hcju/isabelleps huggingface dataset:
final_i_1: shows "enat final_i \<le> llength t"
leftmost_element_property: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains zs :: "'a list" and x' :: "'a" and ys :: "'a list" where "xs = zs @ x' # ys" and "P x'" and "\<forall>z\<in>set zs. \<not> P z"
less_divide_eq_numeral_1: fixes w :: "num"   and b :: "'a"   and c :: "'a" shows "(numeral w < b / c) = (if (0::'a) < c then numeral w * c < b else if c < (0::'a) then b < numeral w * c else numeral w < (0::'a))"
orderE: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" obtains "a = sup a b"
assms: shows "s \<sqsubseteq> t \<cdot> \<theta>"
t_stop_nonneg: shows "0 \<le> ego2.t_stop"
checked_invariant_cs: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" assumes "checked_simplex' cs J s" shows "invariant_simplex' cs J s"
reachable_s_finite_thm_1_a: fixes s :: "('b, bool) fmap"   and PROB :: "(('b, bool) fmap \<times> ('b, bool) fmap) set" assumes "s \<in> valid_states PROB" shows "\<forall>l\<in>reachable_s PROB s. l \<in> valid_states PROB"
inf_concat_surj: fixes j :: "nat"   and n :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "j < n i" shows "\<exists>k. inf_concat n k = (i, j)"
has_all_wt_instances_ofD': fixes N :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s N"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"   and "has_all_wt_instances_of \<Gamma> N M"   and "t \<in> N" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
squarefree_factorial_semiring': fixes n :: "'b" assumes "n \<noteq> (0::'b)" shows "squarefree n = (\<forall>p\<in>#prime_factorization n. multiplicity p n = 1)"
obj_openz_lc: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and p :: "sterm"   and l :: "Label" assumes "lc (Obj f T)"   and "lc p"   and "l \<in> dom f" shows "lc (the (f l)\<^bsup>[Obj f T,p]\<^esup>)"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (cod \<mu> \<star> cod \<nu>)"
empty_Collect_eq: fixes P :: "'a \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
LeftDerives1_append_suffix: fixes v :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "LeftDerives1 v i r w"   and "is_sentence u" shows "LeftDerives1 (v @ u) i r (w @ u)"
IH_1: assumes "poly p x\<^sub>0 \<noteq> 0"   and "poly q x\<^sub>0 = 0"   and "poly (hd (r # ps'')) x\<^sub>0 \<noteq> 0"   and "ps' \<in> set (split_sign_changes (r # ps'') x\<^sub>0)" shows "\<forall>\<^sub>F x in at x\<^sub>0. sign_changes ps' x = sign_changes ps' x\<^sub>0"
rec_inseparableI: fixes A :: "nat set"   and B :: "nat set" assumes "\<And>X. \<lbrakk>A \<subseteq> X; B \<subseteq> - X; computable X\<rbrakk> \<Longrightarrow> False" shows "rec_inseparable A B"
same_level_path_inner_path: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'" obtains as' :: "'edge list" where "n -as'\<rightarrow>\<^sub>\<iota>* n'" and "set (sourcenodes as') \<subseteq> set (sourcenodes as)"
y_neq_new_last: shows "y \<noteq> new_last"
sublens_iff_sublens': fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "vwb_lens X"   and "vwb_lens Y" shows "X \<subseteq>\<^sub>L Y = X \<subseteq>\<^sub>L' Y"
in_var_semi_uvar: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "mwb_lens ($x)"
pair_ord_def: fixes x :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and y :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" shows "pair_ord x y = (sig_of_pair x \<preceq>\<^sub>t sig_of_pair y)"
spair_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "spair p q = (let t1 = punit.lt (rep_list p); t2 = punit.lt (rep_list q); l = lcs t1 t2 in monom_mult ((1::'b) / punit.lc (rep_list p)) (l - t1) p - monom_mult ((1::'b) / punit.lc (rep_list q)) (l - t2) q)"
