(* Title: ZF/upair.thy ID: $Id: upair.thy,v 1.18 2007/10/07 19:19:33 wenzelm Exp $ Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory Copyright 1993 University of Cambridge Observe the order of dependence: Upair is defined in terms of Replace Un is defined in terms of Upair and Union (similarly for Int) cons is defined in terms of Upair and Un Ordered pairs and descriptions are defined using cons ("set notation") *) header{*Unordered Pairs*} theory upair imports ZF uses "Tools/typechk.ML" begin setup TypeCheck.setup lemma atomize_ball [symmetric, rulify]: "(!!x. x:A ==> P(x)) == Trueprop (ALL x:A. P(x))" by (simp add: Ball_def atomize_all atomize_imp) subsection{*Unordered Pairs: constant @{term Upair}*} lemma Upair_iff [simp]: "c : Upair(a,b) <-> (c=a | c=b)" by (unfold Upair_def, blast) lemma UpairI1: "a : Upair(a,b)" by simp lemma UpairI2: "b : Upair(a,b)" by simp lemma UpairE: "[| a : Upair(b,c); a=b ==> P; a=c ==> P |] ==> P" by (simp, blast) subsection{*Rules for Binary Union, Defined via @{term Upair}*} lemma Un_iff [simp]: "c : A Un B <-> (c:A | c:B)" apply (simp add: Un_def) apply (blast intro: UpairI1 UpairI2 elim: UpairE) done lemma UnI1: "c : A ==> c : A Un B" by simp lemma UnI2: "c : B ==> c : A Un B" by simp declare UnI1 [elim?] UnI2 [elim?] lemma UnE [elim!]: "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P" by (simp, blast) (*Stronger version of the rule above*) lemma UnE': "[| c : A Un B; c:A ==> P; [| c:B; c~:A |] ==> P |] ==> P" by (simp, blast) (*Classical introduction rule: no commitment to A vs B*) lemma UnCI [intro!]: "(c ~: B ==> c : A) ==> c : A Un B" by (simp, blast) subsection{*Rules for Binary Intersection, Defined via @{term Upair}*} lemma Int_iff [simp]: "c : A Int B <-> (c:A & c:B)" apply (unfold Int_def) apply (blast intro: UpairI1 UpairI2 elim: UpairE) done lemma IntI [intro!]: "[| c : A; c : B |] ==> c : A Int B" by simp lemma IntD1: "c : A Int B ==> c : A" by simp lemma IntD2: "c : A Int B ==> c : B" by simp lemma IntE [elim!]: "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P" by simp subsection{*Rules for Set Difference, Defined via @{term Upair}*} lemma Diff_iff [simp]: "c : A-B <-> (c:A & c~:B)" by (unfold Diff_def, blast) lemma DiffI [intro!]: "[| c : A; c ~: B |] ==> c : A - B" by simp lemma DiffD1: "c : A - B ==> c : A" by simp lemma DiffD2: "c : A - B ==> c ~: B" by simp lemma DiffE [elim!]: "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" by simp subsection{*Rules for @{term cons}*} lemma cons_iff [simp]: "a : cons(b,A) <-> (a=b | a:A)" apply (unfold cons_def) apply (blast intro: UpairI1 UpairI2 elim: UpairE) done (*risky as a typechecking rule, but solves otherwise unconstrained goals of the form x : ?A*) lemma consI1 [simp,TC]: "a : cons(a,B)" by simp lemma consI2: "a : B ==> a : cons(b,B)" by simp lemma consE [elim!]: "[| a : cons(b,A); a=b ==> P; a:A ==> P |] ==> P" by (simp, blast) (*Stronger version of the rule above*) lemma consE': "[| a : cons(b,A); a=b ==> P; [| a:A; a~=b |] ==> P |] ==> P" by (simp, blast) (*Classical introduction rule*) lemma consCI [intro!]: "(a~:B ==> a=b) ==> a: cons(b,B)" by (simp, blast) lemma cons_not_0 [simp]: "cons(a,B) ~= 0" by (blast elim: equalityE) lemmas