hcju/isabelleps huggingface dataset:
card_extensional_funcset_not_inj_on: fixes S :: "'c set"   and T :: "'d set" assumes "finite S"   and "finite T"   and "card S \<le> card T" shows "card {f \<in> S \<rightarrow>\<^sub>E T. \<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)"
length_partition_by_median_3: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length l - length r \<le> 1"
put_tog: shows "take (length qs div 2) qs @ drop (length qs div 2) qs = qs"
nb: shows "set (map (\<lambda>(t, op). encode_operator_precondition \<Pi> t op) (List.product [0..<t] (\<Pi>\<^sub>\<O>))) = (\<lambda>(t, op). encode_operator_precondition \<Pi> t op) ` ({0..<t} \<times> set (\<Pi>\<^sub>\<O>))"
3: fixes g :: "'g"   and p :: "'val" assumes "unnecessaryPhi g p"   and "\<not> trivial g p"   and "p \<in> allVars g" obtains q :: "'val" where "phiArg g p q" and "unnecessaryPhi g q" and "\<not> def_dominates g q p"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x" shows "Group.comm_monoid (add_monoid R)"
inv_inter: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "inv t1"   and "inv t2" shows "inv (local.inter t1 t2)"
fw_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i' \<le> n"   and "j' \<le> n" shows "fw m n k i j i' j' \<le> m i' j'"
prv_imp_dsjE: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> \<psi>) (imp (imp \<chi> \<psi>) (imp (dsj \<phi> \<chi>) \<psi>)))"
simps_13: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a" shows "(v;; va \<sqsubseteq> {vb} WHILE vc DO vd {ve}) = False"
finite_lattice_complete_Sup_upper: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> Sup A"
vfsequence_vlrestriction: fixes xs :: "V"   and k :: "V" assumes "vfsequence xs"   and "k \<in>\<^sub>\<circ> \<omega>" shows "vfsequence (xs \<restriction>\<^sup>l\<^sub>\<circ> k)"
lens_plus_assoc: fixes X :: "'a \<Longrightarrow> 'd"   and Y :: "'b \<Longrightarrow> 'd"   and Z :: "'c \<Longrightarrow> 'd" shows "(X +\<^sub>L Y) +\<^sub>L Z \<approx>\<^sub>L X +\<^sub>L Y +\<^sub>L Z"
is_sty_one_def: fixes P :: "cld list"   and ty :: "ty"   and ty' :: "ty" shows "is_sty_one P ty ty' = (case find_path_ty_f P ty of None \<Rightarrow> None | Some ctxclds \<Rightarrow> case ty' of ty_top \<Rightarrow> Some True | ty_def ctx' dcl' \<Rightarrow> Some ((ctx', dcl') \<in> set (map (\<lambda>(ctx, cld). (ctx, class_name_f cld)) ctxclds)))"
lt_plus_eqI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<prec>\<^sub>t lt q" shows "lt (p + q) = lt q"
nhdE: fixes U :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "U \<in> nhds\<^bsub>T\<^esub> x" obtains m :: "'b set" where "U \<subseteq> carrier\<^bsub>T\<^esub>" and "m open\<^bsub>T\<^esub>" and "x \<in> m" and "m \<subseteq> U"
add_le_imp_le_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and i :: "'c"   and k :: "'c"   and n :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n" shows "less_eq i (minus n k)"
assms_4: shows "insertion (nth_default 0 (xs' @ x # xs)) c = C"
degen_acc_run_sound: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" assumes "b_graph.is_acc_run (degeneralize_ext m) r" shows "is_acc_run (fst \<circ> r)"
minus_vec_def: shows "(-) \<equiv> \<lambda>x y. \<chi>i. x $h i - y $h i"
exec_gpv_oracle_bisim': fixes X :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "'b"   and s2 :: "'d"   and oracle1 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and oracle2 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and gpv :: "('g, 'e, 'f) gpv" assumes "X s1 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