zip_replicate_map: fixes xs :: "'a list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "zip (replicate n x) xs = map (Pair x) xs"
TOTAL: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "Total r"
mono_step': fixes S1 :: "(char list \<Rightarrow> 'a) option"   and S2 :: "(char list \<Rightarrow> 'a) option"   and C1 :: "(char list \<Rightarrow> 'a) option acom"   and C2 :: "(char list \<Rightarrow> 'a) option acom" assumes "S1 \<le> S2"   and "C1 \<le> C2" shows "step' S1 C1 \<le> step' S2 C2"
ideals_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.ideals less_eq = {X. X \<noteq> {} \<and> X \<in> ord.downsets less_eq \<and> ord.directed less_eq X}"
less_eq_extended_case: fixes x1a :: "'a extended"   and x2a :: "'a extended" shows "(x1a \<le> x2a) = (case x1a of Fin xda \<Rightarrow> case x2a of Fin x2da \<Rightarrow> xda \<le> x2da | \<infinity> \<Rightarrow> True | -\<infinity> \<Rightarrow> False | \<infinity> \<Rightarrow> case x2a of \<infinity> \<Rightarrow> True | _ \<Rightarrow> False | -\<infinity> \<Rightarrow> case x2a of Fin xba \<Rightarrow> True | _ \<Rightarrow> True)"
iwlsFSbSwTR_qInitAbs_respectsP_alphaAbsGood: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "qInitAbs MOD respectsP alphaAbsGood"
Quotient_antichain: shows "Quotient (eq_onp (\<lambda>A. finite A \<and> incomparable A)) antichain set_antichain cr_antichain"
simps_2: fixes uv :: "'e"   and \<sigma> :: "'g" shows "pol uv (Guard \<sigma>) = Fext"
disjunctiveD: fixes x :: "'b \<Rightarrow> 'b"   and y :: "'b \<Rightarrow> 'b"   and z :: "'b \<Rightarrow> 'b" assumes "x \<in> Comp.disjunctive" shows "x \<circ> y \<squnion> z = (x \<circ> y) \<squnion> (x \<circ> z)"
Int_space_eq1: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and x :: "'b set" assumes "semiring_of_sets \<Omega> M"   and "x \<in> M" shows "\<Omega> \<inter> x = x"
indep_vars_def: fixes M' :: "'e \<Rightarrow> 'f measure"   and X :: "'e \<Rightarrow> 'a \<Rightarrow> 'f"   and I :: "'e set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
sum_carrier_vec: fixes A :: "'b vec set"   and n :: "nat"   and B :: "'b vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n" shows "A + B \<subseteq> carrier_vec n"
proj2_line_through_incident_1: fixes p :: "proj2"   and q :: "proj2" shows "proj2_incident p (proj2_line_through p q)"
lhs_step_bind: fixes m' :: "'b nres"   and m :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres" assumes "nofail m' \<Longrightarrow> nofail m"   and "\<And>x. nf_inres m x \<Longrightarrow> f x \<le> m'" shows "m \<bind> f \<le> m'"
induct: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a"   and a2 :: "nat" assumes "\<And>y x. P y x 0"   and "\<And>y x. P y x (Suc 0)"   and "\<And>n y x. \<lbrakk>n \<noteq> 0; even n; P y (f x x) (n div 2)\<rbrakk> \<Longrightarrow> P y x n"   and "\<And>n y x. \<lbrakk>n \<noteq> 1; odd n; P (f x y) (f x x) (n div 2)\<rbrakk> \<Longrightarrow> P y x n" shows "P a0 a1 a2"
in_hhomE: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "B.in_hhom \<mu> a b" obtains "B.arr \<mu>" and "B.obj a" and "B.obj b" and "src\<^sub>B \<mu> = a" and "trg\<^sub>B \<mu> = b"
exact_decompI: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and m :: "nat" assumes "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> h \<in> P[X]"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> U \<subseteq> X"   and "\<And>h h' U U'. \<lbrakk>(h, U) \<in> set ps; (h', U') \<in> set ps; poly_deg h = poly_deg h'; m < card U; m < card U'\<rbrakk> \<Longrightarrow> (h, U) = (h', U')" shows "exact_decomp m ps"
measurable_lborel1: fixes M :: "'b measure" shows "M \<rightarrow>\<^sub>M lborel = borel_measurable M"