cons_neq_0 = cons_not_0 [THEN notE, standard] declare cons_not_0 [THEN not_sym, simp] subsection{*Singletons*} lemma singleton_iff: "a : {b} <-> a=b" by simp lemma singletonI [intro!]: "a : {a}" by (rule consI1) lemmas singletonE = singleton_iff [THEN iffD1, elim_format, standard, elim!] subsection{*Descriptions*} lemma the_equality [intro]: "[| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a" apply (unfold the_def) apply (fast dest: subst) done (* Only use this if you already know EX!x. P(x) *) lemma the_equality2: "[| EX! x. P(x); P(a) |] ==> (THE x. P(x)) = a" by blast lemma theI: "EX! x. P(x) ==> P(THE x. P(x))" apply (erule ex1E) apply (subst the_equality) apply (blast+) done (*the_cong is no longer necessary: if (ALL y.P(y)<->Q(y)) then (THE x.P(x)) rewrites to (THE x. Q(x)) *) (*If it's "undefined", it's zero!*) lemma the_0: "~ (EX! x. P(x)) ==> (THE x. P(x))=0" apply (unfold the_def) apply (blast elim!: ReplaceE) done (*Easier to apply than theI: conclusion has only one occurrence of P*) lemma theI2: assumes p1: "~ Q(0) ==> EX! x. P(x)" and p2: "!!x. P(x) ==> Q(x)" shows "Q(THE x. P(x))" apply (rule classical) apply (rule p2) apply (rule theI) apply (rule classical) apply (rule p1) apply (erule the_0 [THEN subst], assumption) done lemma the_eq_trivial [simp]: "(THE x. x = a) = a" by blast lemma the_eq_trivial2 [simp]: "(THE x. a = x) = a" by blast subsection{*Conditional Terms: @{text "if-then-else"}*} lemma if_true [simp]: "(if True then a else b) = a" by (unfold if_def, blast) lemma if_false [simp]: "(if False then a else b) = b" by (unfold if_def, blast) (*Never use with case splitting, or if P is known to be true or false*) lemma if_cong: "[| P<->Q; Q ==> a=c; ~Q ==> b=d |] ==> (if P then a else b) = (if Q then c else d)" by (simp add: if_def cong add: conj_cong) (*Prevents simplification of x and y: faster and allows the execution of functional programs. NOW THE DEFAULT.*) lemma if_weak_cong: "P<->Q ==> (if P then x else y) = (if Q then x else y)" by simp (*Not needed for rewriting, since P would rewrite to True anyway*) lemma if_P: "P ==> (if P then a else b) = a" by (unfold if_def, blast) (*Not needed for rewriting, since P would rewrite to False anyway*) lemma if_not_P: "~P ==> (if P then a else b) = b" by (unfold if_def, blast) lemma split_if [split]: "P(if Q then x else y) <-> ((Q --> P(x)) & (~Q --> P(y)))" by (case_tac Q, simp_all) (** Rewrite rules for boolean case-splitting: faster than addsplits[split_if] **) lemmas split_if_eq1 = split_if [of "%x. x = b", standard] lemmas split_if_eq2 = split_if [of "%x. a = x", standard] lemmas split_if_mem1 = split_if [of "%x. x : b", standard] lemmas split_if_mem2 = split_if [of "%x. a : x", standard] lemmas split_ifs = split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 (*Logically equivalent to split_if_mem2*) lemma if_iff: "a: (if P then x else y) <-> P & a:x | ~P & a:y" by simp lemma if_type [TC]: "[| P ==> a: A; ~P ==> b: A |] ==> (if P then a else b): A" by simp (** Splitting IFs in the assumptions **) lemma split_if_asm: "P(if Q then x else y) <-> (~((Q & ~P(x)) | (~Q & ~P(y))))" by simp lemmas if_splits = split_if split_if_asm subsection{*Consequences of Foundation*} (*was called mem_anti_sym*) lemma mem_asym: "[| a:b; ~P ==> b:a |] ==> P" apply (rule classical) apply (rule_tac A1 = "{a,b}" in foundation [THEN disjE]) apply (blast elim!: equalityE)+ done (*was called mem_anti_refl*) lemma mem_irrefl: "a:a ==> P" by (blast intro: mem_asym) (*mem_irrefl should NOT be added to default databases: it would be tried on most goals, making proofs slower!