hcju/isabelleps huggingface dataset:
mono_update: fixes a :: "'b"   and a' :: "'b"   and S :: "'c \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "a \<sqsubseteq> a'"   and "S \<sqsubseteq> S'" shows "S(x := a) \<sqsubseteq> S'(x := a')"
\<Gamma>_sat_preserving: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" assumes "sound_inference_system \<Gamma>"   and "satisfiable N" shows "satisfiable (N \<union> concls_of (inference_system.inferences_from \<Gamma> N))"
iT_Div_mem_iff: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k" shows "(x \<in> I \<oslash> k) = (\<exists>y\<in>I. y div k = x)"
map_def: fixes gf :: "'c \<times> 'c" shows "HomC.map gf = (if CopxC.arr gf then S.mkArr (HomC.set (CopxC.dom gf)) (HomC.set (CopxC.cod gf)) (inC \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>C h \<cdot>\<^sub>C fst gf) \<circ> HomC.\<psi> (CopxC.dom gf)) else S.null)"
\<omega>consistent_iffStd2: assumes "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> prv (imp (neg (neg \<phi>)) \<phi>)" shows "\<omega>consistent = \<omega>consistentStd2"
gfp_upperbound: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and X :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq X (f X)" shows "less_eq X (complete_lattice.gfp Sup less_eq f)"
is_functor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "identity_functor C" shows "functor C C (identity_functor.map C)"
distinct_13: fixes x5 :: "nat \<times> char list list" shows "Null \<noteq> Ref x5"
colossless_converterD: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and a :: "'h" assumes "colossless_converter \<I> \<I>' conv"   and "a \<in> outs_\<I> \<I>" shows "colossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). colossless_converter \<I> \<I>' conv')"
uminus_mult_mat_vec: fixes v :: "'b vec"   and A :: "'b mat" assumes "dim_vec v = dim_col A" shows "- A *\<^sub>v v = - (A *\<^sub>v v)"
ruler_le: fixes ps :: "'b list"   and xs :: "'b list"   and qs :: "'b list" assumes "ps \<le>p xs"   and "qs \<le>p xs"   and "\<^bold>|ps\<^bold>| \<le> \<^bold>|qs\<^bold>|" shows "ps \<le>p qs"
c_def: fixes xs :: "'b list"   and fs :: "'b list"   and i :: "nat" shows "c xs fs i = xij_f xs fs 0 i"
PrioMap_axioms: shows "PrioMap PST_RBT.empty PST_RBT.update PST_RBT.delete PM.invar lookup rbt_is_empty pst_getmin"
divide_less_ennreal: fixes b :: "ennreal"   and a :: "ennreal"   and c :: "ennreal" assumes "b \<noteq> 0"   and "b < top" shows "(a / b < c) = (a < c * b)"
lim_inverse_n: shows "(\<lambda>n. inverse (of_nat n)) \<longlonglongrightarrow> (0::'a)"
silent_moves_into_\<tau>rtrancl3p: fixes s :: "'s1"   and s' :: "'s1" assumes "s -\<tau>1\<rightarrow>* s'" shows "s -\<tau>1-[]\<rightarrow>* s'"
simps_14: fixes x1 :: "'a"   and x41 :: "'a local.term"   and x42 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> x41 \<^bold>\<cdot> x42"
init_fin_\<tau>diverge_conv: fixes t :: "'t"   and status :: "status"   and x :: "'x2"   and m :: "'m2" shows "\<tau>trsys.\<tau>diverge (r2.init_fin t) r2.init_fin_\<tau>move ((status, x), m) = (r2.\<tau>diverge t (x, m) \<and> status = Running)"
atLeast_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> {k..}) = (k \<le> i)"
dma_\<C>: fixes mem\<^sub>1 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "\<forall>x\<in>\<C>\<^sub>C. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma\<^sub>C mem\<^sub>1 = dma\<^sub>C mem\<^sub>2"
fixpoint_sol_2: fixes x :: "'a"   and t :: "real" assumes "x \<in> X"   and "f x = (0::'a)" shows "flow0 x t = x"
E_def: fixes c :: "nat \<times> nat \<Rightarrow> 'c" shows "Graph.E c \<equiv> {(u, v). c (u, v) \<noteq> (0::'c)}"