bd: fixes s :: "real"   and t :: "real"   and C :: "nat \<Rightarrow> real \<Rightarrow> real"   and D :: "nat \<Rightarrow> real \<Rightarrow> real"   and q :: "nat"   and p :: "nat" assumes "s \<le> t"   and "rho_bound1 C"   and "rho_bound2 D"   and "D q t - D q s \<le> C p t - C p s"   and "correct p t"   and "correct q t" shows "\<bar>C p t - D q t\<bar> \<le> \<bar>C p s - D q s\<bar> + 2 * \<rho> * (t - s)"
inv_diff: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "inv t1"   and "inv t2" shows "inv (diff t1 t2)"
rel_ipurge_aux_single_dom: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b" shows "rel_ipurge_aux P I D {u} = rel_ipurge P I D u"
dSS: fixes k :: "nat" assumes "k < Suc (Suc n)" shows "f k \<in> carrier_mat d d"
cases: fixes x :: "int list" obtains xs :: "int list" where "x = xs"
graph_accs_1: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "finite V"   and "finite E" shows "nodes (graph V E) = V \<union> fst ` E \<union> snd ` E"
weakBisimOutputPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<approx> M\<langle>N\<rangle>.Q"
atoms\<^sub>0_def: shows "atoms\<^sub>0 = ATOM.atoms\<^sub>0 (\<lambda>a. hd_coeff a \<noteq> 0)"
inv_M_def: shows "inv_M = (\<lambda>x. if x + x \<le> m then x else x - m)"
inner_Basis_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "less_eq a b"   and "c \<in> Basis" shows "inner a c \<le> inner b c"
adapt_stoch_proc_def: fixes F :: "'a \<Rightarrow> 'b measure"   and X :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure" shows "adapt_stoch_proc F X N = (\<forall>t. X t \<in> F t \<rightarrow>\<^sub>M N)"
sub_supp_empty: fixes s :: "'a \<Rightarrow> 'b" shows "(supp_fun s \<subseteq> {}) = (s = 0)"
br'_invar_abs: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_\<alpha> ` br'_invar \<delta> \<subseteq> br_invar \<delta>"
exhaust_sel: fixes node :: "'b node" obtains "node = SOURCE"   | "node = Inner (node.inner node)"
strategyproofness_1: shows "(pmf (sds R28) a < pmf (sds R32) d \<or> pmf (sds R28) a + (pmf (sds R28) c + pmf (sds R28) d) < pmf (sds R32) d + (pmf (sds R32) b + pmf (sds R32) a)) \<or> pmf (sds R28) a = pmf (sds R32) d \<and> pmf (sds R28) a + (pmf (sds R28) c + pmf (sds R28) d) = pmf (sds R32) d + (pmf (sds R32) b + pmf (sds R32) a)"
simple_wfvarsoccs\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>n\<^sub>d: fixes S :: "('c, 'd) strand_step list" assumes "simple S" shows "wfvarsoccs\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S))"
Am0: shows "A $ m $ mod_type_class.from_nat k \<noteq> (0::'a)"
eq_refl: fixes less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l"   and y :: "'l" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
assms_1: shows "\<forall>a\<in>S \<union> {P1, P2}. \<forall>b\<in>S \<union> {P1, P2}. a \<noteq> b \<longrightarrow> a \<inter> b = {}"
targetsD_1: fixes b :: "'a"   and \<mu> :: "'a" assumes "b \<in> targets \<mu>" shows "ide b"
case_prod_eta: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). f (x, y)) = f"
IH: fixes k :: "nat" assumes "m + n \<noteq> k" shows "prv (neg (eql (pls (Num m) (Num n)) (Num k)))"
ideal_generated_subset2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "ideal_generated {a} \<subseteq> ideal_generated {c}"   and "ideal_generated {b} \<subseteq> ideal_generated {c}" shows "ideal_generated {a, b} \<subseteq> ideal_generated {c}"
aa_bb_pos_2: fixes k :: "nat" shows "0 < bb k"
distinct_24: fixes x13 :: "Syntax.term" shows "Roll x13 \<noteq> Unit"
hle: fixes y :: "'b"   and x' :: "'a" assumes "y \<in> T"   and "x' \<in> S"   and "f x' = y" shows "h (k y) \<le> h x'"
valid_two_cube_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "valid_two_cube twoC \<equiv> card (boundary twoC) = 4"
