds up to homotopy (propositional equality).

M-types are dual to W-types, they represent coinductive (potentially infinite) data such as streams. M-types can be derived from W-types.

Mutually inductive definitions 

This technique allows some definitions of multiple types that depend on each other. For example, defining two parity predicates on natural numbers using two mutually inductive types in Coq:

Inductive even : nat -> Prop :=
  | zero_is_even : even O
  | S_of_odd_is_even : (forall n:nat, odd n -> even (S n))
with odd : nat -> Prop :=
  | S_of_even_is_odd : (forall n:nat, even n -> odd (S n)).

Induction-recursion 

Induction-recursion started as a study into the limits of ITT.  Once found, the limits were turned into rules that allowed defining new inductive types.  These types could depend upon a function and the function on the type, as long as both were defined simultaneously.

Universe types can be defined using induction-recursion.

Induction-induction 

Induction-induction allows definition of a type and a family of types at the same time.  So, a type  and a family of types .

Higher inductive types 

This is a current research area in Homotopy Type Theory (HoTT).  HoTT differs from ITT by its identity type (equality).  Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them.

A simple example is the  type, which is defined with two constructors, a basepoint;

and a loop;

The existence of a new constructor for the identity type makes  a higher inductive type.

See also 
 Coinduction permits (effectively) infinite structures in type theory.

References

External links 
 Induction-Recursion Slides
 Induction-Induction Slides
 Higher Inductive Types: a tour of the menagerie

Category:Type theory