CH_def: fixes X :: "X5 set" shows "BSI.CH X = (\<Union>h. CX5h h X)"
cases: fixes r :: "'b stateful_list_policy" obtains hostsL :: "'b list" and flows_fixL :: "('b \<times> 'b) list" and flows_stateL :: "('b \<times> 'b) list" where "r = \<lparr>hostsL = hostsL, flows_fixL = flows_fixL, flows_stateL = flows_stateL\<rparr>"
l10: fixes x :: "'a"   and y :: "'a" assumes "- x \<sqinter> y = bot" shows "- x \<le> - y"
read_read_nonsync: fixes c :: "'b \<Rightarrow> 'c"   and C :: "'c set"   and d :: "'d \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> 'c process"   and Q :: "'d \<Rightarrow> 'c process" assumes "\<And>y. c y \<notin> C"   and "\<And>y. d y \<in> C" shows "(read c UNIV P \<lbrakk>C\<rbrakk> read d UNIV Q) = (c`?`x \<rightarrow> (P x \<lbrakk>C\<rbrakk> read d UNIV Q))"
simps_2: fixes Ts :: "typ list"   and i :: "nat" shows "typ_of1 Ts (Bv i) = (if i < length Ts then Some (Ts ! i) else None)"
freeword_funlift_surj': fixes ys :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "ys \<in> lists (f ` S \<union> uminus ` f ` S)" shows "sum_list ys \<in> freeword_funlift f ` FreeGroup S"
hb_consistent_elim_2: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and y :: "'c" assumes "happens_before hb_weak hb"   and "happens_before.hb_consistent hb (xs @ [y])" obtains "happens_before.hb_consistent hb xs" and "\<forall>x\<in>set xs. \<not> hb y x"
add_less_same_cancel2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a + b \<subset># b) = (a \<subset># {#})"
closure_of_subtopology: fixes X :: "'b topology"   and U :: "'b set"   and S :: "'b set" shows "subtopology X U closure_of S = U \<inter> X closure_of (U \<inter> S)"
fid_on_vinsert: fixes a :: "V"   and A :: "V" shows "vinsert [a, a]\<^sub>\<circ> (fid_on A) = fid_on (vinsert a A)"
simps_3: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'b"   and g21 :: "'a \<Rightarrow> 'c"   and g22 :: "'a \<Rightarrow> 'a" assumes "p a" shows "terminal (unfold_tllist p g1 g21 g22 a) = g1 a"
3: shows "(F \<squnion> e \<squnion> e\<^sup>T) * F \<le> F"
finite_dimensional_vector_space_pair_1_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>1 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" shows "finite_dimensional_vector_space_pair_1_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
dghm_singleton_def: fixes j :: "V"   and \<CC> :: "V" shows "dghm_singleton j \<CC> = [\<lambda>a\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. ZFC_in_HOL.set {\<langle>j, a\<rangle>}, \<lambda>f\<in>\<^sub>\<circ>\<CC>\<lparr>Arr\<rparr>. ZFC_in_HOL.set {\<langle>j, f\<rangle>}, \<CC>, \<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>]\<^sub>\<circ>"
split_cong: fixes q :: "'k \<times> 'l"   and f :: "'k \<Rightarrow> 'l \<Rightarrow> 'm"   and g :: "'k \<Rightarrow> 'l \<Rightarrow> 'm"   and p :: "'k \<times> 'l" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
hcomp_obj_arr: fixes b :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" assumes "S.obj b"   and "S.arr f"   and "b = S.trg f" shows "b \<star>\<^sub>S f = f"
simps_5: fixes \<phi> :: "'a fm"   and \<psi> :: "'a fm" shows "depth (Or \<phi> \<psi>) = max (depth \<phi>) (depth \<psi>) + 1"
aux_qInit_ignoreFirst_def: fixes K :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model \<times> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm + ('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model \<times> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" shows "aux_qInit_ignoreFirst K = (case K of Inl (MOD, x) \<Rightarrow> termIn x | Inr (MOD, x) \<Rightarrow> absIn x)"