*) lemma mem_not_refl: "a ~: a" apply (rule notI) apply (erule mem_irrefl) done (*Good for proving inequalities by rewriting*) lemma mem_imp_not_eq: "a:A ==> a ~= A" by (blast elim!: mem_irrefl) lemma eq_imp_not_mem: "a=A ==> a ~: A" by (blast intro: elim: mem_irrefl) subsection{*Rules for Successor*} lemma succ_iff: "i : succ(j) <-> i=j | i:j" by (unfold succ_def, blast) lemma succI1 [simp]: "i : succ(i)" by (simp add: succ_iff) lemma succI2: "i : j ==> i : succ(j)" by (simp add: succ_iff) lemma succE [elim!]: "[| i : succ(j); i=j ==> P; i:j ==> P |] ==> P" apply (simp add: succ_iff, blast) done (*Classical introduction rule*) lemma succCI [intro!]: "(i~:j ==> i=j) ==> i: succ(j)" by (simp add: succ_iff, blast) lemma succ_not_0 [simp]: "succ(n) ~= 0" by (blast elim!: equalityE) lemmas succ_neq_0 = succ_not_0 [THEN notE, standard, elim!] declare succ_not_0 [THEN not_sym, simp] declare sym [THEN succ_neq_0, elim!] (* succ(c) <= B ==> c : B *) lemmas succ_subsetD = succI1 [THEN [2] subsetD] (* succ(b) ~= b *) lemmas succ_neq_self = succI1 [THEN mem_imp_not_eq, THEN not_sym, standard] lemma succ_inject_iff [simp]: "succ(m) = succ(n) <-> m=n" by (blast elim: mem_asym elim!: equalityE) lemmas succ_inject = succ_inject_iff [THEN iffD1, standard, dest!] subsection{*Miniscoping of the Bounded Universal Quantifier*} lemma ball_simps1: "(ALL x:A. P(x) & Q) <-> (ALL x:A. P(x)) & (A=0 | Q)" "(ALL x:A. P(x) | Q) <-> ((ALL x:A. P(x)) | Q)" "(ALL x:A. P(x) --> Q) <-> ((EX x:A. P(x)) --> Q)" "(~(ALL x:A. P(x))) <-> (EX x:A. ~P(x))" "(ALL x:0.P(x)) <-> True" "(ALL x:succ(i).P(x)) <-> P(i) & (ALL x:i. P(x))" "(ALL x:cons(a,B).P(x)) <-> P(a) & (ALL x:B. P(x))" "(ALL x:RepFun(A,f). P(x)) <-> (ALL y:A. P(f(y)))" "(ALL x:Union(A).P(x)) <-> (ALL y:A. ALL x:y. P(x))" by blast+ lemma ball_simps2: "(ALL x:A. P & Q(x)) <-> (A=0 | P) & (ALL x:A. Q(x))" "(ALL x:A. P | Q(x)) <-> (P | (ALL x:A. Q(x)))" "(ALL x:A. P --> Q(x)) <-> (P --> (ALL x:A. Q(x)))" by blast+ lemma ball_simps3: "(ALL x:Collect(A,Q).P(x)) <-> (ALL x:A. Q(x) --> P(x))" by blast+ lemmas ball_simps [simp] = ball_simps1 ball_simps2 ball_simps3 lemma ball_conj_distrib: "(ALL x:A. P(x) & Q(x)) <-> ((ALL x:A. P(x)) & (ALL x:A. Q(x)))" by blast subsection{*Miniscoping of the Bounded Existential Quantifier*} lemma bex_simps1: "(EX x:A. P(x) & Q) <-> ((EX x:A. P(x)) & Q)" "(EX x:A. P(x) | Q) <-> (EX x:A. P(x)) | (A~=0 & Q)" "(EX x:A. P(x) --> Q) <-> ((ALL x:A. P(x)) --> (A~=0 & Q))" "(EX x:0.P(x)) <-> False" "(EX x:succ(i).P(x)) <-> P(i) | (EX x:i. P(x))" "(EX x:cons(a,B).P(x)) <-> P(a) | (EX x:B. P(x))" "(EX x:RepFun(A,f). P(x)) <-> (EX y:A. P(f(y)))" "(EX x:Union(A).P(x)) <-> (EX y:A. EX x:y. P(x))" "(~(EX x:A. P(x))) <-> (ALL x:A. ~P(x))" by blast+ lemma bex_simps2: "(EX x:A. P & Q(x)) <-> (P & (EX x:A. Q(x)))" "(EX x:A. P | Q(x)) <-> (A~=0 & P) | (EX x:A. Q(x))" "(EX x:A. P --> Q(x)) <-> ((A=0 | P) --> (EX x:A. Q(x)))" by blast+ lemma bex_simps3: "(EX x:Collect(A,Q).P(x)) <-> (EX x:A. Q(x) & P(x))" by blast lemmas bex_simps [simp] = bex_simps1 bex_simps2 bex_simps3 lemma bex_disj_distrib: "(EX x:A. P(x) | Q(x)) <-> ((EX x:A. P(x)) | (EX x:A. Q(x)))" by blast (** One-point rule for bounded quantifiers: see HOL/Set.ML **) lemma bex_triv_one_point1 [simp]: "(EX x:A. x=a) <-> (a:A)" by blast lemma bex_triv_one_point2 [simp]: "(EX x:A. a=x) <-> (a:A)" by blast lemma bex_one_point1 [simp]: "(EX x:A. x=a & P(x)) <-> (a:A & P(a))" by blast lemma bex_one_point2 [simp]: "(EX x:A. a=x & P(x)) <-> (a:A & P(a))" by blast lemma ball_one_point1 [simp]: "(ALL x:A. x=a --> P(x)) <-> (a:A --> P(a))" by blast lemma ball_one_point2 [simp]: "(ALL x:A. a=x --> P(x)) <-> (a:A --> P(a))" by blast subsection{*Miniscoping of the Replacement Operator*} text{*These cover both @{term Replace} and @{term Collect}*} lemma Rep_simps [simp]: "{x. y:0, R(x,y)} = 0" "{x:0. P(x)} = 0" "{x:A. Q} = (if Q then A else 0)" "RepFun(0,f) = 0" "RepFun(succ(i),f) = cons(f(i), RepFun(i,f))" "RepFun(cons(a,B),f) = cons(f(a), RepFun(B,f))" by (simp_all, blast+) subsection{*Miniscoping of Unions*} lemma UN_simps1: "(UN x:C. cons(a, B(x))) = (if C=0 then 0 else cons(a, UN x:C. B(x)))" "(UN x:C. A(x) Un B') = (if C=0 then 0 else (UN x:C. A(x)) Un B')" "(UN x:C. A' Un B(x)) = (if C=0 then 0 else A' Un (UN x:C. B(x)))" "(UN x:C. A(x) Int B') = ((UN x:C. A(x)) Int B')" "(UN x:C. A' Int B(x)) = (A' Int (UN x:C. B(x)))" "(UN x:C. A(x) - B') = ((UN x:C. A(x)) - B')" "(UN x:C. A' - B(x)) = (if C=0 then 0 else A' - (INT x:C. B(x)))" apply (simp_all add: Inter_def) apply (blast intro!: equalityI )+ done lemma UN_simps2: "(UN x: Union(A). B(x)) = (UN y:A. UN x:y. B(x))" "(UN z: (UN x:A. B(x)). C(z)) = (UN x:A. UN z: B(x). C(z))" "(UN x: RepFun(A,f). B(x)) = (UN a:A. B(f(a)))" by blast+ lemmas UN_simps [simp] = UN_simps1 UN_simps2 text{*Opposite of miniscoping: pull the operator out*} lemma UN_extend_simps1: "(UN x:C. A(x)) Un B = (if C=0 then B else (UN x:C. A(x) Un B))" "((UN x:C. A(x)) Int B) = (UN x:C. A(x) Int B)" "((UN x:C. A(x)) - B) = (UN x:C. A(x) - B)" apply simp_all apply blast+ done lemma UN_extend_simps2: "cons(a, UN x:C. B(x)) = (if C=0 then {a} else (UN x:C. cons(a, B(x))))" "A Un (UN x:C. B(x)) = (if C=0 then A else (UN x:C. A Un B(x)))" "(A Int (UN x:C. B(x))) = (UN x:C. A Int B(x))" "A - (INT x:C. B(x)) = (if C=0 then A else (UN x:C. A - B(x)))" "(UN y:A. UN x:y. B(x)) = (UN x: Union(A). B(x))" "(UN a:A. B(f(a))) = (UN x: RepFun(A,f). B(x))" apply (simp_all add: Inter_def) apply (blast intro!: equalityI)+ done lemma UN_UN_extend: "(UN x:A. UN z: B(x). C(z)) = (UN z: (UN x:A. B(x)). C(z))" by blast lemmas UN_extend_simps = UN_extend_simps1 UN_extend_simps2 UN_UN_extend subsection{*Miniscoping of Intersections*} lemma INT_simps1: "(INT x:C. A(x) Int B) = (INT x:C. A(x)) Int B" "(INT x:C. A(x) - B) = (INT x:C. A(x)) - B" "(INT x:C. A(x) Un B) = (if C=0 then 0 else (INT x:C. A(x)) Un B)" by (simp_all add: Inter_def, blast+) lemma INT_simps2: "(INT x:C. A Int B(x)) = A Int (INT x:C. B(x))" "(INT x:C. A - B(x)) = (if C=0 then 0 else A - (UN x:C. B(x)))" "(INT x:C. cons(a, B(x))) = (if