simple_match_port_alt: fixes m :: "16 word \<times> 16 word"   and p :: "16 word" shows "simple_match_port m p = (p \<in> wordinterval_to_set (uncurry WordInterval m))"
le_supI2: fixes x :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and a :: "'c \<times> 'd" assumes "less_eq_prod x b" shows "less_eq_prod x (sup_prod a b)"
QEI: fixes x :: "('a, 'b) BsSkewElem" assumes "x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt qy))" shows "elem_invar x"
integral_less_AE_space: fixes M :: "'b measure"   and X :: "'b \<Rightarrow> real"   and Y :: "'b \<Rightarrow> real" assumes "finite_measure M"   and "integrable M X"   and "integrable M Y"   and "AE x in M. X x < Y x"   and "emeasure M (space M) \<noteq> 0" shows "integral\<^sup>L M X < integral\<^sup>L M Y"
finfun_apply: fixes x :: "'d \<Rightarrow>f 'e" shows "($) x \<in> finfun"
satReport: fixes F0 :: "Literal list list"   and Vbl :: "nat set"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" assumes "vars F0 \<subseteq> Vbl"   and "InvariantVarsF F F0 Vbl"   and "InvariantConsistent M"   and "InvariantEquivalent F0 F"   and "\<not> formulaFalse F (elements M)"   and "Vbl \<subseteq> vars (elements M)" shows "model (elements M) F0"
assms_3: shows "is_left_adjoint (g \<star> w)"
while_eval_elim: fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   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 "\<langle>While e c \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<oplus> annos \<and> mem' = mem"
SOC_iff_indRelRTPO_is_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> (\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*))) \<and> strong_reduction_simulation (TRel\<^sup>+) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>+ \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>+))) = ((\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q')) \<and> (\<forall>P Q Q'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q')))"
prefix_append: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<le>p ys \<cdot> zs) = (xs \<le>p ys \<or> (\<exists>us. xs = ys \<cdot> us \<and> us \<le>p zs))"
assms_2: assumes "\<not> b" shows "m2 \<le> m2'"
bit_imp_le_length: fixes w :: "'b word"   and n :: "nat" assumes "bit w n" shows "n < LENGTH('b)"
least_n_eq: shows "least_non_zero_position_of_vector_from_index (vec_to_iarray (column j A)) (mod_type_class.to_nat i) = mod_type_class.to_nat (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
lemma2b'': fixes p :: "bool list"   and q :: "bool list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps R)\<^sup>*" shows "(False # p, False # q) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"
test_preserve: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p \<cdot> q \<cdot> p = q \<cdot> p"
card_leq_pref: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "finite carrier"   and "x \<in> carrier"   and "y \<in> carrier" shows "(card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation)) = y \<succeq>[relation] x"
has_derivative_inner: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x \<bullet> g x) has_derivative (\<lambda>h. f x \<bullet> g' h + f' h \<bullet> g x)) (at x within s)"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "inv x \<in> carrier G"
sel_rews_3: fixes a :: "'a"   and Stream :: "'a Stream" shows "sthead\<cdot>(a && Stream) = a"
reaches1_steps: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^sup>+ y" shows "\<exists>xs. steps (x # xs @ [y])"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'k list"   and a :: "'k" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
3: shows "Can (Inv (DOM f\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>rep f\<^bold>\<rfloor> \<^bold>\<cdot> COD f\<^bold>\<down>)"