sig_red_single_red_single: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "punit.red_single (rep_list p) (rep_list q) (rep_list f) t"
Legendre_mult: fixes p :: "int"   and a :: "int"   and b :: "int" assumes "prime p" shows "Legendre (a * b) p = Legendre a p * Legendre b p"
some_elem_ne: fixes s :: "'a set" assumes "s \<noteq> {}" shows "some_elem s \<in> s"
eq_0_size: fixes t :: "'a tree" shows "(0 = size t) = (t = \<langle>\<rangle>)"
\<sigma>\<^sub>s\<^sub>y\<^sub>m'_def: shows "\<sigma>\<^sub>s\<^sub>y\<^sub>m' = \<sigma>\<^sub>s\<^sub>y\<^sub>m(sv := e \<sigma>)"
cases: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and a :: "'c stream" assumes "alw \<phi> a" obtains xs :: "'c stream" where "a = xs" and "\<phi> xs" and "alw \<phi> (stl xs)"
evalEffect_gState_progress_rel: fixes prog :: "program"   and g :: "gState\<^sub>I"   and p :: "pState"   and e :: "edgeEffect" assumes "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p"   and "cl_inv (g, p)" shows "(g, fst (evalEffect e prog g p)) \<in> gState_progress_rel prog"
Sup_apply: fixes A :: "('b \<Rightarrow> 'a) set"   and x :: "'b" shows "(\<Squnion> A) x = (\<Squnion>f\<in>A. f x)"
1: assumes "\<not> (x = (0::'a) \<or> is_unit p \<or> \<not> p dvd x)" shows "divide_out_aux p (x div p, acc + 1) = (fst (x div p, acc + 1) div p ^ multiplicity p (fst (x div p, acc + 1)), snd (x div p, acc + 1) + multiplicity p (fst (x div p, acc + 1)))"
uw\<theta>\<omega>\<nu>: shows "ide w \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> g \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w) \<cdot> \<nu> = \<omega>"
F_strict_imp_F_inj: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n"   and "f \<in> Cauchy_Binet.F_strict n m" shows "f \<in> Cauchy_Binet.F_inj n m"
outer: fixes q1 :: "nat"   and q2 :: "nat" shows "\<exists>N. partn_lst {..<N} [q1, q2] r"
c_failures_tr: fixes xps :: "('e \<times> 'f) list"   and X :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" assumes "(xps, X) \<in> c_failures step out s\<^sub>0" shows "xps = c_tr step out s\<^sub>0 (map fst xps)"
conv: fixes xy :: "'v \<times> 'v" assumes "xy \<in> \<^bold>V \<times> \<^bold>V" shows "convergent (\<lambda>n. h (k n) xy)"
prv_imp_cnj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<psi>)"   and "prv (imp \<phi> \<chi>)" shows "prv (imp \<phi> (cnj \<psi> \<chi>))"
algebraic_power: fixes x :: "'a"   and n :: "nat" assumes "algebraic x" shows "algebraic (x ^ n)"
simps_4: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and w :: "'b"   and p :: "('c, 'd) fm"   and q :: "('c, 'd) fm" shows "(M, g, w \<Turnstile> p \<^bold>\<or> q) = (M, g, w \<Turnstile> p \<or> M, g, w \<Turnstile> q)"
len_hchop_right_right_border: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> right (len v2 ts c) = right (ext v2)"
OUT_g_B: fixes x :: "'v" assumes "x \<notin> A \<Gamma>" shows "d_OUT g x = 0"
continuous_on_minus: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. - f x)"
is_Link_def: fixes ctree :: "('d, 'e, 'f) ctree" shows "is_Link ctree = (\<exists>x1. ctree = Link x1)"
std_normal_moment_odd: shows "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k + 1)) 0"
unat_eq_of_nat: fixes n :: "nat"   and x :: "'a word" assumes "n < 2 ^ LENGTH('a)" shows "(unat x = n) = (x = word_of_nat n)"
eq_\<I>_generat_into_rel_generat: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and D :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and generat :: "('a, 'c, 'd \<Rightarrow> 'e) generat"   and generat' :: "('b, 'c, 'd \<Rightarrow> 'f) generat" assumes "eq_\<I>_generat A \<I>_full D generat generat'" shows "rel_generat A (=) (rel_fun (=) D) generat generat'"
diff_add_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
