#theorem-proving — Public Fediverse posts

I'm trying out Aristotle and Lean Copilot for formalizing and proving theorems, and, so far, Aristotle feels way more powerful. Lean Copilot's suggestions aren't always right. Aristotle, on the other hand, can bring in previous lemmas, figure out missing hypotheses, and so on –besides, of course, finishing the proofs themselves.

Disclaimer: this is probably mostly down to the underlying model: Aristotle uses a specialized in-house model (within an agentic architecture), while Lean Copilot is using a more general-purpose one (and not a particularly strong one, at least in my setup... DeepSeek-R1, I think).

#lean #theoremproving #aristotle

I'm trying out Aristotle and Lean Copilot for formalizing and proving theorems, and, so far, Aristotle feels way more powerful. Lean Copilot's suggestions aren't always right. Aristotle, on the other hand, can bring in previous lemmas, figure out missing hypotheses, and so on –besides, of course, finishing the proofs themselves.

Disclaimer: this is probably mostly down to the underlying model: Aristotle uses a specialized in-house model (within an agentic architecture), while Lean Copilot is using a more general-purpose one (and not a particularly strong one, at least in my setup... DeepSeek-R1, I think).

#lean #theoremproving #aristotle

I'm trying out Aristotle and Lean Copilot for formalizing and proving theorems, and, so far, Aristotle feels way more powerful. Lean Copilot's suggestions aren't always right. Aristotle, on the other hand, can bring in previous lemmas, figure out missing hypotheses, and so on –besides, of course, finishing the proofs themselves.

Disclaimer: this is probably mostly down to the underlying model: Aristotle uses a specialized in-house model (within an agentic architecture), while Lean Copilot is using a more general-purpose one (and not a particularly strong one, at least in my setup... DeepSeek-R1, I think).

#lean #theoremproving #aristotle

I'm trying out Aristotle and Lean Copilot for formalizing and proving theorems, and, so far, Aristotle feels way more powerful. Lean Copilot's suggestions aren't always right. Aristotle, on the other hand, can bring in previous lemmas, figure out missing hypotheses, and so on –besides, of course, finishing the proofs themselves.

Disclaimer: this is probably mostly down to the underlying model: Aristotle uses a specialized in-house model (within an agentic architecture), while Lean Copilot is using a more general-purpose one (and not a particularly strong one, at least in my setup... DeepSeek-R1, I think).

#lean #theoremproving #aristotle

Want to learn more about the latest developments on using AI and (interactive) theorem proving in Mathematics? Wait no longer!

We have a great line-up of speakers at our online Workshop on AI and Theorem Provers in Mathematics. The workshop will be held online from 8th to 10th of April and attendance is free (registration required). For more details, visit the workshop website: https://aitpm.github.io/

#math #itp #theoremProving #isabelleHOL #lean #llm #ai #formal_mehods #agda #hol #mathematics

Want to learn more about the latest developments on using AI and (interactive) theorem proving in Mathematics? Wait no longer!

We have a great line-up of speakers at our online Workshop on AI and Theorem Provers in Mathematics. The workshop will be held online from 8th to 10th of April and attendance is free (registration required). For more details, visit the workshop website: https://aitpm.github.io/

#math #itp #theoremProving #isabelleHOL #lean #llm #ai #formal_mehods #agda #hol #mathematics

Want to learn more about the latest developments on using AI and (interactive) theorem proving in Mathematics? Wait no longer!

We have a great line-up of speakers at our online Workshop on AI and Theorem Provers in Mathematics. The workshop will be held online from 8th to 10th of April and attendance is free (registration required). For more details, visit the workshop website: https://aitpm.github.io/

#math #itp #theoremProving #isabelleHOL #lean #llm #ai #formal_mehods #agda #hol #mathematics

Want to learn more about the latest developments on using AI and (interactive) theorem proving in Mathematics? Wait no longer!