C=0 then 0 else cons(a, INT x:C. B(x)))" "(INT x:C. A Un B(x)) = (if C=0 then 0 else A Un (INT x:C. B(x)))" apply (simp_all add: Inter_def) apply (blast intro!: equalityI)+ done lemmas INT_simps [simp] = INT_simps1 INT_simps2 text{*Opposite of miniscoping: pull the operator out*} lemma INT_extend_simps1: "(INT x:C. A(x)) Int B = (INT x:C. A(x) Int B)" "(INT x:C. A(x)) - B = (INT x:C. A(x) - B)" "(INT x:C. A(x)) Un B = (if C=0 then B else (INT x:C. A(x) Un B))" apply (simp_all add: Inter_def, blast+) done lemma INT_extend_simps2: "A Int (INT x:C. B(x)) = (INT x:C. A Int B(x))" "A - (UN x:C. B(x)) = (if C=0 then A else (INT x:C. A - B(x)))" "cons(a, INT x:C. B(x)) = (if C=0 then {a} else (INT x:C. cons(a, B(x))))" "A Un (INT x:C. B(x)) = (if C=0 then A else (INT x:C. A Un B(x)))" apply (simp_all add: Inter_def) apply (blast intro!: equalityI)+ done lemmas INT_extend_simps = INT_extend_simps1 INT_extend_simps2 subsection{*Other simprules*} (*** Miniscoping: pushing in big Unions, Intersections, quantifiers, etc. ***) lemma misc_simps [simp]: "0 Un A = A" "A Un 0 = A" "0 Int A = 0" "A Int 0 = 0" "0 - A = 0" "A - 0 = A" "Union(0) = 0" "Union(cons(b,A)) = b Un Union(A)" "Inter({b}) = b" by blast+ end
lemma atomize_ball:
∀x∈A. P(x) == (!!x. x ∈ A ==> P(x))
lemma Upair_iff:
c ∈ Upair(a, b) <-> c = a ∨ c = b
lemma UpairI1:
a ∈ Upair(a, b)
lemma UpairI2:
b ∈ Upair(a, b)
lemma UpairE:
[| a ∈ Upair(b, c); a = b ==> P; a = c ==> P |] ==> P
lemma Un_iff:
c ∈ A ∪ B <-> c ∈ A ∨ c ∈ B
lemma UnI1:
c ∈ A ==> c ∈ A ∪ B
lemma UnI2:
c ∈ B ==> c ∈ A ∪ B
lemma UnE:
[| c ∈ A ∪ B; c ∈ A ==> P; c ∈ B ==> P |] ==> P
lemma UnE':
[| c ∈ A ∪ B; c ∈ A ==> P; [| c ∈ B; c ∉ A |] ==> P |] ==> P
lemma UnCI:
(c ∉ B ==> c ∈ A) ==> c ∈ A ∪ B
lemma Int_iff:
c ∈ A ∩ B <-> c ∈ A ∧ c ∈ B
lemma IntI:
[| c ∈ A; c ∈ B |] ==> c ∈ A ∩ B
lemma IntD1:
c ∈ A ∩ B ==> c ∈ A
lemma IntD2:
c ∈ A ∩ B ==> c ∈ B
lemma IntE:
[| c ∈ A ∩ B; [| c ∈ A; c ∈ B |] ==> P |] ==> P
lemma Diff_iff:
c ∈ A - B <-> c ∈ A ∧ c ∉ B
lemma DiffI:
[| c ∈ A; c ∉ B |] ==> c ∈ A - B
lemma DiffD1:
c ∈ A - B ==> c ∈ A
lemma DiffD2:
c ∈ A - B ==> c ∉ B
lemma DiffE:
[| c ∈ A - B; [| c ∈ A; c ∉ B |] ==> P |] ==> P
lemma cons_iff:
a ∈ cons(b, A) <-> a = b ∨ a ∈ A
lemma consI1:
a ∈ cons(a, B)
lemma consI2:
a ∈ B ==> a ∈ cons(b, B)
lemma consE:
[| a ∈ cons(b, A); a = b ==> P; a ∈ A ==> P |] ==> P
lemma consE':
[| a ∈ cons(b, A); a = b ==> P; [| a ∈ A; a ≠ b |] ==> P |] ==> P
lemma consCI:
(a ∉ B ==> a = b) ==> a ∈ cons(b, B)
lemma cons_not_0:
cons(a, B) ≠ 0
lemma cons_neq_0:
cons(a, B) = 0 ==> R
lemma singleton_iff:
a ∈ {b} <-> a = b
lemma singletonI:
a ∈ {a}
lemma singletonE:
[| a ∈ {b}; a = b ==> PROP W |] ==> PROP W
lemma the_equality:
[| P(a); !!x. P(x) ==> x = a |] ==> (THE x. P(x)) = a
lemma the_equality2:
[| ∃!x. P(x); P(a) |] ==> (THE x. P(x)) = a
lemma theI:
∃!x. P(x) ==> P(THE x. P(x))
lemma the_0:
¬ (∃!x. P(x)) ==> (THE x. P(x)) = 0
lemma theI2:
[| ¬ Q(0) ==> ∃!x. P(x); !!x. P(x) ==> Q(x) |] ==> Q(THE x. P(x))
lemma the_eq_trivial:
(THE x. x = a) = a
lemma the_eq_trivial2:
(THE x. a = x) = a
lemma if_true:
(if True then a else b) = a
lemma if_false:
(if False then a else b) = b
lemma if_cong:
[| P <-> Q; Q ==> a = c; ¬ Q ==> b = d |]
