hcju/isabelleps huggingface dataset:
eq_fract_1: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "(Fract a b = Fract c d) = (a * d = c * b)"
impl_of_empty: shows "rbt.impl_of RBT.empty = rbt.Empty"
WhileBodyThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and c :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
fdim_def: shows "fdim = R_scalar_mult.dim_R UNIV (aezfun_scalar_mult.fsmult fgsmult)"
fbrelation_vintersectionI: fixes s :: "V" shows "fbrelation (r \<inter>\<^sub>\<circ> s)"
subst_apply_fv_subset_strand2: fixes P :: "('c, 'd) strand_step \<Rightarrow> 'd set"   and ac :: "poscheckvariant"   and x :: "('c, 'd) strand_step"   and S :: "('c, 'd) strand_step list"   and V :: "'d set"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q \<or> P = fv_r\<^sub>e\<^sub>q ac"   and "P x \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V"   and "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"   and "set (bvars\<^sub>s\<^sub>t\<^sub>p x) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "P (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta>) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<union> fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)"
assms_1: shows "continuous_on (convex hull {a, b, c}) f"
quadratic_sub_leq_helper: fixes var :: "nat"   and L :: "real list"   and Cv :: "real"   and C :: "real mpoly"   and A :: "real mpoly"   and Av :: "real"   and B :: "real mpoly"   and Bv :: "real" assumes "var < length L"   and "0 \<le> Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) C = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) A = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) B = Bv" shows "aEval (Leq (A + B * Var var)) (L[var := sqrt Cv]) = eval (Or (And (fm.Atom (Leq A)) (fm.Atom (Leq (B\<^sup>2 * C - A\<^sup>2)))) (And (fm.Atom (Leq B)) (fm.Atom (Leq (A\<^sup>2 - B\<^sup>2 * C))))) (L[var := sqrt Cv])"
listrel_Cons1: fixes y :: "'b"   and ys :: "'b list"   and xs :: "'c list"   and r :: "('b \<times> 'c) set" assumes "(y # ys, xs) \<in> listrel r" obtains y :: "'c" and ys :: "'c list" where "xs = y # ys" and "(yb__, y) \<in> r" and "(ysa__, ys) \<in> listrel r"
simps_8: fixes l :: "'b"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(AssignR\<^sub>\<E> l \<E>) [e] = Assign (VE (Loc l)) (\<E> [e])"
total_recursive_def: shows "total_recursive = (\<lambda>f. graph f \<in> ce_rels)"
in_read_only_convs_2: fixes a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and R :: "nat set" shows "(a \<in> read_only (\<S> \<oplus>\<^bsub>W\<^esub> R)) = (if a \<in> R then a \<notin> W else a \<in> read_only \<S>)"
cd_closed: fixes n' :: "'node"   and S :: "'node set"   and n :: "'node" assumes "n' \<in> PDG_BS S"   and "n weakly controls n'" shows "n \<in> PDG_BS S"
neq_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "(x \<noteq> y) = (less x y \<or> less y x)"
simps_1: fixes n :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Load n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length LT \<and> LT ! n \<noteq> Err \<and> length ST < mxs)"
simps_1: fixes x :: "nat \<Rightarrow> bool" shows "count x 0 = (if x 0 then 1 else 0)"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "'e" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M2" shows "a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M2\<^esub> x = a \<odot>\<^bsub>M2\<^esub> (b \<odot>\<^bsub>M2\<^esub> x)"
summable_ignore_initial_segment: fixes k :: "nat" assumes "summable f" shows "summable (\<lambda>n. f (n + k))"
sel_1: fixes x1 :: "'d" shows "generat.result (Pure x1) = x1"
pos_recurrent_def: fixes x :: "'s option" shows "K2.pos_recurrent x = (K2.recurrent x \<and> K2.U' x x \<noteq> \<infinity>)"
l11_24: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P InAngle A B C" shows "P InAngle C B A"
has_integral_Fejer_kernel_half: fixes n :: "nat" shows "has_bochner_integral (lebesgue_on {0..pi}) (Fejer_kernel n) (if n = 0 then 0 else pi / 2)"
\<sigma>\<^sub>X_def: fixes x :: "'Var" shows "\<sigma>\<^sub>X x = (if x \<in> X then if x \<in> dom \<sigma> then \<sigma> x else Some (mem x) else None)"
simps_9: fixes m :: "nat"   and p :: "fm"   and q :: "fm" shows "bound m (Or p q) = (bound m p \<and> bound m q)"