MOST_coeff_eq_0: fixes p :: "'b poly" shows "\<forall>\<^sub>\<infinity>n. coeff p n = (0::'b)"
remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "mset_set A = add_mset x (mset_set (A - {x}))"
prv_exi_inst: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm" shows "bprv (imp (subst \<phi> t x) (exi x \<phi>))"
assms_2: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "residue (p ^ m) (f n) = f m"
chine_simps_1: shows "Maps.arr SPN_r_SPN_s.chine"
sturm_squarefree_def: fixes p :: "real poly" shows "sturm_squarefree p = sturm (p div gcd p (pderiv p))"
Ints_det: fixes A :: "'b mat" assumes "\<And>i j. \<lbrakk>i < dim_row A; j < dim_col A\<rbrakk> \<Longrightarrow> A $$ (i, j) \<in> \<int>" shows "det A \<in> \<int>"
termination_lemma: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ss :: "'b list"   and n :: "nat"   and qs :: "(nat \<times> 'b) list"   and p :: "nat"   and w :: "nat set" assumes "Semilat A r f"   and "ss \<in> list n A"   and "\<forall>(q, t)\<in>set qs. q < n \<and> t \<in> A"   and "p \<in> w" shows "ss [\<sqsubset>\<^bsub>r\<^esub>] merges f qs ss \<or> merges f qs ss = ss \<and> {q. \<exists>t. (q, t) \<in> set qs \<and> t \<squnion>\<^bsub>f\<^esub> ss ! q \<noteq> ss ! q} \<union> (w - {p}) \<subset> w"
true_imp: fixes p :: "'a upred" shows "(true \<Rightarrow> p) = p"
QP: shows "Q * P = 1\<^sub>m (dim_row A)"
enum_add_le: fixes xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and mi :: "'d"   and ma :: "'d" shows "foldl (+) 0 (enum xs index key n mi ma) \<le> length xs"
rel_fundefs_def: fixes f :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e) Inca.instr fundef option"   and g :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e, 'g, 'h, 'i) Ubx.instr fundef option" shows "rel_fundefs f g = (\<forall>x. rel_option (rel_fundef (\<lambda>y z. y = norm_instr z)) (f x) (g x))"
Sum1: fixes P :: "pi"   and Rs :: "residual"   and Q :: "pi" assumes "P \<longmapsto> Rs" shows "P \<oplus> Q \<longmapsto> Rs"
lfp_def: 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 f :: "'c \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.lfp Inf less_eq f = Inf {u. less_eq (f u) u}"
intro: fixes qp :: "('a, 'b seq_action) automaton" assumes "init qp \<noteq> {}" shows "openproc_parq_axioms qp"
surjective: fixes r :: "('b, 'c) routing_rule_scheme" shows "r = \<lparr>routing_match = routing_match r, metric = metric r, routing_action = routing_action r, \<dots> = routing_rule.more r\<rparr>"
assms_2: shows "f \<in> space\<^sub>N (\<LL> p M)"
gauge_integral_by_substitution: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "g a \<le> g b"   and "\<forall>x\<in>{a..b}. (g has_vector_derivative g' x) (at x within {a..b})"   and "continuous_on {a..b} g'"   and "continuous_on (g ` {a..b}) f" shows "integral {g a..g b} f = integral {a..b} (\<lambda>x. f (g x) * g' x)"
zero_discr_def: shows "0 \<equiv> Discr (0::'a)"
select_convs_2: fixes carrier :: "'b set"   and eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and more :: "'a" shows "gorder.more \<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr> = more"
assms: fixes x :: "'a"   and x' :: "'b" shows "P x x' = \<up> (P' x x')"
extend_sum: fixes S :: "'b set"   and A :: "'b set"   and a :: "'b \<Rightarrow> 'a" assumes "finite S"   and "A \<subseteq> S" shows "(\<Sum>x\<in>S. if x \<in> A then a x else (0::'a)) = sum a A"
