theory ProofAutomation
imports Main
begin

lemma "\<forall>x. \<exists>y. x = y"
by auto

lemma "A \<subseteq> B \<inter> C \<Longrightarrow> A \<subseteq> B \<union> C"
by auto

lemma "\<lbrakk> \<forall> xs \<in> A. \<exists> ys. xs = ys @ ys; us \<in> A \<rbrakk> \<Longrightarrow> \<exists>n. length us = n+n"
by fastforce

lemma "\<lbrakk>xs @ ys = ys @ xs; length xs = length ys \<rbrakk> \<Longrightarrow> xs = ys"
by auto

lemma "\<lbrakk> (a::nat) \<le> x + b; 2*x < c\<rbrakk> \<Longrightarrow> 2*a + 1 \<le> 2*b + c"
by arith

lemma "\<lbrakk> (a::nat) \<le> b; b \<le> c; c \<le> d; d \<le> e \<rbrakk> \<Longrightarrow> a \<le> e"
by (blast intro: le_trans)

lemma "Suc(Suc(Suc a)) \<le> b \<Longrightarrow> a \<le> b"
by(blast dest: Suc_leD)

inductive ev :: "nat \<Rightarrow> bool" where
ev0: "ev 0" |
evSS: "ev n \<Longrightarrow> ev (n + 2)"

fun even :: "nat \<Rightarrow> bool" where
"even 0 = True" |
"even (Suc 0) = False" |
"even (Suc(Suc n)) = even n"

lemma "ev m \<Longrightarrow> even m"