==> (if P then a else b) = (if Q then c else d)
lemma if_weak_cong:
P <-> Q ==> (if P then x else y) = (if Q then x else y)
lemma if_P:
P ==> (if P then a else b) = a
lemma if_not_P:
¬ P ==> (if P then a else b) = b
lemma split_if:
P(if Q then x else y) <-> (Q --> P(x)) ∧ (¬ Q --> P(y))
lemma split_if_eq1:
(if Q then x else y) = b <-> (Q --> x = b) ∧ (¬ Q --> y = b)
lemma split_if_eq2:
a = (if Q then x else y) <-> (Q --> a = x) ∧ (¬ Q --> a = y)
lemma split_if_mem1:
(if Q then x else y) ∈ b <-> (Q --> x ∈ b) ∧ (¬ Q --> y ∈ b)
lemma split_if_mem2:
a ∈ (if Q then x else y) <-> (Q --> a ∈ x) ∧ (¬ Q --> a ∈ y)
lemma split_ifs:
(if Q then x else y) = b <-> (Q --> x = b) ∧ (¬ Q --> y = b)
a = (if Q then x else y) <-> (Q --> a = x) ∧ (¬ Q --> a = y)
(if Q then x else y) ∈ b <-> (Q --> x ∈ b) ∧ (¬ Q --> y ∈ b)
a ∈ (if Q then x else y) <-> (Q --> a ∈ x) ∧ (¬ Q --> a ∈ y)
lemma if_iff:
a ∈ (if P then x else y) <-> P ∧ a ∈ x ∨ ¬ P ∧ a ∈ y
lemma if_type:
[| P ==> a ∈ A; ¬ P ==> b ∈ A |] ==> (if P then a else b) ∈ A
lemma split_if_asm:
P(if Q then x else y) <-> ¬ (Q ∧ ¬ P(x) ∨ ¬ Q ∧ ¬ P(y))
lemma if_splits:
P(if Q then x else y) <-> (Q --> P(x)) ∧ (¬ Q --> P(y))
P(if Q then x else y) <-> ¬ (Q ∧ ¬ P(x) ∨ ¬ Q ∧ ¬ P(y))
lemma mem_asym:
[| a ∈ b; ¬ P ==> b ∈ a |] ==> P
lemma mem_irrefl:
a ∈ a ==> P
lemma mem_not_refl:
a ∉ a
lemma mem_imp_not_eq:
a ∈ A ==> a ≠ A
lemma eq_imp_not_mem:
a = A ==> a ∉ A
lemma succ_iff:
i ∈ succ(j) <-> i = j ∨ i ∈ j
lemma succI1:
i ∈ succ(i)
lemma succI2:
i ∈ j ==> i ∈ succ(j)
lemma succE:
[| i ∈ succ(j); i = j ==> P; i ∈ j ==> P |] ==> P
lemma succCI:
(i ∉ j ==> i = j) ==> i ∈ succ(j)
lemma succ_not_0:
succ(n) ≠ 0
lemma succ_neq_0:
succ(n) = 0 ==> R
lemma succ_subsetD:
succ(c) ⊆ B ==> c ∈ B
lemma succ_neq_self:
succ(b) ≠ b
lemma succ_inject_iff:
succ(m) = succ(n) <-> m = n
lemma succ_inject:
succ(m) = succ(n) ==> m = n
lemma ball_simps1:
(∀x∈A. P(x) ∧ Q) <-> (∀x∈A. P(x)) ∧ (A = 0 ∨ Q)
(∀x∈A. P(x) ∨ Q) <-> (∀x∈A. P(x)) ∨ Q
(∀x∈A. P(x) --> Q) <-> (∃x∈A. P(x)) --> Q
¬ (∀x∈A. P(x)) <-> (∃x∈A. ¬ P(x))
(∀x∈0. P(x)) <-> True
(∀x∈succ(i). P(x)) <-> P(i) ∧ (∀x∈i. P(x))
(∀x∈cons(a, B). P(x)) <-> P(a) ∧ (∀x∈B. P(x))
(∀x∈RepFun(A, f). P(x)) <-> (∀y∈A. P(f(y)))
(∀x∈\<Union>A. P(x)) <-> (∀y∈A. ∀x∈y. P(x))
lemma ball_simps2:
(∀x∈A. P ∧ Q(x)) <-> (A = 0 ∨ P) ∧ (∀x∈A. Q(x))
(∀x∈A. P ∨ Q(x)) <-> P ∨ (∀x∈A. Q(x))
(∀x∈A. P --> Q(x)) <-> P --> (∀x∈A. Q(x))
lemma ball_simps3:
(∀x∈Collect(A, Q). P(x)) <-> (∀x∈A. Q(x) --> P(x))
lemma ball_simps:
(∀x∈A. P(x) ∧ Q) <-> (∀x∈A. P(x)) ∧ (A = 0 ∨ Q)
(∀x∈A. P(x) ∨ Q) <-> (∀x∈A. P(x)) ∨ Q
(∀x∈A. P(x) --> Q) <-> (∃x∈A. P(x)) --> Q
¬ (∀x∈A. P(x)) <-> (∃x∈A. ¬ P(x))
(∀x∈0. P(x)) <-> True
(∀x∈succ(i). P(x)) <-> P(i) ∧ (∀x∈i. P(x))
(∀x∈cons(a, B). P(x)) <-> P(a) ∧ (∀x∈B. P(x))
(∀x∈RepFun(A, f). P(x)) <-> (∀y∈A. P(f(y)))
(∀x∈\<Union>A. P(x)) <-> (∀y∈A. ∀x∈y. P(x))
(∀x∈A. P ∧ Q(x)) <-> (A = 0 ∨ P) ∧ (∀x∈A. Q(x))
(∀x∈A. P ∨ Q(x)) <-> P ∨ (∀x∈A. Q(x))
(∀x∈A. P --> Q(x)) <-> P --> (∀x∈A. Q(x))
(∀x∈Collect(A, Q). P(x)) <-> (∀x∈A. Q(x) --> P(x))
lemma ball_conj_distrib:
(∀x∈A. P(x) ∧ Q(x)) <-> (∀x∈A. P(x)) ∧ (∀x∈A. Q(x))
lemma bex_simps1:
(∃x∈A. P(x) ∧ Q) <-> (∃x∈A. P(x)) ∧ Q
(∃x∈A. P(x) ∨ Q) <-> (∃x∈A. P(x)) ∨ A ≠ 0 ∧ Q
(∃x∈A. P(x) --> Q) <-> (∀x∈A. P(x)) --> A ≠ 0 ∧ Q