llexord_lappend_leftD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "llexord r (lappend xs ys) (lappend xs zs)"   and "lfinite xs"   and "\<And>x. x \<in> lset xs \<Longrightarrow> \<not> r x x" shows "llexord r ys zs"
flatten_sequence_id: fixes cs :: "('d, 'e, 'f) com list" assumes "cs \<noteq> []"   and "\<forall>c\<in>set cs. flatten c = [c]" shows "flatten (sequence Seq cs) = cs"
assms: fixes s :: "'b"   and a :: "'a"   and s' :: "'b" assumes "run_state m s = (a, s')" shows "P s s'"
wf\<^sub>1: shows "wf_J\<^sub>1_prog (compP\<^sub>1 P)"
pwd_spy_enc_pubk_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_enc_pubk n)"
asymp_equiv_altdef: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sim>[F] g) = ((\<lambda>x. f x - g x) \<in> o[F](g))"
circline_type_unit_circle: shows "circline_type unit_circle = - 1"
tau_transition_eqvt: fixes P :: "'state"   and P' :: "'state"   and p :: "perm" assumes "P \<Rightarrow> P'" shows "p \<bullet> P \<Rightarrow> p \<bullet> P'"
rec_conj_def: shows "rec_conj = CN rec_sign [rec_mult]"
assms_3: shows "\<forall>x y z. snd (g x y) \<longrightarrow> fst (g y z) \<longrightarrow> fst (g x z)"
simps_4: fixes C :: "ty"   and D :: "ty"   and X :: "ntrm set" shows "NEGc (C OR D) X = AXIOMSc (C OR D) \<union> BINDINGc (C OR D) X \<union> ORRIGHT1 (C OR D) (\<parallel><C>\<parallel>) \<union> ORRIGHT2 (C OR D) (\<parallel><D>\<parallel>)"
diff_fun_space_eq: shows "sub.diff_fun_space = {f. diff_fun k (charts_submanifold S) f} \<inter> {f. extensional0 sub.carrier f}"
snapshot_algorithm_is_correct: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "\<exists>t' i. distributed_system.trace channel trans send recv init t' final \<and> t' <~~> t \<and> computation.state_equal_to_snapshot channel (distributed_system.s channel trans send recv init t' i) final \<and> i \<le> length t'"
DisjointPaths_extend: fixes P :: "'a list" assumes "v0 \<leadsto>P\<leadsto>\<^bsub>H_x\<^esub> v1"   and "\<And>xs v. \<lbrakk>xs \<in> Q; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths H_x v0 v1 (insert P Q)"
axioms_3: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme" assumes "simulation R A B" shows "simulation_axioms R A B"
tagrel_e: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>"
independent_Basis: shows "vector_space_over_itself.independent {1::'a}"
simps_1: fixes M :: "'d"   and N :: "'d" shows "subject (M\<lparr>N\<rparr>) = Some M"
primitive_extractor_correct_3: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "\<not> has_disc disc ms"
Lambert_W_neg_iff: fixes x :: "real" shows "(Lambert_W x < 0) = (x < 0)"
order_on_defs_3: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
jvm_trans: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state"   and \<sigma>'' :: "jvm_state" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''"
wr: shows "well_related_set (Collect derivable) (\<sqsubseteq>)"
etranE: fixes c :: "'b com option \<times> 'b"   and c' :: "'b com option \<times> 'b" assumes "c -e\<rightarrow> c'" obtains P :: "'b com option" and s :: "'b" and t :: "'b" where "c = (P, s)" and "c' = (P, t)"
IH: fixes n :: "nat"   and c :: "int" assumes "m \<le> n + 1" shows "(pderiv ^^ m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int m + 1) m * c) (n - m)"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'g list"   and x :: "'g" assumes "i < length xs" shows "xs[i := x] ! i = x"
Abs_cases: fixes x :: "'b" obtains y :: "'a" where "x = Abs y" and "y \<in> A"
min_def: shows "min = (LEAST n. sdrop n s = sconst x)"