evaluate_list_singletonI: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r :: "(v, v) result" assumes "evaluate ck env s e (s', r)" shows "evaluate_list ck env s [e] (s', list_result r)"
C: shows "steps' init qs Strat n = steps' init (qs @ r2) (Strat @ r1) n"
elims: fixes x :: "'b equation"   and xa :: "('b \<times> 'b trm) list"   and y :: "'b equation" assumes "subst_equation x xa = y" obtains u :: "'b trm" and v :: "'b trm" and s :: "('b \<times> 'b trm) list" where "x = Eq u v" and "xa = s" and "y = Eq (u \<lhd> s) (v \<lhd> s)"
left_commute: fixes b :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" shows "(b::'a::refinement_lattice) \<iinter> ((a::'a::refinement_lattice) \<iinter> (c::'a::refinement_lattice)) = a \<iinter> (b \<iinter> c)"
amin_le_plusTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K"   and "v x \<noteq> \<infinity>"   and "v y \<noteq> \<infinity>"   and "v x \<le> v y" shows "amin (v x) (v y) \<le> v (x \<plusminus> y)"
strict_implies_order: fixes a :: "'g"   and b :: "'g" assumes "a < b" shows "a \<le> b"
preservation: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "\<exists>\<Gamma>'''. \<turnstile> \<Gamma>''' {c'} \<Gamma>' \<and> (mds_consistent mds \<Gamma> \<longrightarrow> mds_consistent mds' \<Gamma>''')"
simps: fixes fs :: "int vec iarray"   and fi :: "int vec"   and i :: "nat"   and dmus :: "int iarray iarray"   and j :: "nat" shows "dmu_array_row_main fs fi i dmus j = (if j = i then dmus else let sj = Suc j; dmus_i = dmus !! i; djj = dmus !! j !! j; dmu_ij = djj * (fi \<bullet> fs !! sj) - sigma_array dmus dmus_i (dmus !! sj) djj j; dmus' = iarray_update dmus i (iarray_append dmus_i dmu_ij) in dmu_array_row_main fs fi i dmus' sj)"
x: shows "x \<in> mat_kernel (berlekamp_resulting_mat u)"
InvariantEquivalentZLAfterApplyUnitPropagate: fixes state :: "State"   and Phi :: "Literal list list" assumes "InvariantEquivalentZL (getF state) (getM state) Phi"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "\<not> getConflictFlag state"   and "getQ state \<noteq> []" shows "let state' = applyUnitPropagate state in InvariantEquivalentZL (getF state') (getM state') Phi"
set_of_pow_inc: fixes A :: "'a interval"   and A' :: "'a interval"   and n :: "nat" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A ^ n) \<subseteq> set_of (A' ^ n)"
rInvim_def: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e"   and K :: "'e set" shows "rInvim A R f K = {a \<in> carrier A. f a \<in> K}"
reds: shows "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es'',s''\<rangle>"
fmimage_alt_def: fixes m :: "('d, 'c) fmap"   and S :: "'d fset" shows "fmimage m S = fmran (fmrestrict_fset S m)"
\<theta>_in_hom_2: shows "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>"
qFreshBinp_preserves_alphaBinp_aux: fixes qbinp :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs option"   and qbinp' :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs option"   and xs :: "'varSort"   and x :: "'var" assumes "qGoodBinp qbinp \<or> qGoodBinp qbinp'"   and "(qbinp, qbinp') \<in> alphaBinp"   and "qFreshBinp xs x qbinp" shows "qFreshBinp xs x qbinp'"
path_invar: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and ns :: "'b list" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path \<alpha>n invar inEdges' g ns" shows "invar g"
I: shows "R * F \<Longrightarrow>\<^sub>A Q"
synth_simps_4: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "(\<lbrace>X, Y\<rbrace> \<in> synth H) = (\<lbrace>X, Y\<rbrace> \<in> H \<or> X \<in> synth H \<and> Y \<in> synth H)"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case IntVir of Static \<Rightarrow> f1 | SuperM \<Rightarrow> f2 | IntVir \<Rightarrow> f3) = f3"