(∃x∈0. P(x)) <-> False
(∃x∈succ(i). P(x)) <-> P(i) ∨ (∃x∈i. P(x))
(∃x∈cons(a, B). P(x)) <-> P(a) ∨ (∃x∈B. P(x))
(∃x∈RepFun(A, f). P(x)) <-> (∃y∈A. P(f(y)))
(∃x∈\<Union>A. P(x)) <-> (∃y∈A. ∃x∈y. P(x))
¬ (∃x∈A. P(x)) <-> (∀x∈A. ¬ P(x))
lemma bex_simps2:
(∃x∈A. P ∧ Q(x)) <-> P ∧ (∃x∈A. Q(x))
(∃x∈A. P ∨ Q(x)) <-> A ≠ 0 ∧ P ∨ (∃x∈A. Q(x))
(∃x∈A. P --> Q(x)) <-> A = 0 ∨ P --> (∃x∈A. Q(x))
lemma bex_simps3:
(∃x∈Collect(A, Q). P(x)) <-> (∃x∈A. Q(x) ∧ P(x))
lemma bex_simps:
(∃x∈A. P(x) ∧ Q) <-> (∃x∈A. P(x)) ∧ Q
(∃x∈A. P(x) ∨ Q) <-> (∃x∈A. P(x)) ∨ A ≠ 0 ∧ Q
(∃x∈A. P(x) --> Q) <-> (∀x∈A. P(x)) --> A ≠ 0 ∧ Q
(∃x∈0. P(x)) <-> False
(∃x∈succ(i). P(x)) <-> P(i) ∨ (∃x∈i. P(x))
(∃x∈cons(a, B). P(x)) <-> P(a) ∨ (∃x∈B. P(x))
(∃x∈RepFun(A, f). P(x)) <-> (∃y∈A. P(f(y)))
(∃x∈\<Union>A. P(x)) <-> (∃y∈A. ∃x∈y. P(x))
¬ (∃x∈A. P(x)) <-> (∀x∈A. ¬ P(x))
(∃x∈A. P ∧ Q(x)) <-> P ∧ (∃x∈A. Q(x))
(∃x∈A. P ∨ Q(x)) <-> A ≠ 0 ∧ P ∨ (∃x∈A. Q(x))
(∃x∈A. P --> Q(x)) <-> A = 0 ∨ P --> (∃x∈A. Q(x))
(∃x∈Collect(A, Q). P(x)) <-> (∃x∈A. Q(x) ∧ P(x))
lemma bex_disj_distrib:
(∃x∈A. P(x) ∨ Q(x)) <-> (∃x∈A. P(x)) ∨ (∃x∈A. Q(x))
lemma bex_triv_one_point1:
(∃x∈A. x = a) <-> a ∈ A
lemma bex_triv_one_point2:
(∃x∈A. a = x) <-> a ∈ A
lemma bex_one_point1:
(∃x∈A. x = a ∧ P(x)) <-> a ∈ A ∧ P(a)
lemma bex_one_point2:
(∃x∈A. a = x ∧ P(x)) <-> a ∈ A ∧ P(a)
lemma ball_one_point1:
(∀x∈A. x = a --> P(x)) <-> a ∈ A --> P(a)
lemma ball_one_point2:
(∀x∈A. a = x --> P(x)) <-> a ∈ A --> P(a)
lemma Rep_simps:
{x . y ∈ 0, R(x, y)} = 0
{x ∈ 0 . P(x)} = 0
{x ∈ A . Q} = (if Q then A else 0)
RepFun(0, f) = 0
RepFun(succ(i), f) = cons(f(i), RepFun(i, f))
RepFun(cons(a, B), f) = cons(f(a), RepFun(B, f))
lemma UN_simps1:
(\<Union>x∈C. cons(a, B(x))) = (if C = 0 then 0 else cons(a, \<Union>x∈C. B(x)))
(\<Union>x∈C. A(x) ∪ B') = (if C = 0 then 0 else (\<Union>x∈C. A(x)) ∪ B')
(\<Union>x∈C. A' ∪ B(x)) = (if C = 0 then 0 else A' ∪ (\<Union>x∈C. B(x)))
(\<Union>x∈C. A(x) ∩ B') = (\<Union>x∈C. A(x)) ∩ B'
(\<Union>x∈C. A' ∩ B(x)) = A' ∩ (\<Union>x∈C. B(x))
(\<Union>x∈C. A(x) - B') = (\<Union>x∈C. A(x)) - B'
(\<Union>x∈C. A' - B(x)) = (if C = 0 then 0 else A' - (\<Inter>x∈C. B(x)))
lemma UN_simps2:
(\<Union>x∈\<Union>A. B(x)) = (\<Union>y∈A. \<Union>x∈y. B(x))
(\<Union>z∈\<Union>x∈A. B(x). C(z)) = (\<Union>x∈A. \<Union>z∈B(x). C(z))
(\<Union>x∈RepFun(A, f). B(x)) = (\<Union>a∈A. B(f(a)))
lemma UN_simps:
(\<Union>x∈C. cons(a, B(x))) = (if C = 0 then 0 else cons(a, \<Union>x∈C. B(x)))
(\<Union>x∈C. A(x) ∪ B') = (if C = 0 then 0 else (\<Union>x∈C. A(x)) ∪ B')
(\<Union>x∈C. A' ∪ B(x)) = (if C = 0 then 0 else A' ∪ (\<Union>x∈C. B(x)))
(\<Union>x∈C. A(x) ∩ B') = (\<Union>x∈C. A(x)) ∩ B'
(\<Union>x∈C. A' ∩ B(x)) = A' ∩ (\<Union>x∈C. B(x))
(\<Union>x∈C. A(x) - B') = (\<Union>x∈C. A(x)) - B'
(\<Union>x∈C. A' - B(x)) = (if C = 0 then 0 else A' - (\<Inter>x∈C. B(x)))
(\<Union>x∈\<Union>A. B(x)) = (\<Union>y∈A. \<Union>x∈y. B(x))
(\<Union>z∈\<Union>x∈A. B(x). C(z)) = (\<Union>x∈A. \<Union>z∈B(x). C(z))
(\<Union>x∈RepFun(A, f). B(x)) = (\<Union>a∈A. B(f(a)))
lemma UN_extend_simps1:
(\<Union>x∈C. A(x)) ∪ B = (if C = 0 then B else \<Union>x∈C. A(x) ∪ B)
(\<Union>x∈C. A(x)) ∩ B = (\<Union>x∈C. A(x) ∩ B)
(\<Union>x∈C. A(x)) - B = (\<Union>x∈C. A(x) - B)