We have a great line-up of speakers at our online Workshop on AI and Theorem Provers in Mathematics. The workshop will be held online from 8th to 10th of April and attendance is free (registration required). For more details, visit the workshop website: https://aitpm.github.io/

#math #itp #theoremProving #isabelleHOL #lean #llm #ai #formal_mehods #agda #hol #mathematics

Want to learn more about the latest developments on using AI and (interactive) theorem proving in Mathematics? Wait no longer!

We have a great line-up of speakers at our online Workshop on AI and Theorem Provers in Mathematics. The workshop will be held online from 8th to 10th of April and attendance is free (registration required). For more details, visit the workshop website: https://aitpm.github.io/

#math #itp #theoremProving #isabelleHOL #lean #llm #ai #formal_mehods #agda #hol #mathematics

[New Blog Post] State of Knuckledragger III: Kernel Changes, Symbolic Union, AI, and more https://www.philipzucker.com/state_o_knuck_3/ #python #logic #theoremproving

[New Blog Post] State of Knuckledragger III: Kernel Changes, Symbolic Union, AI, and more https://www.philipzucker.com/state_o_knuck_3/ #python #logic #theoremproving

[New Blog Post] State of Knuckledragger III: Kernel Changes, Symbolic Union, AI, and more https://www.philipzucker.com/state_o_knuck_3/ #python #logic #theoremproving

[New Blog Post] State of Knuckledragger III: Kernel Changes, Symbolic Union, AI, and more https://www.philipzucker.com/state_o_knuck_3/ #python #logic #theoremproving

Terminal now can help you with formal proofs and theorem provers 🤯

📐 **lean-tui** — A TUI for visualizing Lean programs and proofs

💯 Live proof trees, data/effect flow views & real-time updates from your editor

🦀 Written in Rust & built with @ratatui_rs

⭐ Source: https://codeberg.org/wvhulle/lean-tui

#rustlang #ratatui #tui #lean #theoremproving #cli #devtools #terminal

Terminal now can help you with formal proofs and theorem provers 🤯

📐 **lean-tui** — A TUI for visualizing Lean programs and proofs

💯 Live proof trees, data/effect flow views & real-time updates from your editor

🦀 Written in Rust & built with @ratatui_rs

⭐ Source: https://codeberg.org/wvhulle/lean-tui

#rustlang #ratatui #tui #lean #theoremproving #cli #devtools #terminal

Terminal now can help you with formal proofs and theorem provers 🤯

📐 **lean-tui** — A TUI for visualizing Lean programs and proofs

💯 Live proof trees, data/effect flow views & real-time updates from your editor

🦀 Written in Rust & built with @ratatui_rs

⭐ Source: https://codeberg.org/wvhulle/lean-tui

#rustlang #ratatui #tui #lean #theoremproving #cli #devtools #terminal

Terminal now can help you with formal proofs and theorem provers 🤯

📐 **lean-tui** — A TUI for visualizing Lean programs and proofs

💯 Live proof trees, data/effect flow views & real-time updates from your editor

🦀 Written in Rust & built with @ratatui_rs

⭐ Source: https://codeberg.org/wvhulle/lean-tui

#rustlang #ratatui #tui #lean #theoremproving #cli #devtools #terminal

Terminal now can help you with formal proofs and theorem provers 🤯

📐 **lean-tui** — A TUI for visualizing Lean programs and proofs

💯 Live proof trees, data/effect flow views & real-time updates from your editor

🦀 Written in Rust & built with @ratatui_rs

⭐ Source: https://codeberg.org/wvhulle/lean-tui

#rustlang #ratatui #tui #lean #theoremproving #cli #devtools #terminal

Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated

Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated

Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated

Lean 4 is apparently the new secret sauce of #AI dominance, because who knew that theorem proving could be so *riveting*? 🤔✨ But don't worry, before you can learn how to take over the world with math, you'll need to pass the Vercel Security Checkpoint IQ test, where only the chosen ones with #JavaScript enabled may proceed. 🛂🔒
https://venturebeat.com/ai/lean4-how-the-theorem-prover-works-and-why-its-the-new-competitive-edge-in #Lean4 #TheoremProving #VercelSecurity #HackerNews #ngated

