author | wenzelm |
Sat, 29 Mar 2008 22:55:49 +0100 | |
changeset 26496 | 49ae9456eba9 |
parent 26411 | cd74690f3bfb |
child 27115 | 0dcafa5c9e3f |
permissions | -rw-r--r-- |
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(* Title: FOL/FOL.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson and Markus Wenzel |
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*) |
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header {* Classical first-order logic *} |
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theory FOL |
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imports IFOL |
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uses |
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"~~/src/Provers/classical.ML" |
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"~~/src/Provers/blast.ML" |
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"~~/src/Provers/clasimp.ML" |
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"~~/src/Tools/induct.ML" |
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("cladata.ML") |
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("blastdata.ML") |
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("simpdata.ML") |
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begin |
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||
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subsection {* The classical axiom *} |
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axioms |
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classical: "(~P ==> P) ==> P" |
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subsection {* Lemmas and proof tools *} |
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lemma ccontr: "(\<not> P \<Longrightarrow> False) \<Longrightarrow> P" |
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by (erule FalseE [THEN classical]) |
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(*** Classical introduction rules for | and EX ***) |
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lemma disjCI: "(~Q ==> P) ==> P|Q" |
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apply (rule classical) |
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apply (assumption | erule meta_mp | rule disjI1 notI)+ |
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apply (erule notE disjI2)+ |
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done |
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(*introduction rule involving only EX*) |
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lemma ex_classical: |
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assumes r: "~(EX x. P(x)) ==> P(a)" |
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shows "EX x. P(x)" |
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apply (rule classical) |
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apply (rule exI, erule r) |
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done |
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(*version of above, simplifying ~EX to ALL~ *) |
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lemma exCI: |
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assumes r: "ALL x. ~P(x) ==> P(a)" |
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shows "EX x. P(x)" |
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apply (rule ex_classical) |
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apply (rule notI [THEN allI, THEN r]) |
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apply (erule notE) |
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apply (erule exI) |
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done |
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lemma excluded_middle: "~P | P" |
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apply (rule disjCI) |
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apply assumption |
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done |
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(*For disjunctive case analysis*) |
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ML {* |
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fun excluded_middle_tac sP = |
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res_inst_tac [("Q",sP)] (@{thm excluded_middle} RS @{thm disjE}) |
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*} |
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lemma case_split_thm: |
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assumes r1: "P ==> Q" |
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and r2: "~P ==> Q" |
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shows Q |
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apply (rule excluded_middle [THEN disjE]) |
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apply (erule r2) |
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apply (erule r1) |
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done |
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lemmas case_split = case_split_thm [case_names True False] |
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(*HOL's more natural case analysis tactic*) |
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ML {* |
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fun case_tac a = res_inst_tac [("P",a)] @{thm case_split_thm} |
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*} |
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(*** Special elimination rules *) |
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(*Classical implies (-->) elimination. *) |
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lemma impCE: |
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assumes major: "P-->Q" |
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and r1: "~P ==> R" |
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and r2: "Q ==> R" |
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shows R |
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apply (rule excluded_middle [THEN disjE]) |
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apply (erule r1) |
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apply (rule r2) |
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apply (erule major [THEN mp]) |
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done |
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(*This version of --> elimination works on Q before P. It works best for |
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those cases in which P holds "almost everywhere". Can't install as |
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default: would break old proofs.*) |
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lemma impCE': |
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assumes major: "P-->Q" |
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and r1: "Q ==> R" |
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and r2: "~P ==> R" |
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shows R |
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apply (rule excluded_middle [THEN disjE]) |
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apply (erule r2) |
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apply (rule r1) |
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apply (erule major [THEN mp]) |
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done |
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(*Double negation law*) |
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lemma notnotD: "~~P ==> P" |
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apply (rule classical) |
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apply (erule notE) |
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apply assumption |
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done |
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lemma contrapos2: "[| Q; ~ P ==> ~ Q |] ==> P" |
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apply (rule classical) |
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apply (drule (1) meta_mp) |
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apply (erule (1) notE) |
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done |
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(*** Tactics for implication and contradiction ***) |
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(*Classical <-> elimination. Proof substitutes P=Q in |