lang_eq_ext_Nil_fold_Deriv: fixes r :: "'c rexp"   and s :: "'c rexp" shows "(lang r = lang s) = (\<forall>(K, L)\<in>{(fold Deriv w (lang r), fold Deriv w (lang s)) |w. w \<in> lists (atoms r \<union> atoms s)}. ([] \<in> K) = ([] \<in> L))"
add_pos_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less_eq zero b" shows "less zero (plus a b)"
fresh_1: fixes a :: "vrs"   and x2 :: "vrs"   and x1 :: "ty" shows "a \<sharp> VarB x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
edges_delete_edge: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and G :: "('c, 'd, 'e) graph_scheme" shows "edges (delete_edge v e v' G) = edges G - {(v, e, v')}"
mr_charn: fixes a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "a \<in> set p"   and "x \<in> dom (C a)"   and "\<And>r. r \<in> set p \<and> x \<in> dom (C r) \<Longrightarrow> r = a" shows "applied_rule_rev C x p = \<lfloor>a\<rfloor>"
vector_choose_dist: fixes c :: "real"   and x :: "'b" assumes "0 \<le> c" obtains y :: "'b" where "dist x y = c"
simps_12: fixes I :: "\<I>"   and \<phi> :: "Formula.formula" shows "Formula.future_bounded (formula.Prev I \<phi>) = Formula.future_bounded \<phi>"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree"   and a1 :: "('b \<times> nat) tree" assumes "P \<langle>\<rangle> \<langle>\<rangle>"   and "\<And>v va vb. P \<langle>\<rangle> \<langle>v, va, vb\<rangle>"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>v va vb vc vd ve. P \<langle>v, va, vb\<rangle> \<langle>vc, vd, ve\<rangle>"   and "\<And>vc vd ve v va vb. P \<langle>vc, vd, ve\<rangle> \<langle>v, va, vb\<rangle>" shows "P a0 a1"
synPrefix: fixes s :: "'c event list"   and t :: "'c event list"   and ta :: "'c event list"   and A :: "'c event set"   and u :: "'c event list" shows "\<exists>t1 u1. s @ t setinterleaves ((ta, u), A) \<longrightarrow> t1 \<le> ta \<and> u1 \<le> u \<and> s setinterleaves ((t1, u1), A)"
nat_filtration_Suc_sets: fixes n :: "nat" shows "sets (nat_filtration n) \<subseteq> sets (nat_filtration (Suc n))"
vsv_vinsert_no_match_appI: fixes a :: "V"   and c :: "V"   and d :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> xs"   and "c \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs"   and "xs\<lparr>c\<rparr> = d" shows "vinsert \<langle>a, b\<rangle> xs\<lparr>c\<rparr> = d"
bigo_mult2: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f *o O(g) \<subseteq> O(f * g)"
safe_rpd_fv_regex: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Past Strict r"   and "s \<in> rpd test i r" shows "fv_regex fv s = fv_regex fv r"
ShuffleLR_ZeroR: fixes r :: "'a language" shows "ShuffleLR r Zero = Zero"
cases: fixes x :: "('fsym, 'a) efsym" obtains f :: "'fsym" where "x = Oldf f"   | \<sigma> :: "'a" where "x = Wit \<sigma>"
ang_const: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p" assumes "QCongA a"   and "A \<noteq> B" shows "\<exists>C. a A B C"
oa_H: shows "out_arcs H x = out_arcs G x - {a'} - {a}"
fpxs_root_order_nonzero: fixes f :: "'b fpxs" shows "fpxs_root_order f \<noteq> 0"
F_def: shows "F = (\<lambda>p x. P x \<and> p (stl x))"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "atom fm"   and y :: "atom fm" assumes "lowerFm x xa xb = y" obtains d :: "nat" and amount :: "nat" and f :: "atom fm" where "x = d" and "xa = amount" and "xb = f" and "y = map_fm_binders (\<lambda>a x. lowerAtom (d + x) amount a) f 0"
simps: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "force g s = (case generator g s of Done \<Rightarrow> None | Skip x \<Rightarrow> force g x | Yield x s' \<Rightarrow> Some (x, s'))"
