hcju/isabelleps huggingface dataset:
PP_PPf_eqv: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (Syntax_with_Connectives.eqv cnj imp (HBL1.PP var subst P (enc \<phi>)) (exi (Generic_Syntax.Variable var 11) (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf (Var (Generic_Syntax.Variable var 11)) (enc \<phi>))))"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> e + (b \<cdot> e + c) = (a + b) \<cdot> e + c"
left_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "monoid f z" shows "f z a = a"
fraction_uminus: fixes f :: "('a, 'b) IEEE.float" shows "fraction (- f) = fraction f"
l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs \<equiv> (known_ptr = DocumentClass.known_ptr \<and> type_wf = DocumentClass.type_wf) \<and> set_child_nodes = l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes \<and> set_child_nodes_locs = l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes_locs"
small_nstep_Rec2: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "small_nstep P \<sigma> (Suc n) = {\<sigma>2. \<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>2 \<in> small_nstep P \<sigma>1 n}"
bangChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<parallel> !P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<noteq> P \<parallel> !P" shows "!P \<Longrightarrow>\<^sub>\<tau> P'"
chainFreshFresh: fixes x :: "name"   and xvec :: "name list"   and y :: "name"   and p :: "'d" assumes "x \<sharp> xvec"   and "y \<sharp> xvec" shows "xvec \<sharp>* ([(x, y)] \<bullet> p) = xvec \<sharp>* p"
pochhammer_0_left: fixes n :: "nat" shows "pochhammer (0::'b) n = (if n = 0 then 1::'b else (0::'b))"
sub: shows "fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> punit.dgrad_p_set d m"
fpairwiseD: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c fset"   and m :: "'c"   and n :: "'c" assumes "fpairwise P M"   and "m |\<in>| M"   and "n |\<in>| M" shows "P m n"
fps_lr_inverse_zero_1: fixes x :: "'a" shows "fps_left_inverse 0 x = fps_const x"
intros: fixes p :: "'c \<times> 'b" shows "snd p \<in> Basic_BNFs.snds p"
sat_models_invariant: fixes A :: "'c ltl set"   and \<phi> :: "'c ltl" shows "(A \<in> sat_models (Abs \<phi>)) = A \<Turnstile>\<^sub>P \<phi>"
correctCompositionDiffLevelsA3: shows "correctCompositionDiffLevels sA3"
finite_columns: fixes A :: "(('b, 'd) vec, 'c) vec" shows "finite (columns A)"
fut_spec_def: fixes \<phi>\<Phi> :: "'b \<times> ('a, 'k) aformula list" shows "fut_spec \<phi>\<Phi> \<equiv> (\<forall>\<phi>\<in>set (snd \<phi>\<Phi>). local.wf idx \<phi>) \<and> (\<forall>\<AA>. #\<^sub>V \<AA> = idx \<longrightarrow> (if b then (\<exists>k. (SNOC (zero idx) ^^ k) \<AA> \<Turnstile>\<^sub>b \<psi>) = (\<exists>\<phi>\<in>set (snd \<phi>\<Phi>). \<AA> \<Turnstile>\<^sub>b \<phi>) else (\<forall>k. (SNOC (zero idx) ^^ k) \<AA> \<Turnstile>\<^sub>b \<psi>) = (\<forall>\<phi>\<in>set (snd \<phi>\<Phi>). \<AA> \<Turnstile>\<^sub>b \<phi>)))"
closure_of_eq_diff: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S = topspace X - \<Union> {T. openin X T \<and> disjnt S T}"
OC_wrt_preorder_iff_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_corresponding TRel \<and> preorder TRel \<and> weak_reduction_bisimulation TRel Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel) \<and> preorder Rel \<and> weak_reduction_bisimulation Rel (STCal Source Target))"
supporters_nonagent1: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt"   and y :: "'alt" assumes "is_pref_profile R"   and "x \<notin> alts" shows "supporters R x y = {}"
intros: fixes q :: "'d"   and l :: "'e"   and qs :: "'d list"   and \<delta> :: "('d, 'e) ta_rule set"   and q' :: "'d" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "q' \<in> set qs" shows "(q, q') \<in> f_succ \<delta>"