OclAnd_true2: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and true) = X"
msb_non_min: fixes w :: "32 Word.word" assumes "uint w \<noteq> 0"   and "uint w \<noteq> 2 ^ LENGTH(31)" shows "msb (- w) = (\<not> msb w)"
unset_bit_negative_int_iff: fixes n :: "nat"   and k :: "int" shows "(unset_bit n k < 0) = (k < 0)"
le_less_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
nextElem_suc2: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "last (vertices f) = v"   and "v \<in> \<V> f" shows "f \<bullet> v = hd (vertices f)"
ab: shows "A1 + A2 \<noteq> B1 + B2"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and sl :: "name"   and sl' :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name"   and x :: "tm"   and x' :: "tm" assumes "atom l \<sharp> (s, k, sl, sl', m, n, sm, sm', sn, sn')"   and "atom sl \<sharp> (s, sl', m, n, sm, sm', sn, sn')"   and "atom sl' \<sharp> (s, m, n, sm, sm', sn, sn')"   and "atom m \<sharp> (s, n, sm, sm', sn, sn')"   and "atom n \<sharp> (s, sm, sm', sn, sn')"   and "atom sm \<sharp> (s, sm', sn, sn')"   and "atom sm' \<sharp> (s, sn, sn')"   and "atom sn \<sharp> (s, sn')"   and "atom sn' \<sharp> s" shows "SeqHRP x x' s k = LstSeqP s k (HPair x x') AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (OrdP (Var sl) AND WRP (Var sl) (Var sl') OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ HPair (Var sm) (Var sn) AND Var sl' EQ Q_HPair (Var sm') (Var sn')))))))))))"
Abs_unit_induct: fixes P :: "unit \<Rightarrow> bool"   and x :: "unit" assumes "\<And>y. y \<in> {True} \<Longrightarrow> P (Abs_unit y)" shows "P x"
4: shows "kronecker_product (M_mat_R signs1 subs1) (M_mat_R signs2 subs2) $$ (i, j) = z_R (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z_R (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))"
zip_with_altdef': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" shows "zip_with f xs ys = map (\<lambda>i. f (xs ! i) (ys ! i)) [0..<min (length xs) (length ys)]"
eq_sym: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "\<psi> \<sim>\<^sub>P \<phi>"
rsp: shows "rel_fun (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'a))) (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'a))) uminus uminus"
AE_restrict_space_iff: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "\<Omega> \<inter> space M \<in> sets M" shows "almost_everywhere (restrict_space M \<Omega>) P = set_almost_everywhere \<Omega> M P"
components_eq: fixes c :: "'a set"   and s :: "'a set"   and c' :: "'a set" assumes "c \<in> components s"   and "c' \<in> components s" shows "(c = c') = (c \<inter> c' \<noteq> {})"
delayed_other_t_stop_eq: assumes "\<delta> \<le> other.t_stop" shows "delayed_safe_distance.other.t_stop + \<delta> = other.t_stop"
lower_closed: fixes x :: "'c" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>" shows "\<pi>\<^sup>*\<^bsub>F\<^esub> x \<in> carrier \<Y>\<^bsub>F\<^esub>"
allocationFinite: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G"   and "finite G" shows "finite a"
binary_in_sigma_sets: fixes a :: "'c set"   and sp :: "'c set"   and A :: "'c set set"   and b :: "'c set"   and i :: "nat" assumes "a \<in> sigma_sets sp A"   and "b \<in> sigma_sets sp A" shows "binary a b i \<in> sigma_sets sp A"
fermat_liar_witness_of_composition_1: shows "local.fermat_liar \<noteq> local.fermat_witness"
exposed_face_of_def: fixes T :: "'b set"   and S :: "'b set" shows "(T exposed_face_of S) = (T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b}))"
nat_pow_eone: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "x [^]\<^bsub>K [X]\<^esub> 1 = x"
col_permutation_5: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col A C B"
exhaust: fixes y :: "C" obtains "y = \<bottom>"   | Ca :: "C" where "y = C\<cdot>Ca"
all: fixes i :: "nat" assumes "i < j" shows "P (xs ! i)"
SReal_hypreal_of_real: fixes x :: "real" shows "hypreal_of_real x \<in> \<real>"
measurable_Sup1: fixes m :: "'c measure"   and M :: "'c measure set"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "m \<in> M"   and "f \<in> m \<rightarrow>\<^sub>M N"   and "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> space m = space n" shows "f \<in> Sup M \<rightarrow>\<^sub>M N"