Agentic system design paper:
https://althofer.de/agentic_strategy_design_for_math_proofs.pdf
#1stProof #TheoremProving

Agentic system design paper:
https://althofer.de/agentic_strategy_design_for_math_proofs.pdf
#1stProof #TheoremProving

First Proof (#1stProof): We ran an AI-only workflow (no human mathematical input) and published a writeup + outputs.
Report: https://althofer.de/first-proof-competition/first-proof-report.html
Official: https://1stproof.org/
I’d appreciate critique—especially rigor/correctness checks and suggestions for better verification.
#1stProof #Mathematics #TheoremProving #AI

First Proof (#1stProof): We ran an AI-only workflow (no human mathematical input) and published a writeup + outputs.
Report: https://althofer.de/first-proof-competition/first-proof-report.html
Official: https://1stproof.org/
I’d appreciate critique—especially rigor/correctness checks and suggestions for better verification.
#1stProof #Mathematics #TheoremProving #AI

First Proof (#1stProof): We ran an AI-only workflow (no human mathematical input) and published a writeup + outputs.
Report: https://althofer.de/first-proof-competition/first-proof-report.html
Official: https://1stproof.org/
I’d appreciate critique—especially rigor/correctness checks and suggestions for better verification.
#1stProof #Mathematics #TheoremProving #AI

First Proof (#1stProof): We ran an AI-only workflow (no human mathematical input) and published a writeup + outputs.
Report: https://althofer.de/first-proof-competition/first-proof-report.html
Official: https://1stproof.org/
I’d appreciate critique—especially rigor/correctness checks and suggestions for better verification.
#1stProof #Mathematics #TheoremProving #AI

First Proof (#1stProof): We ran an AI-only workflow (no human mathematical input) and published a writeup + outputs.
Report: https://althofer.de/first-proof-competition/first-proof-report.html
Official: https://1stproof.org/
I’d appreciate critique—especially rigor/correctness checks and suggestions for better verification.
#1stProof #Mathematics #TheoremProving #AI

Lean Theorem Prover Mathlib

https://github.com/leanprover-community/mathlib4

#HackerNews #Lean #Theorem #Prover #Mathlib #mathlib4 #LeanProver #theoremProving #functionalProgramming

Lean Theorem Prover Mathlib

https://github.com/leanprover-community/mathlib4

#HackerNews #Lean #Theorem #Prover #Mathlib #mathlib4 #LeanProver #theoremProving #functionalProgramming

Lean Theorem Prover Mathlib

https://github.com/leanprover-community/mathlib4

#HackerNews #Lean #Theorem #Prover #Mathlib #mathlib4 #LeanProver #theoremProving #functionalProgramming

Lean Theorem Prover Mathlib

https://github.com/leanprover-community/mathlib4

#HackerNews #Lean #Theorem #Prover #Mathlib #mathlib4 #LeanProver #theoremProving #functionalProgramming

Lean Theorem Prover Mathlib

https://github.com/leanprover-community/mathlib4

#HackerNews #Lean #Theorem #Prover #Mathlib #mathlib4 #LeanProver #theoremProving #functionalProgramming

Does anyone know if an inductive Nat datatype defined as a place-value system could replace the need to rewrite the PA definition to bigints in the compiler?

#theoremProving #types #functional_programming

Does anyone know if an inductive Nat datatype defined as a place-value system could replace the need to rewrite the PA definition to bigints in the compiler?

#theoremProving #types #functional_programming

Does anyone know if an inductive Nat datatype defined as a place-value system could replace the need to rewrite the PA definition to bigints in the compiler?