insertion_Const: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" shows "insertion f (Const c) = c"
polydivide_aux_properties: fixes p :: "poly"   and n0 :: "nat"   and s :: "poly"   and n1 :: "nat"   and a :: "poly"   and n :: "nat"   and k :: "nat"   and k' :: "nat"   and r :: "poly" assumes "isnpolyh p n0"   and "isnpolyh s n1"   and "head p = a"   and "Reflected_Multivariate_Polynomial.degree p = n"   and "p \<noteq> 0\<^sub>p" shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k \<le> k' \<and> (Reflected_Multivariate_Polynomial.degree r = 0 \<or> Reflected_Multivariate_Polynomial.degree r < Reflected_Multivariate_Polynomial.degree p) \<and> Ex (isnpolyh r) \<and> (\<exists>q n1. isnpolyh q n1 \<and> a ^\<^sub>p k' - k *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
Suc: shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. length xs = k \<and> (\<forall>i<k. P i (xs ! i)))"
pbdom_def: fixes f :: "'a"   and g :: "'a" shows "f \<down>\<down> g \<equiv> C.dom \<p>\<^sub>0[f, g]"
type_def: shows "\<exists>Rep Abs. type_definition Rep Abs {0..<p}"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<AA>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>" shows "\<AA>\<lparr>Cod\<rparr> = []\<^sub>\<circ>"
distinct_1: fixes x21 :: "'d"   and x22 :: "'d list" shows "[] \<noteq> x21 # x22"
lNAct_ex: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> = n \<or> \<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> < n \<and> \<not> (\<exists>k\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. latestAct_cond c t n k)"
adds_zero: fixes s :: "'a" shows "(s adds (0::'a)) = (s = (0::'a))"
dim_span_eq_card_independent: fixes B :: "'b set" assumes "independent B" shows "dim (span B) = card B"
absorb_iff2: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (a \<sqinter> b = b)"
o2: shows "rel_generat A C (rel_fun R (rel_gpv'' A C R)) (IO out' rpv) b2"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'c"   and color :: "color" shows "h (case color of color.R \<Rightarrow> f1 | color.B \<Rightarrow> f2) = (case color of color.R \<Rightarrow> h f1 | color.B \<Rightarrow> h f2)"
preim_of_max_lideal_is_max: fixes I :: "'c set"   and B :: "'c set"   and addB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroB :: "'c"   and oneB :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d" assumes "max_lideal I B addB multB zeroB oneB"   and "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "max_lideal (f \<^sup>\<inverse> A I) (f \<^sup>\<inverse> A B) addA multA zeroA oneA"
fmdom_filter: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmdom (fmfilter P m) = ffilter P (fmdom m)"
assms_2: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option" shows "R e h l e h l"
resolvable_design_axioms_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset" shows "resolvable_design_axioms point_set block_collection partition \<equiv> incidence_system.resolution point_set block_collection partition"
sinvar_def: fixes G :: "'b graph"   and nP :: "'b \<Rightarrow> char list set" shows "SINVAR_Tainting.sinvar G nP \<equiv> \<forall>(v1, v2)\<in>edges G. nP v1 \<subseteq> nP v2"
prj_in_hom_4: shows "Maps.in_hom SPN_fgh.Prj\<^sub>1 (Chn (SPN f \<circ> SPN g \<circ> SPN h)) SPN_fgh.\<mu>.apex"
recvmsg_def: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d seq_action" shows "recvmsg P a \<equiv> case a of receive m \<Rightarrow> P m | _ \<Rightarrow> True"
orbit_thm: fixes m :: "'c"   and rep :: "'a set \<Rightarrow> 'a" assumes "m \<in> M"   and "\<And>U. U \<in> carrier (G Mod stabilizer m) \<Longrightarrow> rep U \<in> U" shows "bij_betw (\<lambda>H. \<phi> (inv rep H) m) (carrier (G Mod stabilizer m)) (orbit m)"
generate_idem: fixes A :: "'a set" assumes "A \<subseteq> carrier G" shows "generate G (generate G A) = generate G A"