\<R>\<^sub>2_bisim_step: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma>' :: "'b \<Rightarrow> Sec option"   and c\<^sub>1' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1' :: "'b \<Rightarrow> 'd" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>'"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>1', mds'), mem\<^sub>1') \<in> sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "\<exists>c\<^sub>2' mem\<^sub>2'. (((c\<^sub>2, mds), mem\<^sub>2), (c\<^sub>2', mds'), mem\<^sub>2') \<in> sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B \<and> (((c\<^sub>1', mds'), mem\<^sub>1'), (c\<^sub>2', mds'), mem\<^sub>2') \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>'"
IFail_def: fixes s :: "Istate"   and s' :: "Istate"   and p :: "Proc" shows "IFail s s' p = (ioutput s' = (ioutput s)(p := NotAnInput) \<and> (\<exists>ip\<in>Inputs. iinput s' = (iinput s)(p := ip) \<and> iallInput s' = iallInput s \<union> {ip}) \<and> ichosen s' = ichosen s)"
mult_pos_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less zero (times a b)"
2: shows "punit.lt (rep_list p') \<preceq> punit.lt (rep_list (monom_mult (1::'b) t p))"
bounded_linear_uniform_limit_intros_13: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and x :: "'d" assumes "uniform_limit X g l F" shows "uniform_limit X (\<lambda>a b. x * g a b) (\<lambda>a. x * l a) F"
borel_measurable_times: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
transpose_\<phi>: shows "tab = \<a>\<^sup>-\<^sup>1[k\<^sup>*, h, p\<^sub>0] \<cdot> k.trnl\<^sub>\<eta> p\<^sub>1 \<phi>"
strong_inducts: fixes P :: "'e \<Rightarrow> 'd prefix \<Rightarrow> bool"   and z :: "'e"   and prefix :: "'d prefix" assumes "\<And>a1 list a2 z. P z (pInput a1 list a2)"   and "\<And>a1 a2 z. P z (pOutput a1 a2)"   and "\<And>z. P z pTau" shows "P z prefix"
ennreal_diff_self: fixes a :: "ennreal" assumes "a \<noteq> \<top>" shows "a - a = 0"
r_neg2: fixes x :: "rat"   and y :: "rat" shows "x + (- x + y) = y"
SemnDeterm: fixes s :: "Var \<Rightarrow> Val"   and c :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val"   and m :: "nat"   and r :: "Var \<Rightarrow> Val" assumes " s , c \<rightarrow>\<^sub>n  t "   and " s , c \<rightarrow>\<^sub>m  r " shows "m = n \<and> r = t"
axioms_typ_of_propT: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term" assumes "wf_theory thy"   and "A \<in> axioms thy" shows "typ_of A = Some propT"
coincidence_formula_cor: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<phi> :: "fml"   and I :: "interp" assumes "Uvariation \<omega> \<omega>' U"   and "FVF \<phi> \<inter> U = {}" shows "(\<omega> \<in> fml_sem I \<phi>) = (\<omega>' \<in> fml_sem I \<phi>)"
simps_2: fixes y :: "'b option" shows "(None \<sqsubseteq> y) = True"
Write\<^sub>s\<^sub>b: fixes \<O>' :: "nat set"   and volatile :: "bool"   and \<O> :: "nat set"   and A :: "nat set"   and R :: "nat set"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and L :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and rs :: "'b memref list" assumes "\<O>' = (if volatile then \<O> \<union> A - R else \<O>)"   and "\<S>' = (if volatile then \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L else \<S>)"   and "\<R>' = (if volatile then Map.empty else \<R>)" shows "(m, Write\<^sub>s\<^sub>b volatile a sop v A L R W # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m(a := v), rs, \<O>', \<R>', \<S>')"
add_col_sub_row_diag: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat"   and k :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat n n"   and "local.inv_all local.uppert A"   and "i < j"   and "j < n"   and "k < n" shows "add_col_sub_row a i j A $$ (k, k) = A $$ (k, k)"
cong_eval_trans: fixes l :: "bdt"   and r :: "bdt"   and t :: "bdt" assumes "l \<sim> r"   and "r \<sim> t" shows "l \<sim> t"
simps_7: fixes v :: "Ferrack.num" shows "not (Ge v) = NOT (Ge v)"
path_det: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and n'' :: "'node" assumes "n -as\<rightarrow>* n'"   and "n -as\<rightarrow>* n''" shows "n' = n''"
intros_2: fixes h :: "'d \<Rightarrow> 'c"   and t :: "('d, 'c) blindable\<^sub>m"   and x :: "'c blindable\<^sub>h"   and bo :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "hash_blindable h t = x" shows "blinding_of_blindable h bo (Blinded x) t"