CCfix_unroll: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "CoCallArityAnalysis.CCfix cccExp \<Gamma>\<cdot>(ae, G) = CoCallAnalysis.ccBindsExtra (CoCallArityAnalysis.CCexp cccExp) \<Gamma>\<cdot>(ae, CoCallArityAnalysis.CCfix cccExp \<Gamma>\<cdot>(ae, G)) \<squnion> G"
ain: fixes x :: "'a" assumes "x \<in> T" shows "a x \<in> {0..1}"
path_connected_arcwise: fixes S :: "'b set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> (\<exists>g. arc g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y))"
mk_poly_sub_dim_1: fixes A :: "'b poly mat"   and l :: "nat"   and j :: "nat" shows "dim_row (mk_poly_sub A l j) = dim_row A"
less_imp_triv: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y" shows "(less y x \<longrightarrow> P) = True"
of_nat_floor: fixes r :: "'a" assumes "(0::'a) \<le> r" shows "of_nat (nat \<lfloor>r\<rfloor>) \<le> r"
unit_div_mult_self: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "b div a * a = b"
len_lexext_rev_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "len_lexext_rev gt ys xs" shows "len_lexext_rev gt' ys xs"
RedCondF: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "redp P E (if (false) e\<^sub>1 else e\<^sub>2) s e\<^sub>2 s"
induct: fixes x :: "formula list"   and rules :: "(formula list \<times> formula list set) set"   and P :: "formula list \<Rightarrow> bool" assumes "x \<in> deductions rules"   and "\<And>conc prems. \<lbrakk>(conc, prems) \<in> rules; prems \<in> Pow (deductions rules \<inter> Collect P)\<rbrakk> \<Longrightarrow> P conc" shows "P x"
transpose_row_def: fixes A :: "(('a, 'c) vec, 'b) vec"   and i :: "'c" shows "transpose_row A i = (\<chi>j. A $ j $ i)"
arr_inv: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.arr (B.VV.inv f)"
distr_return: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and x :: "'b" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "x \<in> space M" shows "distr (return M x) N f = return N (f x)"
intro: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space s1"   and "vector_space s2" shows "vector_space_pair s1 s2"
upper_triangular_def: fixes A :: "(('c, 'd) vec, 'd) vec" shows "upper_triangular A = (\<forall>i j. j < i \<longrightarrow> A $ i $ j = (0::'c))"
cc_imp_c: shows "\<forall>I. I \<Turnstile>m CC \<longrightarrow> I \<Turnstile> C"
convergence: fixes j :: "nat" assumes "t j \<le> t0 + e" shows "dist (solution (t j)) (grid_function (discrete_evolution (euler_increment f)) (solution (t 0)) t j) \<le> euler_C TYPE('a) / B' * (exp (B' * (t0 + e - t 0) + 1) - 1) * max_stepsize j ^ 1"
g: shows "g \<in> arcs G \<rightarrow> arcs G"
permutations_of_set_altdef: fixes A :: "'b set" assumes "finite A" shows "permutations_of_set A = permutations_of_multiset (mset_set A)"
stopping_time_eq_const: fixes T :: "'a \<Rightarrow> 't"   and t :: "'t" assumes "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> = t)"
DFS_invar_def: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" shows "DFS_invar G param s \<equiv> param_DFS G \<and> DFS_invar_axioms G param s"
preallocated_new: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and x :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" assumes "preallocated h"   and "h a = None" shows "preallocated (h(a \<mapsto> x))"
cname_of_xcp: fixes h :: "heap"   and C :: "char list" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "cname_of h (addr_of_sys_xcpt C) = C"
lappend_ltake_ldrop: fixes n :: "enat"   and xs :: "'a llist" shows "lappend (ltake n xs) (ldrop n xs) = xs"
decs_unclocked: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and count1 :: "nat"   and count2 :: "nat" shows "(evaluate_decs False mn env s ds (s', r) \<longrightarrow> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s') \<and> evaluate_decs False mn env (update_clock (\<lambda>_. count1) s) ds (update_clock (\<lambda>_. count1) s', r) = evaluate_decs False mn env (update_clock (\<lambda>_. count2) s) ds (update_clock (\<lambda>_. count2) s', r)"