no_better_thansubset_rel: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "x \<in> carrier"   and "y \<in> carrier"   and "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation" shows "x \<succeq>[relation] y"
fst_add: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "fst (x + y) = fst x + fst y"
intro: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and N :: "'c set" assumes "monoid M composition unit"   and "submonoid_axioms N M composition unit" shows "submonoid N M composition unit"
closed_glob_consistent_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set" shows "abs.closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A) \<in> \<R> \<longrightarrow> (\<forall>A. (\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma\<^sub>A mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma\<^sub>A mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma\<^sub>A mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>\<^sub>A) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x) \<longrightarrow> (\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle>\<^sub>A, \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>\<^sub>A) \<in> \<R>))"
Mp_prod_mset: fixes b :: "int poly multiset" shows "\<Prod>\<^sub># (image_mset Mp b) =m \<Prod>\<^sub># b"
double_diff: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
append_Nil: fixes ys :: "'g list" shows "[] @ ys = ys"
continuous_on_closure_sequentially: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on (closure S) f = (\<forall>x a. a \<in> closure S \<and> (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> x \<sqsubseteq> xa" obtains xb :: "'a" and y :: "'a" where "x = Some xb" and "xa = Some y" and "\<not> xb \<sqsubseteq> y"   | uu_ :: "'a" where "x = Some uu_" and "xa = None"
distinct_snd_tseytin_assmt: fixes F :: "'a formula" shows "distinct (map snd (tseytin_assmt F))"
assms: shows "f piecewise_C1_differentiable_on (\<lambda>x. m * x + c) ` S"
elims_2: fixes x :: "'a atom formula" assumes "is_predAtom x" obtains uu_ :: "predicate" and uv_ :: "'a list" where "x = Atom (predAtm uu_ uv_)"
hyps_2: fixes t' :: "trm" assumes "t \<longmapsto> t'" shows "\<Gamma> \<turnstile> t' : T"
hd_in_set_conv: fixes l :: "'c list" shows "(hd l \<in> set l) = (l \<noteq> [])"
ccHeap_rec_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "ccHeap_rec \<Gamma> e\<cdot>a = CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), ccExp_syn a e)"
simps_2: fixes prim :: "prim"   and \<beta> :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "\<A> (P prim) \<beta> ve = DP prim"
r_inconsist_def: shows "r_inconsist \<equiv> let f = Cn 2 r_length [recf.Id 2 1]; g = Cn 4 r_ifless [recf.Id 4 1, Cn 4 r_length [recf.Id 4 3], recf.Id 4 1, Cn 4 r_ifeq [Cn 4 r_phi [recf.Id 4 2, recf.Id 4 0], Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0], recf.Id 4 1, recf.Id 4 0]] in Cn 2 (Pr 2 f g) [Cn 2 r_length [recf.Id 2 1], recf.Id 2 0, recf.Id 2 1]"
proj_mcont: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" shows "mcont lSup lprefix lSup lprefix (dynamic_component.proj tCMP active c)"
m_max_gt: fixes l :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "l \<le> n" shows "f l \<le> m_max n f"
while_def: fixes b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "while b c s = the (while_option b c s)"
respects_own_guarantees_def: fixes cm :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)" shows "conc.respects_own_guarantees cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> conc.doesnt_read_or_modify (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> conc.doesnt_modify (fst cm) x)"
THE_addr_loc_type: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and a :: "'b"   and al :: "addr_loc"   and T :: "ty" assumes "heap_base.addr_loc_type typeof_addr P h a al T" shows "(THE T. heap_base.addr_loc_type typeof_addr P h a al T) = T"
