Theory Force

theory Force imports Main begin 
  (*Use Divides rather than Main to experiment with the first proof.
    Otherwise, it is done by the nat_divide_cancel_factor simprocs.*)

declare div_mult_self_is_m [simp del]

lemma div_mult_self_is_m: 
      "0<n ⟹ (m*n) div n = (m::nat)"
apply (insert div_mult_mod_eq [of "m*n" n])
apply simp
done


lemma "(∀x. P x) ∧ (∃x. Q x) ⟶ (∀x. P x ∧ Q x)"
apply clarify
oops

text ‹
proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
{\isacharparenleft}{\isasymforall}x{\isachardot}\ P\ x{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}{\isasymexists}x{\isachardot}\ Q\ x{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\isasymforall}x{\isachardot}\ P\ x\ {\isasymand}\ Q\ x{\isacharparenright}\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ xa{\isachardot}\ {\isasymlbrakk}{\isasymforall}x{\isachardot}\ P\ x{\isacharsemicolon}\ Q\ xa{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ {\isasymand}\ Q\ x
›

text ‹
couldn't find a good example of clarsimp

@{thm[display]"someI"}
\rulename{someI}
›

lemma "⟦Q a; P a⟧ ⟹ P (SOME x. P x ∧ Q x) ∧ Q (SOME x. P x ∧ Q x)"
apply (rule someI)
oops

lemma "⟦Q a; P a⟧ ⟹ P (SOME x. P x ∧ Q x) ∧ Q (SOME x. P x ∧ Q x)"
apply (fast intro!: someI)
done

text‹
proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
{\isasymlbrakk}Q\ a{\isacharsemicolon}\ P\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}SOME\ x{\isachardot}\ P\ x\ {\isasymand}\ Q\ x{\isacharparenright}\ {\isasymand}\ Q\ {\isacharparenleft}SOME\ x{\isachardot}\ P\ x\ {\isasymand}\ Q\ x{\isacharparenright}\isanewline
\ \isadigit{1}{\isachardot}\ {\isasymlbrakk}Q\ a{\isacharsemicolon}\ P\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharquery}x\ {\isasymand}\ Q\ {\isacharquery}x
›

end