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src/HOL/Calculation.thy

author | wenzelm |

Sun, 11 Feb 2001 16:31:54 +0100 | |

changeset 11096 | bedfd42db838 |

parent 11089 | 0f6f1cd500e5 |

child 11253 | caabb021ec0f |

permissions | -rw-r--r-- |

tuned trans rules;

(* Title: HOL/Calculation.thy ID: $Id$ Author: Markus Wenzel, TU Muenchen License: GPL (GNU GENERAL PUBLIC LICENSE) Setup transitivity rules for calculational proofs. *) theory Calculation = IntArith: lemma forw_subst: "a = b ==> P b ==> P a" by (rule ssubst) lemma back_subst: "P a ==> a = b ==> P b" by (rule subst) lemma set_rev_mp: "x:A ==> A <= B ==> x:B" by (rule subsetD) lemma set_mp: "A <= B ==> x:A ==> x:B" by (rule subsetD) lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b" by (simp add: order_less_le) lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b" by (simp add: order_less_le) lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P" by (rule order_less_asym) lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" by (rule subst) lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" by (rule ssubst) lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" by (rule subst) lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" by (rule ssubst) lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b < c" finally (order_less_trans) show ?thesis . qed lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_less_trans) show ?thesis . qed lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a < c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b < c" finally (order_le_less_trans) show ?thesis . qed lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a <= f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_le_less_trans) show ?thesis . qed lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b <= c" finally (order_less_le_trans) show ?thesis . qed lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a < f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a < f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_less_le_trans) show ?thesis . qed lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_trans) show ?thesis . qed lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b <= c" finally (order_trans) show ?thesis . qed lemma ord_le_eq_subst: "a <= b ==> f b = c ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b = c" finally (ord_le_eq_trans) show ?thesis . qed lemma ord_eq_le_subst: "a = f b ==> b <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a = f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (ord_eq_le_trans) show ?thesis . qed lemma ord_less_eq_subst: "a < b ==> f b = c ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b = c" finally (ord_less_eq_trans) show ?thesis . qed lemma ord_eq_less_subst: "a = f b ==> b < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a = f b" also assume "b < c" hence "f b < f c" by (rule r) finally (ord_eq_less_trans) show ?thesis . qed text {* Note that this list of rules is in reverse order of priorities. *} lemmas basic_trans_rules [trans] = order_less_subst2 order_less_subst1 order_le_less_subst2 order_le_less_subst1 order_less_le_subst2 order_less_le_subst1 order_subst2 order_subst1 ord_le_eq_subst ord_eq_le_subst ord_less_eq_subst ord_eq_less_subst forw_subst back_subst dvd_trans rev_mp mp set_rev_mp set_mp order_neq_le_trans order_le_neq_trans order_less_asym' order_less_trans order_le_less_trans order_less_le_trans order_trans order_antisym ord_le_eq_trans ord_eq_le_trans ord_less_eq_trans ord_eq_less_trans transitive trans end