extend_hintikka: fixes C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set"   and f :: "nat \<Rightarrow> (nat, 'a) form" assumes "finite_char C"   and "infinite (- \<Union> (params ` S))"   and "\<forall>y. \<exists>n. y = f n"   and "alt_consistency C"   and "S \<in> C" shows "hintikka (Extend S C f)"
has_integral_implies_lebesgue_measurable_cbox: fixes f :: "'c \<Rightarrow> real"   and I :: "real"   and x :: "'c"   and y :: "'c" assumes "(f has_integral I) (cbox x y)" shows "f \<in> borel_measurable (lebesgue_on (cbox x y))"
list_before_rel_def: fixes l :: "'b list" shows "list_before_rel l \<equiv> {(a, b). \<exists>l1 l2 l3. l = l1 @ a # l2 @ b # l3}"
r2f_f2r_galois_var2: fixes f :: "('a \<times> 'b) set \<Rightarrow> 'c"   and R :: "('a \<Rightarrow> 'b set) \<Rightarrow> 'c" shows "(f \<circ> \<R> = R) = (R \<circ> \<F> = f)"
ContractsWithBilateralSubstitutesAndIRC_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<equiv> ContractsWithBilateralSubstitutes Xd Xh Pd Ch \<and> ContractsWithIRC Xd Xh Pd Ch"
map_psubst_new_away: fixes fresh :: "'a"   and z :: "('a, 'b) form list"   and c :: "'a" assumes "news fresh z" shows "map (psubst (id(fresh := c))) (map (psubst (id(c := fresh))) z) = z"
indaut_uniq_aut: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and s :: "'b \<Rightarrow> 'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "ChamberComplexAutomorphism X s"   and "s \<noteq> id"   and "fixespointwise s (C0 \<inter> g ` C0)" shows "s = OpposedThinChamberComplexFoldings.induced_automorphism X f g"
elims: fixes x :: "'a abs_set"   and y :: "atom set" assumes "supp_set x = y" obtains as :: "atom set" and xa :: "'a" where "x = [as]set. xa" and "y = supp xa - as"
fimageI2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A"
TUnit_def: shows "TUnit \<equiv> Abs_ty TUnit_Rep"
cmie: assumes "m\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "m\<inverse> O e = m\<inverse>"
hom_power: fixes x :: "'b"   and n :: "nat" shows "[:x ^ n:] = [:x:] ^ n"
nabla_sum_expand: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> \<nabla>(x + y) +  |y\<rangle> \<nabla>(x + y) = \<nabla>(x + y)"
vcons_eq_vcard_eq: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence_pair xs\<^sub>1 xs\<^sub>2"   and "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "vcard xs\<^sub>1 = vcard xs\<^sub>2"
Bang: fixes P :: "pi"   and Rs :: "Late_Semantics.residual" assumes "P \<parallel> !P \<longmapsto>\<^sub>l Rs" shows "!P \<longmapsto>\<^sub>l Rs"
simps_1: fixes \<rho> :: "(nat \<times> exp) list"   and n :: "nat" shows "psubst \<rho> (ENat n) = ENat n"
rev_done': shows "rev done = map proj_thd (take (length done) (update_until args rel1 rel2 nt auxlist))"
nb\<^sub>4: fixes op\<^sub>1 :: "('a, 'b) sas_plus_operator"   and op\<^sub>2 :: "('a, 'b) sas_plus_operator" assumes "op\<^sub>1 \<in> set ops"   and "op\<^sub>2 \<in> set ops" shows "SAS_Plus_Semantics.are_operator_effects_consistent op\<^sub>1 op\<^sub>2"
The_split_eq: fixes x :: "'a"   and y :: "'b" shows "(THE (x', y'). x = x' \<and> y = y') = (x, y)"
pos_finite: shows "AE M' in M. \<integral>\<^sup>+ x. ennreal (f x) \<partial>M' \<noteq> \<infinity>"
pp_total_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.total (\<ominus> \<ominus> f) = (\<forall>i. \<exists>j. f (i, j) \<noteq> bot)"
0: fixes i :: "nat" assumes "i < n" shows "P $$ (i, j) = 0"
UN_E: fixes b :: "'d"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'e" where "x \<in> A" and "b \<in> B x"
inject_5: fixes x51 :: "'a"   and x52 :: "bexp"   and x53 :: "'a acom"   and x54 :: "'a"   and y51 :: "'a"   and y52 :: "bexp"   and y53 :: "'a acom"   and y54 :: "'a" shows "({x51} WHILE x52 DO x53 {x54} = {y51} WHILE y52 DO y53 {y54}) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53 \<and> x54 = y54)"
