Theory PMLinHOL_shallow_minimal_tests

subsection‹Tests with the minimal shallow embedding›

theory PMLinHOL_shallow_minimal_tests (* Christoph Benzmüller, 2025 *)
  imports PMLinHOL_shallow_minimal
begin 

―‹Hilbert calculus: proving that the schematic axioms and rules implied by the embedding›
lemma H1: "m φ m (ψ m φ)" by auto
lemma H2: "m (φ m (ψ m γ)) m ((φ m ψ) m (φ m γ)) " by auto
lemma H3: "m (¬mφm ¬mψ) m (ψ m φ)" by auto
lemma MP: "m φ  m (φm ψ)  m ψ" by auto

―‹Reasoning with the Hilbert calculus: interactive and fully automated›
lemma HCderived1: "m (φ m φ)" ―‹sledgehammer(HC1 HC2 HC3 MP) returns: by (metis HC1 HC2 MP)›
  proof -
    have 1: "m φ m ((ψ m φ) m φ)" using H1 by auto 
    have 2: "m (φ m ((ψ m φ) m φ)) m ((φ m (ψ m φ)) m (φ m φ))" using H2 by auto 
    have 3: "m (φ m (ψ m φ)) m (φ m φ)" using 1 2 MP by meson
    have 4: "m φ m (ψm φ)" using H1 by auto 
    thus ?thesis using 3 4 MP by meson 
  qed

lemma HCderived2: "m φ m (¬mφm ψ) " by (metis H1 H2 H3 MP) 
lemma HCderived3: "m (¬mφm φ) m φ" by (metis H1 H2 H3 MP) 
lemma HCderived4: "m (φ m ψ) m (¬mψ m ¬mφ) " by auto

―‹Modal logic: the schematic necessitation rule and distribution axiom are implied›
lemma Nec: "m φ  m mφ" by (smt DefM)
lemma Dist:"m m(φ m ψ) m (mφ m mψ) " by auto

―‹Correspondence theory: correct statements› 
lemma cM:"reflexive R  (φ. m mφ m φ)" by auto
lemma cBa: "symmetric R  (φ. m φ m mmφ)" by auto 
lemma cBb: "symmetric R  (φ. m φ m mmφ)" by (metis DefM)
lemma c4a: "transitive R  (φ. m mφ m m(mφ))" by (smt DefM)
lemma c4b: "transitive R  (φ. m mφ m m(mφ))" by auto

―‹Correspondence theory: incorrect statements› 
lemma "reflexive R  (φ. m mφ m m(mφ))" nitpick[card 𝗐=3,show_all] oops ―‹nitpick: Counterexample›

―‹Simple, incorrect validity statements›
lemma "m φ m mφ" nitpick[card 𝗐=2, card 𝒮= 1] oops ―‹nitpick: Counterexample: modal collapse not implied›
lemma "m m( mφ mmψ) m m( mψ mmφ)" nitpick[card 𝗐=3] oops ―‹nitpick: Counterexample› 
lemma "m (m(m φ)) mm(m φ)" nitpick[card 𝗐=2] oops ―‹nitpick: Counterexample› 

―‹Implied axiom schemata in S5›
lemma KB: "symmetric R  (φ ψ.m (m(mφ)) mm(mφ))" by auto
lemma K4B: "symmetric R  transitive R  (φ ψ. m m( mφ mmψ) m m( mψ mmφ))" by (smt DefM)
end