measure_pmf_conv_infsetsum: fixes p :: "'a pmf"   and A :: "'a set" shows "measure_pmf.prob p A = infsetsum (pmf p) A"
inj_on_verts_sccs: shows "inj_on verts sccs"
absorb_iff1: fixes a :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(a \<le> b) = (a \<sqinter> b = a)"
AssertionStatImpEqvt: fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "p \<bullet> \<Psi> \<hookrightarrow> \<Psi>' = p \<bullet> \<Psi> \<hookrightarrow> p \<bullet> \<Psi>'"
casts': shows "P \<turnstile> Ts Casts vs'' to ws "
l2polR_eq_el: fixes N :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "N \<noteq> []" shows "Cp (list2policyR N) p = list2policy (map Cp N) p"
refl: fixes x :: "comb" shows "x \<Rrightarrow>\<^sup>1 x"
fresh_subst': fixes x :: "name"   and u :: "lam"   and t :: "lam" assumes "x \<sharp> u" shows "x \<sharp> t[x::=u]"
elims_2: fixes x :: "('b \<times> nat) tree" assumes "ltree x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and a :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, n), r\<rangle>" and "min_height r \<le> min_height l \<and> n = min_height r + 1 \<and> ltree l \<and> ltree r"
pseudo_rationalizable_onI: fixes Rs :: "('a \<times> 'a) set set"   and A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>r. r \<in> Rs \<Longrightarrow> Linear_order r"   and "\<And>B. B \<subseteq> A \<Longrightarrow> f B = (\<Union>r\<in>Rs. set_option (MaxR.MaxR_opt r B))" shows "pseudo_rationalizable_on A Rs f"
eq_fold: fixes A :: "'b set" shows "sorted_list_of_set A = Finite_Set.fold insort [] A"
Sigma_Un_distrib2: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'c set" shows "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
assms_7: shows "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"
one_or_two_def: shows "one_or_two = {undefined False, undefined True}"
Fun_ide: fixes a :: "'t" assumes "S'.ide a" shows "S'.Fun a = (\<lambda>x\<in>S'.set a. x)"
less_cset_def: fixes xs :: "'a cset"   and ys :: "'a cset" shows "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"
IH: fixes ys :: "'a list" assumes "length xs - length ys = Suc m"   and "set ys \<subseteq> set xs"   and "distinct ys"   and "distinct xs" shows "\<exists>e. e \<in> set xs \<and> e \<notin> set ys"
a_set_shadow_root_def: fixes element_ptr :: "(_) element_ptr" shows "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_shadow_root element_ptr = put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt_update"
block_nominals_branch: fixes block :: "('d, 'e) fm list \<times> 'e"   and branch :: "(('d, 'e) fm list \<times> 'e) list" assumes "block \<in>. branch" shows "block_nominals block \<subseteq> branch_nominals branch"
nths_def: fixes xs :: "'a list"   and A :: "nat set" shows "nths xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<length xs]))"
6: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> y"
abs_eq: fixes xa :: "'d \<Rightarrow> 'c"   and x :: "'d \<times> 'd" shows "map_uprod xa (abs_uprod x) = abs_uprod (map_prod xa xa x)"
K4_imp_Gets: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta X. Gets Tgs \<lbrace>Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>, X\<rbrace> \<in> set evs"
acute_col__out: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C"   and "Acute A B C" shows "B Out A C"
group_comm_groupI: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "comm_group G"
plus_fun_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f + g) x = f x + g x"