#theoremProving #types #functional_programming

Does anyone know if an inductive Nat datatype defined as a place-value system could replace the need to rewrite the PA definition to bigints in the compiler?

#theoremProving #types #functional_programming

Does anyone know if an inductive Nat datatype defined as a place-value system could replace the need to rewrite the PA definition to bigints in the compiler?

#theoremProving #types #functional_programming

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

Emily Riehl: How I became seduced by Univalent Foundations

[tbh I think shes only disclosing her theoretical motivations her; I recall she was posting questions about Linux distros a few years ago, and if we're honest being a nerd is the actual reason 99% of people get into HoTT]
https://www.youtube.com/watch?v=XIYoI5j5Flo&t=486s

#hott #mathematics #categorytheory #theoremproving

DeepSeekMath‑V2 è un AI che dimostra teoremi matematici passo dopo passo.
Genera prove, le verifica con un LLM dedicato e corregge gli errori per migliorarsi continuamente. 🤖📐

#AIperLaMatematica #TheoremProving #VerificaAutomatica

DeepSeekMath‑V2 è un AI che dimostra teoremi matematici passo dopo passo.
Genera prove, le verifica con un LLM dedicato e corregge gli errori per migliorarsi continuamente. 🤖📐

#AIperLaMatematica #TheoremProving #VerificaAutomatica

DeepSeekMath‑V2 è un AI che dimostra teoremi matematici passo dopo passo.
Genera prove, le verifica con un LLM dedicato e corregge gli errori per migliorarsi continuamente. 🤖📐

#AIperLaMatematica #TheoremProving #VerificaAutomatica

DeepSeekMath‑V2 è un AI che dimostra teoremi matematici passo dopo passo.
Genera prove, le verifica con un LLM dedicato e corregge gli errori per migliorarsi continuamente. 🤖📐

#AIperLaMatematica #TheoremProving #VerificaAutomatica

DeepSeekMath‑V2 è un AI che dimostra teoremi matematici passo dopo passo.
Genera prove, le verifica con un LLM dedicato e corregge gli errori per migliorarsi continuamente. 🤖📐

#AIperLaMatematica #TheoremProving #VerificaAutomatica

New research shows how Lean4 can turn large‑language models into AI advisers that pair hypotheses with physics‑consistent proofs. This blend of theorem proving and formal verification promises safer AI and more reliable software. Dive into the details of how AI meets the laws of physics. #Lean4 #TheoremProving #PhysicsConsistent #FormalVerification

🔗 https://aidailypost.com/news/lean4-powers-ai-advisers-pair-hypotheses-physicsconsistent-proofs

New research shows how Lean4 can turn large‑language models into AI advisers that pair hypotheses with physics‑consistent proofs. This blend of theorem proving and formal verification promises safer AI and more reliable software. Dive into the details of how AI meets the laws of physics. #Lean4 #TheoremProving #PhysicsConsistent #FormalVerification

🔗 https://aidailypost.com/news/lean4-powers-ai-advisers-pair-hypotheses-physicsconsistent-proofs

#CondensedDetachment example

Axiom 1: ⊢ (𝜑 → (𝜓 → 𝜑))
Axiom 2: ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Rule of Modus Ponens:
• Major hypothesis: ⊢ (𝜓 → 𝜑)
• Minor hypothesis: ⊢ 𝜓
• Resulting Assertion: ⊢ 𝜑
——
D<major><minor> applies the Rule of Modus Ponens treating the two given tautologies as having metavariables living in different namespaces and returning the normalized result. We extend by using underscore as a placeholder, so D__ recovers the rule of modus ponens.
——
"D2_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → (𝜓 → 𝜒))
• Resulting assertion: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
——
"D21" is a proof which unifies "1" ⊢ (𝜑′ → (𝜓′ → 𝜑′)) with the hypothesis of "D2_" giving the substitution map 𝜎: {𝜑′ ↦ 𝜑, 𝜓′ ↦ 𝜓, 𝜒 ↦ 𝜑} resulting in the tautology: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜑))