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~P ==> ~Q and P ==> Q *) |
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lemma iffCE: |
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assumes major: "P<->Q" |
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and r1: "[| P; Q |] ==> R" |
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and r2: "[| ~P; ~Q |] ==> R" |
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shows R |
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apply (rule major [unfolded iff_def, THEN conjE]) |
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apply (elim impCE) |
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apply (erule (1) r2) |
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apply (erule (1) notE)+ |
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apply (erule (1) r1) |
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done |
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(*Better for fast_tac: needs no quantifier duplication!*) |
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lemma alt_ex1E: |
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assumes major: "EX! x. P(x)" |
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and r: "!!x. [| P(x); ALL y y'. P(y) & P(y') --> y=y' |] ==> R" |
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shows R |
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using major |
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proof (rule ex1E) |
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fix x |
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assume * : "\<forall>y. P(y) \<longrightarrow> y = x" |
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assume "P(x)" |
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then show R |
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proof (rule r) |
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{ fix y y' |
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assume "P(y)" and "P(y')" |
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with * have "x = y" and "x = y'" by - (tactic "IntPr.fast_tac 1")+ |
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then have "y = y'" by (rule subst) |
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} note r' = this |
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show "\<forall>y y'. P(y) \<and> P(y') \<longrightarrow> y = y'" by (intro strip, elim conjE) (rule r') |
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qed |
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qed |
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lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R" |
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by (rule classical) iprover |
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lemma swap: "~ P ==> (~ R ==> P) ==> R" |
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by (rule classical) iprover |
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use "cladata.ML" |
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setup Cla.setup |
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setup cla_setup |
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setup case_setup |
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use "blastdata.ML" |
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setup Blast.setup |
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lemma ex1_functional: "[| EX! z. P(a,z); P(a,b); P(a,c) |] ==> b = c" |
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by blast |
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(* Elimination of True from asumptions: *) |
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lemma True_implies_equals: "(True ==> PROP P) == PROP P" |
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proof |
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assume "True \<Longrightarrow> PROP P" |
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from this and TrueI show "PROP P" . |
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next |
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assume "PROP P" |
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then show "PROP P" . |
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qed |
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lemma uncurry: "P --> Q --> R ==> P & Q --> R" |
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by blast |
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lemma iff_allI: "(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))" |
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by blast |
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lemma iff_exI: "(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))" |
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by blast |
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lemma all_comm: "(ALL x y. P(x,y)) <-> (ALL y x. P(x,y))" by blast |
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lemma ex_comm: "(EX x y. P(x,y)) <-> (EX y x. P(x,y))" by blast |
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(*** Classical simplification rules ***) |
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(*Avoids duplication of subgoals after expand_if, when the true and false |
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cases boil down to the same thing.*) |
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lemma cases_simp: "(P --> Q) & (~P --> Q) <-> Q" by blast |
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(*** Miniscoping: pushing quantifiers in |
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We do NOT distribute of ALL over &, or dually that of EX over | |
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Baaz and Leitsch, On Skolemization and Proof Complexity (1994) |
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show that this step can increase proof length! |
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***) |
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(*existential miniscoping*) |
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lemma int_ex_simps: |
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"!!P Q. (EX x. P(x) & Q) <-> (EX x. P(x)) & Q" |
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"!!P Q. (EX x. P & Q(x)) <-> P & (EX x. Q(x))" |
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"!!P Q. (EX x. P(x) | Q) <-> (EX x. P(x)) | Q" |
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"!!P Q. (EX x. P | Q(x)) <-> P | (EX x. Q(x))" |
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by iprover+ |
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(*classical rules*) |
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lemma cla_ex_simps: |
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"!!P Q. (EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q" |
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"!!P Q. (EX x. P --> Q(x)) <-> P --> (EX x. Q(x))" |
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by blast+ |
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lemmas ex_simps = int_ex_simps cla_ex_simps |
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(*universal miniscoping*) |
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lemma int_all_simps: |
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"!!P Q. (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q" |
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"!!P Q. (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))" |
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"!!P Q. (ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q" |
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"!!P Q. (ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))" |
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by iprover+ |
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(*classical rules*) |
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lemma cla_all_simps: |
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"!!P Q. (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q" |
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"!!P Q. (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))" |
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by blast+ |