mult_less_le_imp_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c \<le> d"   and "(0::'a) \<le> a"   and "(0::'a) < c" shows "a * c < b * d"
permits_acc_def: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and cls :: "qtname"   and accclass :: "qtname" shows "G \<turnstile> membr in cls permits_acc_from accclass = (case accmodi membr of Private \<Rightarrow> declclass membr = accclass | Package \<Rightarrow> pid (declclass membr) = pid accclass | Protected \<Rightarrow> pid (declclass membr) = pid accclass \<or> G\<turnstile>accclass\<prec>\<^sub>C declclass membr \<and> (G\<turnstile>cls\<preceq>\<^sub>C accclass \<or> is_static membr) | Public \<Rightarrow> True)"
fmfilter_comp: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter (\<lambda>x. P x \<and> Q x) m"
UINF_all_cong: fixes F :: "'a \<Rightarrow> ('b, 'c) uexpr"   and G :: "'a \<Rightarrow> ('b, 'c) uexpr" assumes "\<And>P. F P = G P" shows "(\<Sqinter> P \<bullet> F P) = (\<Sqinter> P \<bullet> G P)"
tab_in_hom_1: shows "\<guillemotleft>\<rho> : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C r\<guillemotright>"
pos_recurrent_iffI_communicating: fixes y :: "'s"   and x :: "'s" assumes "(y, x) \<in> communicating" shows "pos_recurrent y = pos_recurrent x"
assms_2: shows "y \<in> set_spmf (sample_set_excl prime_set {x})"
less_Agent_def: fixes x :: "Agent"   and y :: "Agent" shows "x < y \<equiv> False"
map_fun_def: fixes f :: "'p \<Rightarrow> 'r"   and g :: "'s \<Rightarrow> 'q"   and h :: "'r \<Rightarrow> 's" shows "map_fun f g h = g \<circ> h \<circ> f"
typeof_sys_xcp: fixes h :: "'heap"   and C :: "String.literal" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt C)) = \<lfloor>Class C\<rfloor>"
emptyClauseIsFalse: fixes valuation :: "Literal list" shows "clauseFalse [] valuation"
set_integrable_scaleR_left: fixes a :: "'c"   and M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> real" assumes "a \<noteq> (0::'c) \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. f t *\<^sub>R a)"
assms_1: shows "H = {x \<in> A. (x, f x) \<in> r}"
Lazy_tllist_inverse: fixes xs :: "unit \<Rightarrow> 'a \<times> ('a, 'b) tllist + 'b" shows "force (Lazy_tllist xs) = xs ()"
max_distrib: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c" shows "c * max a b = max (c * a) (c * b)"
gselect_edge_impl_def: fixes s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gselect_edge_impl s \<equiv> let (a, s) = s in select_edge_impl s \<bind> (\<lambda>(vo, s). RETURN (vo, a, s))"
inf_mono: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset"   and d :: "'b multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
lm011: fixes R :: "('a \<times> 'b) set"   and x :: "'a" shows "R \<subseteq> R +< {x} \<times> R `` {x}"
ts_inf_make_untimed_inf_tl: fixes x :: "nat \<Rightarrow> 'd list"   and i :: "nat" assumes "ts x" shows "inf_make_untimed (inf_tl x) i = inf_make_untimed x (Suc i)"
r_shrink: fixes f :: "recf"   and x :: "nat" assumes "recfn 2 f" shows "eval (r_shrink f) [x] = eval f [x, x]"
Inf_level: fixes L :: "level set" shows "Inf L \<equiv> if Lo \<in> L then Lo else Hi"
nz: fixes i :: "nat" assumes "M \<le> i" shows "f i \<noteq> (0::'a)"
rep_eq: shows "generator rbt_entries_generator = (rbt_has_next, rbt_entries_next)"
sys_xcpts_defs_6: shows "NoSuchFieldError \<equiv> ''NoSuchFieldError''"
map_comp: fixes g1 :: "'i \<Rightarrow> 'g"   and g2 :: "'j \<Rightarrow> 'h"   and f1 :: "'k \<Rightarrow> 'i"   and f2 :: "'l \<Rightarrow> 'j"   and v :: "('k, 'l) tllist" shows "tmap g1 g2 (tmap f1 f2 v) = tmap (g1 \<circ> f1) (g2 \<circ> f2) v"
top_add: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "top + x = top"