(Note that unification can map variables from either side, but when faced with a variable matching a term has to match the variable to that term.)
——
"DD21_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → 𝜓)
• Resulting assertion: ⊢ ((𝜑 → 𝜑)
——
"DD211" is a proof which unifies "1" ⊢ (𝜑″ → (𝜓″ → 𝜑″)) with the hypothesis of "DD21_" giving the substitution map 𝜎: {𝜑″ ↦ 𝜑, 𝜓 ↦ (𝜓″ → 𝜑)} resulting in the tautology: ⊢ (𝜑 → 𝜑)

This has been adapted and expanded from a run of my symbolic-mgu pre-release crate. https://crates.io/crates/symbolic-mgu

cargo run -r --bin compact -- --wide D__ 1 2 D2_ D21 DD21_ DD211

#math #logic #theoremProving #rust #mostGeneralUnifier #mgu

#CondensedDetachment example

Axiom 1: ⊢ (𝜑 → (𝜓 → 𝜑))
Axiom 2: ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Rule of Modus Ponens:
• Major hypothesis: ⊢ (𝜓 → 𝜑)
• Minor hypothesis: ⊢ 𝜓
• Resulting Assertion: ⊢ 𝜑
——
D<major><minor> applies the Rule of Modus Ponens treating the two given tautologies as having metavariables living in different namespaces and returning the normalized result. We extend by using underscore as a placeholder, so D__ recovers the rule of modus ponens.
——
"D2_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → (𝜓 → 𝜒))
• Resulting assertion: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
——
"D21" is a proof which unifies "1" ⊢ (𝜑′ → (𝜓′ → 𝜑′)) with the hypothesis of "D2_" giving the substitution map 𝜎: {𝜑′ ↦ 𝜑, 𝜓′ ↦ 𝜓, 𝜒 ↦ 𝜑} resulting in the tautology: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜑))

(Note that unification can map variables from either side, but when faced with a variable matching a term has to match the variable to that term.)
——
"DD21_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → 𝜓)
• Resulting assertion: ⊢ ((𝜑 → 𝜑)
——
"DD211" is a proof which unifies "1" ⊢ (𝜑″ → (𝜓″ → 𝜑″)) with the hypothesis of "DD21_" giving the substitution map 𝜎: {𝜑″ ↦ 𝜑, 𝜓 ↦ (𝜓″ → 𝜑)} resulting in the tautology: ⊢ (𝜑 → 𝜑)

This has been adapted and expanded from a run of my symbolic-mgu pre-release crate. https://crates.io/crates/symbolic-mgu

cargo run -r --bin compact -- --wide D__ 1 2 D2_ D21 DD21_ DD211

#math #logic #theoremProving #rust #mostGeneralUnifier #mgu

#CondensedDetachment example

Axiom 1: ⊢ (𝜑 → (𝜓 → 𝜑))
Axiom 2: ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Rule of Modus Ponens:
• Major hypothesis: ⊢ (𝜓 → 𝜑)
• Minor hypothesis: ⊢ 𝜓
• Resulting Assertion: ⊢ 𝜑
——
D<major><minor> applies the Rule of Modus Ponens treating the two given tautologies as having metavariables living in different namespaces and returning the normalized result. We extend by using underscore as a placeholder, so D__ recovers the rule of modus ponens.
——
"D2_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → (𝜓 → 𝜒))
• Resulting assertion: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
——
"D21" is a proof which unifies "1" ⊢ (𝜑′ → (𝜓′ → 𝜑′)) with the hypothesis of "D2_" giving the substitution map 𝜎: {𝜑′ ↦ 𝜑, 𝜓′ ↦ 𝜓, 𝜒 ↦ 𝜑} resulting in the tautology: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜑))