run_ipath: fixes G :: "('a, 'b) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "graph_defs.is_run G r" shows "ipath (g_E G) r"
Confidentiality_B: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list"   and authK :: "nat"   and servTicket :: "msg"   and Ta :: "nat"   and authTicket :: "msg" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "Key servK \<notin> analz (knows Spy evs)"
inj2: shows "inj_on (\<lambda>p i. p (i - 1)) {p. p partitions m - k \<and> sum p {..m - k} = Suc k}"
wf\<^sub>t\<^sub>r\<^sub>m_def: fixes t :: "('fun, 'c) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m t \<equiv> \<forall>f T. Fun f T \<sqsubseteq> t \<longrightarrow> length T = arity f"
AxT: shows "aAxioms Ax \<subseteq> {Vx : A \<turnstile> e1\<equiv>e2 : B |A B e1 e2. Sig aS \<triangleright> Vx : A \<turnstile> e1\<equiv>e2 : B}"
Re_linepath: fixes a :: "real"   and b :: "real"   and x :: "real" shows "Re (linepath (complex_of_real a) (complex_of_real b) x) = (1 - x) * a + x * b"
cont_intersect1: fixes y :: "'b ttree" shows "cont (\<lambda>x. x \<inter>\<inter> y)"
analz_generators: fixes s :: "l2_state" shows "analz (generators s) = generators s"
integer_winding_number: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<notin> path_image \<gamma>" shows "winding_number \<gamma> z \<in> \<int>"
l11_44_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.CongA Bet Cong B A C B C A = Cong B A B C"
assms: shows "P (\<sigma> |\<^sub>s Suc 0)"
bind_fv_def: fixes vT :: "variable \<times> typ"   and t :: "Core.term" shows "bind_fv vT t = bind_fv2 vT 0 t"
llist_all2_lconcatI: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xss :: "'a llist llist"   and yss :: "'b llist llist" assumes "llist_all2 (llist_all2 A) xss yss" shows "llist_all2 A (lconcat xss) (lconcat yss)"
matrix_to_iarray_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "matrix_to_iarray A = IArray (map (vec_to_iarray \<circ> ($) A \<circ> mod_type_class.from_nat) [0..<CARD('e)])"
max_rank_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d" shows "semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0 = card (reach \<Sigma> \<delta> q\<^sub>0 - {q. semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q})"
InvariantImpliedLiteralsAfterLearn: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and F' :: "Literal list list"   and C :: "Literal list" assumes "InvariantImpliedLiterals F M"   and "F' = F @ [C]" shows "InvariantImpliedLiterals F' M"
superadditive_def: fixes u :: "nat \<Rightarrow> real" shows "superadditive u = (\<forall>m n. u m + u n \<le> u (m + n))"
USdBsym: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" shows "sym (Strong_Security.USdB SR DA d)"
bv_uminus_length: fixes w :: "bit list" shows "length (bv_uminus w) \<le> Suc (length w)"
continuous_imp_mono: fixes f :: "'a set \<Rightarrow> 'b set" assumes "continuous f" shows "mono f"
trans_resolve2: fixes n2 :: "nat"   and n1 :: "nat"   and x :: "nat"   and y :: "nat" assumes "n2 < n1"   and "x + n1 \<le> y"   and "y \<le> x + n2" shows "False"
ltlc_to_ltln_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltlc" shows "w \<Turnstile>\<^sub>n ltlc_to_ltln \<phi> = w \<Turnstile>\<^sub>c \<phi>"
infsetsum_cdiv: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "c \<noteq> (0::'a) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. f x / c) = infsetsum f A / c"
False: shows "\<not> fst a \<subseteq>\<^sub>f s"
D: shows "Min (offs_set_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (x # xs) index key mi ma i) < n"
insert_prefix_trie_same: fixes ps :: "bool list"   and b :: "bool"   and lr :: "trie \<times> trie" shows "insert ps (prefix_trie ps (Nd b lr)) = prefix_trie ps (Nd True lr)"
b_e_type_checker_sound: fixes \<C> :: "t_context"   and es :: "b_e list"   and tn :: "t list"   and tm :: "t list" assumes "b_e_type_checker \<C> es (tn _> tm)" shows "\<C> \<turnstile> es : tn _> tm"
l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr \<equiv> (l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<and> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs) \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs get_disconnected_nodes remove_shadow_root remove_shadow_root_locs"