lemma UN_extend_simps2:
cons(a, \<Union>x∈C. B(x)) = (if C = 0 then {a} else \<Union>x∈C. cons(a, B(x)))
A ∪ (\<Union>x∈C. B(x)) = (if C = 0 then A else \<Union>x∈C. A ∪ B(x))
A ∩ (\<Union>x∈C. B(x)) = (\<Union>x∈C. A ∩ B(x))
A - (\<Inter>x∈C. B(x)) = (if C = 0 then A else \<Union>x∈C. A - B(x))
(\<Union>y∈A. \<Union>x∈y. B(x)) = (\<Union>x∈\<Union>A. B(x))
(\<Union>a∈A. B(f(a))) = (\<Union>x∈RepFun(A, f). B(x))
lemma UN_UN_extend:
(\<Union>x∈A. \<Union>z∈B(x). C(z)) = (\<Union>z∈\<Union>x∈A. B(x). C(z))
lemma UN_extend_simps:
(\<Union>x∈C. A(x)) ∪ B = (if C = 0 then B else \<Union>x∈C. A(x) ∪ B)
(\<Union>x∈C. A(x)) ∩ B = (\<Union>x∈C. A(x) ∩ B)
(\<Union>x∈C. A(x)) - B = (\<Union>x∈C. A(x) - B)
cons(a, \<Union>x∈C. B(x)) = (if C = 0 then {a} else \<Union>x∈C. cons(a, B(x)))
A ∪ (\<Union>x∈C. B(x)) = (if C = 0 then A else \<Union>x∈C. A ∪ B(x))
A ∩ (\<Union>x∈C. B(x)) = (\<Union>x∈C. A ∩ B(x))
A - (\<Inter>x∈C. B(x)) = (if C = 0 then A else \<Union>x∈C. A - B(x))
(\<Union>y∈A. \<Union>x∈y. B(x)) = (\<Union>x∈\<Union>A. B(x))
(\<Union>a∈A. B(f(a))) = (\<Union>x∈RepFun(A, f). B(x))
(\<Union>x∈A. \<Union>z∈B(x). C(z)) = (\<Union>z∈\<Union>x∈A. B(x). C(z))
lemma INT_simps1:
(\<Inter>x∈C. A(x) ∩ B) = (\<Inter>x∈C. A(x)) ∩ B
(\<Inter>x∈C. A(x) - B) = (\<Inter>x∈C. A(x)) - B
(\<Inter>x∈C. A(x) ∪ B) = (if C = 0 then 0 else (\<Inter>x∈C. A(x)) ∪ B)
lemma INT_simps2:
(\<Inter>x∈C. A ∩ B(x)) = A ∩ (\<Inter>x∈C. B(x))
(\<Inter>x∈C. A - B(x)) = (if C = 0 then 0 else A - (\<Union>x∈C. B(x)))
(\<Inter>x∈C. cons(a, B(x))) = (if C = 0 then 0 else cons(a, \<Inter>x∈C. B(x)))
(\<Inter>x∈C. A ∪ B(x)) = (if C = 0 then 0 else A ∪ (\<Inter>x∈C. B(x)))
lemma INT_simps:
(\<Inter>x∈C. A(x) ∩ B) = (\<Inter>x∈C. A(x)) ∩ B
(\<Inter>x∈C. A(x) - B) = (\<Inter>x∈C. A(x)) - B
(\<Inter>x∈C. A(x) ∪ B) = (if C = 0 then 0 else (\<Inter>x∈C. A(x)) ∪ B)
(\<Inter>x∈C. A ∩ B(x)) = A ∩ (\<Inter>x∈C. B(x))
(\<Inter>x∈C. A - B(x)) = (if C = 0 then 0 else A - (\<Union>x∈C. B(x)))
(\<Inter>x∈C. cons(a, B(x))) = (if C = 0 then 0 else cons(a, \<Inter>x∈C. B(x)))
(\<Inter>x∈C. A ∪ B(x)) = (if C = 0 then 0 else A ∪ (\<Inter>x∈C. B(x)))
lemma INT_extend_simps1:
(\<Inter>x∈C. A(x)) ∩ B = (\<Inter>x∈C. A(x) ∩ B)
(\<Inter>x∈C. A(x)) - B = (\<Inter>x∈C. A(x) - B)
(\<Inter>x∈C. A(x)) ∪ B = (if C = 0 then B else \<Inter>x∈C. A(x) ∪ B)
lemma INT_extend_simps2:
A ∩ (\<Inter>x∈C. B(x)) = (\<Inter>x∈C. A ∩ B(x))
A - (\<Union>x∈C. B(x)) = (if C = 0 then A else \<Inter>x∈C. A - B(x))
cons(a, \<Inter>x∈C. B(x)) = (if C = 0 then {a} else \<Inter>x∈C. cons(a, B(x)))
A ∪ (\<Inter>x∈C. B(x)) = (if C = 0 then A else \<Inter>x∈C. A ∪ B(x))
lemma INT_extend_simps:
(\<Inter>x∈C. A(x)) ∩ B = (\<Inter>x∈C. A(x) ∩ B)
(\<Inter>x∈C. A(x)) - B = (\<Inter>x∈C. A(x) - B)
(\<Inter>x∈C. A(x)) ∪ B = (if C = 0 then B else \<Inter>x∈C. A(x) ∪ B)
A ∩ (\<Inter>x∈C. B(x)) = (\<Inter>x∈C. A ∩ B(x))
A - (\<Union>x∈C. B(x)) = (if C = 0 then A else \<Inter>x∈C. A - B(x))
cons(a, \<Inter>x∈C. B(x)) = (if C = 0 then {a} else \<Inter>x∈C. cons(a, B(x)))
A ∪ (\<Inter>x∈C. B(x)) = (if C = 0 then A else \<Inter>x∈C. A ∪ B(x))
lemma misc_simps:
0 ∪ A = A
A ∪ 0 = A
0 ∩ A = 0
A ∩ 0 = 0
0 - A = 0
A - 0 = A
\<Union>0 = 0
\<Union>cons(b, A) = b ∪ \<Union>A
\<Inter>{b} = b