(Note that unification can map variables from either side, but when faced with a variable matching a term has to match the variable to that term.)
——
"DD21_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → 𝜓)
• Resulting assertion: ⊢ ((𝜑 → 𝜑)
——
"DD211" is a proof which unifies "1" ⊢ (𝜑″ → (𝜓″ → 𝜑″)) with the hypothesis of "DD21_" giving the substitution map 𝜎: {𝜑″ ↦ 𝜑, 𝜓 ↦ (𝜓″ → 𝜑)} resulting in the tautology: ⊢ (𝜑 → 𝜑)

This has been adapted and expanded from a run of my symbolic-mgu pre-release crate. https://crates.io/crates/symbolic-mgu

cargo run -r --bin compact -- --wide D__ 1 2 D2_ D21 DD21_ DD211

#math #logic #theoremProving #rust #mostGeneralUnifier #mgu

#CondensedDetachment example

Axiom 1: ⊢ (𝜑 → (𝜓 → 𝜑))
Axiom 2: ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Rule of Modus Ponens:
• Major hypothesis: ⊢ (𝜓 → 𝜑)
• Minor hypothesis: ⊢ 𝜓
• Resulting Assertion: ⊢ 𝜑
——
D<major><minor> applies the Rule of Modus Ponens treating the two given tautologies as having metavariables living in different namespaces and returning the normalized result. We extend by using underscore as a placeholder, so D__ recovers the rule of modus ponens.
——
"D2_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → (𝜓 → 𝜒))
• Resulting assertion: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
——
"D21" is a proof which unifies "1" ⊢ (𝜑′ → (𝜓′ → 𝜑′)) with the hypothesis of "D2_" giving the substitution map 𝜎: {𝜑′ ↦ 𝜑, 𝜓′ ↦ 𝜓, 𝜒 ↦ 𝜑} resulting in the tautology: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜑))

(Note that unification can map variables from either side, but when faced with a variable matching a term has to match the variable to that term.)
——
"DD21_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → 𝜓)
• Resulting assertion: ⊢ ((𝜑 → 𝜑)
——
"DD211" is a proof which unifies "1" ⊢ (𝜑″ → (𝜓″ → 𝜑″)) with the hypothesis of "DD21_" giving the substitution map 𝜎: {𝜑″ ↦ 𝜑, 𝜓 ↦ (𝜓″ → 𝜑)} resulting in the tautology: ⊢ (𝜑 → 𝜑)

This has been adapted and expanded from a run of my symbolic-mgu pre-release crate. https://crates.io/crates/symbolic-mgu

cargo run -r --bin compact -- --wide D__ 1 2 D2_ D21 DD21_ DD211

#math #logic #theoremProving #rust #mostGeneralUnifier #mgu

#CondensedDetachment example

Axiom 1: ⊢ (𝜑 → (𝜓 → 𝜑))
Axiom 2: ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Rule of Modus Ponens:
• Major hypothesis: ⊢ (𝜓 → 𝜑)
• Minor hypothesis: ⊢ 𝜓
• Resulting Assertion: ⊢ 𝜑
——
D<major><minor> applies the Rule of Modus Ponens treating the two given tautologies as having metavariables living in different namespaces and returning the normalized result. We extend by using underscore as a placeholder, so D__ recovers the rule of modus ponens.
——
"D2_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → (𝜓 → 𝜒))
• Resulting assertion: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
——
"D21" is a proof which unifies "1" ⊢ (𝜑′ → (𝜓′ → 𝜑′)) with the hypothesis of "D2_" giving the substitution map 𝜎: {𝜑′ ↦ 𝜑, 𝜓′ ↦ 𝜓, 𝜒 ↦ 𝜑} resulting in the tautology: ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜑))