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lemmas all_simps = int_all_simps cla_all_simps |
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(*** Named rewrite rules proved for IFOL ***) |
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lemma imp_disj1: "(P-->Q) | R <-> (P-->Q | R)" by blast |
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lemma imp_disj2: "Q | (P-->R) <-> (P-->Q | R)" by blast |
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lemma de_Morgan_conj: "(~(P & Q)) <-> (~P | ~Q)" by blast |
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lemma not_imp: "~(P --> Q) <-> (P & ~Q)" by blast |
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lemma not_iff: "~(P <-> Q) <-> (P <-> ~Q)" by blast |
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lemma not_all: "(~ (ALL x. P(x))) <-> (EX x.~P(x))" by blast |
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lemma imp_all: "((ALL x. P(x)) --> Q) <-> (EX x. P(x) --> Q)" by blast |
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lemmas meta_simps = |
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triv_forall_equality (* prunes params *) |
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True_implies_equals (* prune asms `True' *) |
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lemmas IFOL_simps = |
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refl [THEN P_iff_T] conj_simps disj_simps not_simps |
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imp_simps iff_simps quant_simps |
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lemma notFalseI: "~False" by iprover |
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lemma cla_simps_misc: |
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"~(P&Q) <-> ~P | ~Q" |
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"P | ~P" |
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"~P | P" |
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"~ ~ P <-> P" |
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"(~P --> P) <-> P" |
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"(~P <-> ~Q) <-> (P<->Q)" by blast+ |
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lemmas cla_simps = |
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de_Morgan_conj de_Morgan_disj imp_disj1 imp_disj2 |
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not_imp not_all not_ex cases_simp cla_simps_misc |
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use "simpdata.ML" |
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setup simpsetup |
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setup "Simplifier.method_setup Splitter.split_modifiers" |
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setup Splitter.setup |
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setup clasimp_setup |
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setup EqSubst.setup |
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subsection {* Other simple lemmas *} |
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lemma [simp]: "((P-->R) <-> (Q-->R)) <-> ((P<->Q) | R)" |
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by blast |
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lemma [simp]: "((P-->Q) <-> (P-->R)) <-> (P --> (Q<->R))" |
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by blast |
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lemma not_disj_iff_imp: "~P | Q <-> (P-->Q)" |
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by blast |
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(** Monotonicity of implications **) |
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lemma conj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)" |
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by fast (*or (IntPr.fast_tac 1)*) |
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lemma disj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)" |
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by fast (*or (IntPr.fast_tac 1)*) |
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lemma imp_mono: "[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)" |
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by fast (*or (IntPr.fast_tac 1)*) |
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lemma imp_refl: "P-->P" |
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by (rule impI, assumption) |
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(*The quantifier monotonicity rules are also intuitionistically valid*) |
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lemma ex_mono: "(!!x. P(x) --> Q(x)) ==> (EX x. P(x)) --> (EX x. Q(x))" |
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by blast |
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lemma all_mono: "(!!x. P(x) --> Q(x)) ==> (ALL x. P(x)) --> (ALL x. Q(x))" |
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by blast |
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subsection {* Proof by cases and induction *} |
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text {* Proper handling of non-atomic rule statements. *} |
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constdefs |
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induct_forall where "induct_forall(P) == \<forall>x. P(x)" |
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induct_implies where "induct_implies(A, B) == A \<longrightarrow> B" |
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induct_equal where "induct_equal(x, y) == x = y" |
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induct_conj where "induct_conj(A, B) == A \<and> B" |
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lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(\<lambda>x. P(x)))" |
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unfolding atomize_all induct_forall_def . |
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lemma induct_implies_eq: "(A ==> B) == Trueprop(induct_implies(A, B))" |
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unfolding atomize_imp induct_implies_def . |
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lemma induct_equal_eq: "(x == y) == Trueprop(induct_equal(x, y))" |
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unfolding atomize_eq induct_equal_def . |
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lemma induct_conj_eq: |
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includes meta_conjunction_syntax |
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shows "(A && B) == Trueprop(induct_conj(A, B))" |
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unfolding atomize_conj induct_conj_def . |
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lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq |
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lemmas induct_rulify [symmetric, standard] = induct_atomize |
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lemmas induct_rulify_fallback = |
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induct_forall_def induct_implies_def induct_equal_def induct_conj_def |
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hide const induct_forall induct_implies induct_equal induct_conj |
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365 |
text {* Method setup. *} |
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367 |
ML {* |
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368 |
structure Induct = InductFun |
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|
369 |
( |
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val cases_default = @{thm case_split} |
371 |
val atomize = @{thms induct_atomize} |
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val rulify = @{thms induct_rulify} |
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val rulify_fallback = @{thms induct_rulify_fallback} |
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parents:
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diff
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); |
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*} |
376 |
||
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parents:
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diff
changeset
|
377 |
setup Induct.setup |
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378 |
declare case_split [cases type: o] |
11678 | 379 |
|
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end |