subterm_subst_not_img_subterm: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'e) Term.term"   and \<I> :: "'e \<Rightarrow> ('c, 'd) Term.term" assumes "t \<sqsubseteq> s \<cdot> \<I>"   and "\<not> (\<exists>w\<in>fv s. t \<sqsubseteq> \<I> w)" shows "\<exists>f T. Fun f T \<sqsubseteq> s \<and> t = Fun f T \<cdot> \<I>"
cl_inI: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> carrier"   and "X \<subseteq> \<E>"   and "x \<in> \<E>"   and "rank_of (insert x X) = rank_of X" shows "x \<in> cl_in \<E> X"
"1_1": fixes k :: "nat" shows "P k (w k) (w\<^sub>1 k)"
count_roots_with_times_root: fixes p :: "'a poly"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "p \<noteq> 0"   and "P a" shows "count_roots_with P ([:a, - (1::'a):] * p) = Suc (count_roots_with P p)"
simple_extension_of_transcendental: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R"   and "(transcendental over K) x" shows "K [X] \<simeq> R\<lparr>carrier := simple_extension K x\<rparr>"
LT13: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> G) \<longrightarrow> (G \<leadsto> H) \<longrightarrow> F \<leadsto> H"
simps: fixes status :: "status"   and x :: "'x1" shows "r1.init_fin_final (status, x) = (status = Finished \<and> final1 x)"
collapse_1: fixes llist :: "'a llist" assumes "lnull llist" shows "llist = LNil"
unique_oid: fixes opset :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and op1 :: "'d"   and op2 :: "'d" assumes "opset opset deps"   and "(oid, op1) \<in> opset"   and "(oid, op2) \<in> opset" shows "op1 = op2"
lim_h_plus_finite: fixes e :: "'v \<times> 'v" shows "(\<Squnion>n. h_plus n e) \<noteq> \<top>"
to_from_view: fixes x :: "view" shows "view_merkle.to_view (view_merkle.from_view x) = x"
ListAif1: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat" assumes "bval b s" shows "preList upds (IF b THEN C1 ELSE C2) l s = preList upds C1 l s"
ground_lit: shows "is_ground_lit (subst_lit L \<sigma>)"
intros_17: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> \<Rightarrow> Abrupt s"
mset_eq_permutation: fixes xs :: "'d list"   and ys :: "'d list" assumes "mset xs = mset ys" obtains p :: "nat \<Rightarrow> nat" where "p permutes {..<length ys}" and "permute_list p ys = xs"
assoc_naturality: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<tau> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.arr \<tau>"   and "Span.src \<mu> = Span.trg \<nu>"   and "Span.src \<nu> = Span.trg \<tau>" shows "Span.assoc (Span.cod \<mu>) (Span.cod \<nu>) (Span.cod \<tau>) \<bullet> ((\<mu> \<circ> \<nu>) \<circ> \<tau>) = (\<mu> \<circ> \<nu> \<circ> \<tau>) \<bullet> Span.assoc (Span.dom \<mu>) (Span.dom \<nu>) (Span.dom \<tau>)"
in_path_weight: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b"   and loc1 :: "'a"   and loc2 :: "'a" assumes "graph weights" shows "(s \<in>\<^sub>A graph.path_weight weights loc1 loc2) = (s \<in> minimal_antichain {s. graph.path_weightp weights loc1 loc2 s})"
ker_def: fixes f :: "'c \<Rightarrow> 'd" shows "ker f = {a. f a = (0::'d)}"
Prog_valid_def: shows "Prog_valid = (\<exists>G. G_valid G \<and> MST_valid)"
carrier_integer_mod_group: fixes n :: "nat" shows "carrier (integer_mod_group n) = (if n = 0 then UNIV else {0..<int n})"
take_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'b" shows "take i (replicate k x) = replicate (min i k) x"
is_binqueue_normalize: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l xs" shows "is_binqueue l (Binomial_Queue.normalize xs)"
distrib_3: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
core_nhds: fixes U :: "'a set"   and x :: "'a" assumes "U \<subseteq> carrier"   and "x \<in> core U" shows "U \<in> nhds x"