(Note that unification can map variables from either side, but when faced with a variable matching a term has to match the variable to that term.)
——
"DD21_" is proof of the rule:
• Hypothesis: ⊢ (𝜑 → 𝜓)
• Resulting assertion: ⊢ ((𝜑 → 𝜑)
——
"DD211" is a proof which unifies "1" ⊢ (𝜑″ → (𝜓″ → 𝜑″)) with the hypothesis of "DD21_" giving the substitution map 𝜎: {𝜑″ ↦ 𝜑, 𝜓 ↦ (𝜓″ → 𝜑)} resulting in the tautology: ⊢ (𝜑 → 𝜑)

This has been adapted and expanded from a run of my symbolic-mgu pre-release crate. https://crates.io/crates/symbolic-mgu

cargo run -r --bin compact -- --wide D__ 1 2 D2_ D21 DD21_ DD211

#math #logic #theoremProving #rust #mostGeneralUnifier #mgu

An abbreviated run for examining sub-proofs of propositional logic from Russell and Whitehead, and proving that they are all tautologies:

```
% cargo test --features serde,bigint -r --test pmproofs_validation -- --include-ignored --no-capture

running 1 test
Validating PM subproofs...
Variable limit: unlimited (bigint feature enabled)
Total subproofs in database: 2997
Processed 100/2997 subproofs...
Processed 200/2997 subproofs...

...

Processed 2800/2997 subproofs...
Processed 2900/2997 subproofs...

========================================
PM SUBPROOF VALIDATION RESULTS
========================================
Total subproofs: 2997
Parse failures: 0
Skipped (too many variables): 0
Validation errors: 0
Not tautologies: 0
Successfully validated: 2997

✓ Successfully validated 2997 subproofs!
test all_pm_subproofs_are_tautologies ... ok

test result: ok. 1 passed; 0 failed; 0 ignored; 0 measured; 0 filtered out; finished in 2.60s

```

#Rust #logic #math #theoremProving #condensedDetachment #mostGeneralUnifier #mgu

An abbreviated run for examining sub-proofs of propositional logic from Russell and Whitehead, and proving that they are all tautologies:

```
% cargo test --features serde,bigint -r --test pmproofs_validation -- --include-ignored --no-capture

running 1 test
Validating PM subproofs...
Variable limit: unlimited (bigint feature enabled)
Total subproofs in database: 2997
Processed 100/2997 subproofs...
Processed 200/2997 subproofs...

...

Processed 2800/2997 subproofs...
Processed 2900/2997 subproofs...

========================================
PM SUBPROOF VALIDATION RESULTS
========================================
Total subproofs: 2997
Parse failures: 0
Skipped (too many variables): 0
Validation errors: 0
Not tautologies: 0
Successfully validated: 2997

✓ Successfully validated 2997 subproofs!
test all_pm_subproofs_are_tautologies ... ok

test result: ok. 1 passed; 0 failed; 0 ignored; 0 measured; 0 filtered out; finished in 2.60s

```

#Rust #logic #math #theoremProving #condensedDetachment #mostGeneralUnifier #mgu

An abbreviated run for examining sub-proofs of propositional logic from Russell and Whitehead, and proving that they are all tautologies:

```
% cargo test --features serde,bigint -r --test pmproofs_validation -- --include-ignored --no-capture

running 1 test
Validating PM subproofs...
Variable limit: unlimited (bigint feature enabled)
Total subproofs in database: 2997
Processed 100/2997 subproofs...
Processed 200/2997 subproofs...

...

Processed 2800/2997 subproofs...
Processed 2900/2997 subproofs...

========================================
PM SUBPROOF VALIDATION RESULTS
========================================
Total subproofs: 2997
Parse failures: 0
Skipped (too many variables): 0
Validation errors: 0
Not tautologies: 0
Successfully validated: 2997

✓ Successfully validated 2997 subproofs!
test all_pm_subproofs_are_tautologies ... ok

test result: ok. 1 passed; 0 failed; 0 ignored; 0 measured; 0 filtered out; finished in 2.60s

```

#Rust #logic #math #theoremProving #condensedDetachment #mostGeneralUnifier #mgu