hcju/isabelleps huggingface dataset:
bcomp_bcomp: fixes b\<^sub>1 :: "bexp"   and f\<^sub>1 :: "bool"   and i\<^sub>1 :: "int"   and b\<^sub>2 :: "bexp"   and f\<^sub>2 :: "bool"   and i\<^sub>2 :: "int"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "bcomp (b\<^sub>1, f\<^sub>1, i\<^sub>1) @ bcomp (b\<^sub>2, f\<^sub>2, i\<^sub>2) \<Turnstile> cfs\<box>"   and "\<And>cfs. bcomp (b\<^sub>1, f\<^sub>1, i\<^sub>1) \<Turnstile> cfs\<box> \<Longrightarrow> bpred (b\<^sub>1, f\<^sub>1, i\<^sub>1) (cfs ! 0) (cfs ! (length cfs - 1))"   and "\<And>cfs. bcomp (b\<^sub>2, f\<^sub>2, i\<^sub>2) \<Turnstile> cfs\<box> \<Longrightarrow> bpred (b\<^sub>2, f\<^sub>2, i\<^sub>2) (cfs ! 0) (cfs ! (length cfs - 1))" shows "case cfs ! 0 of (pc, s, stk) \<Rightarrow> pc = 0 \<and> (bval b\<^sub>1 s \<noteq> f\<^sub>1 \<longrightarrow> (\<exists>k<length cfs. cfs ! k = (size (bcomp (b\<^sub>1, f\<^sub>1, i\<^sub>1) @ bcomp (b\<^sub>2, f\<^sub>2, i\<^sub>2)) + (if bval b\<^sub>2 s = f\<^sub>2 then i\<^sub>2 else 0), s, stk)))"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "exhaustive (x # xs) = (\<not> proper_interval None (Some x) \<and> exhaustive_above x xs)"
combine_c0_ne_c1: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and l :: "int" assumes "p\<^sub>0\<^sub>L \<noteq> p\<^sub>1\<^sub>L"   and "p\<^sub>0\<^sub>R \<noteq> p\<^sub>1\<^sub>R"   and "distinct ps"   and "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "c\<^sub>0 \<noteq> c\<^sub>1"
tag_Times_injI: fixes A :: "'a list set"   and B :: "'a list set"   and x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "tag_Times A B x = tag_Times A B y"   and "x @ z \<in> A \<cdot> B" shows "y @ z \<in> A \<cdot> B"
simps_1: fixes e :: "(name, hf) finfun"   and t :: "tm"   and u :: "tm" shows "eval_fm e (t IN u) = (\<lbrakk>t\<rbrakk>e \<^bold>\<in> \<lbrakk>u\<rbrakk>e)"
map_closed: fixes a :: "'b" assumes "a \<in> \<FF> (fst x)" shows "\<rho> (fst x) z a \<in> \<FF> z"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "A.arr f"   and "A.dom f = a" shows "f \<cdot>\<^sub>A a = f"
of_nat_le_numeral_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "nat"   and i :: "num"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (semiring_1.of_nat one plus zero x) (power.power one times (numeral.numeral one plus i) n) = (x \<le> numeral i ^ n)"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>g \<cdot> f : a \<rightarrow>\<^sub>C c\<guillemotright>" obtains b :: "'a" where "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>"
assms_1: shows "l \<equiv> \<lambda>f \<omega>. g (shd \<omega>) (f (stl \<omega>))"
exists_co_path_decomp1: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "pair_sym_digraph G"   and "e \<in> parcs G"   and "w \<notin> pverts G"   and "pre_digraph.apath (with_proj (subdivide G e w)) u p v"   and "(fst e, w) \<in> set p"   and "w \<noteq> v" shows "\<exists>p1 p2. p = p1 @ (fst e, w) # (w, snd e) # p2"
assms_4: shows "[a ^ (n - 1) = 1] (mod n)"
simps: fixes ps :: "('c, 'b) fm list"   and i :: "'b" shows "block_nominals (ps, i) = {i} \<union> list_nominals ps"
env_restr_belowI2: fixes S :: "'d set"   and m1 :: "'d \<Rightarrow> 'e"   and m2 :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> S \<Longrightarrow> m1 x \<sqsubseteq> m2 x" shows "m1 f|` S \<sqsubseteq> m2"
f: shows "f \<in> M \<rightarrow>\<^sub>M N \<Otimes>\<^sub>M P"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRel Enc"
Rosser_Form_Proofs_def: 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 fls :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and Lq :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and "proof" :: "'d set"   and prfOf :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c"   and S :: "'c"   and encPf :: "'d \<Rightarrow> 'b"   and Pf :: "'c" shows "Rosser_Form_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi fls prv bprv Lq dsj proof prfOf enc N S encPf Pf \<equiv> (Deduct2_with_PseudoOrder var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv Lq \<and> Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N) \<and> Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S \<and> CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf"
kD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt" shows "\<exists>dsn dsk flag hops nhip. rt dip = Some (dsn, dsk, flag, hops, nhip)"
sel_1: fixes x1 :: "'d"   and x2 :: "'d stream" shows "shd (x1 ## x2) = x1"
distinct_hd_tl: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs"   and "x = hd xs" shows "x \<notin> set (tl xs)"
mergeable_3: shows "\<forall>x\<in>A. x < n \<and> v1 ! x = f x"
cos_npi2: fixes n :: "nat" shows "cos (pi * real n) = (- 1) ^ n"
ltake_infinite: fixes w :: "'b llist" shows "ltake \<infinity> w = w"
solution_iff_nth_solution': fixes a :: "int"   and b :: "int" shows "solution (a, b) = ((nat \<bar>a\<bar>, nat \<bar>b\<bar>) \<in> range nth_solution)"
DeMoivre2: fixes r :: "real"   and a :: "real"   and n :: "nat" shows "rcis r a ^ n = rcis (r ^ n) (real n * a)"
finite_kmin: fixes f :: "nat \<Rightarrow> real" shows "finite (kmin f PR)"
rep_nat_pp_pp_inverse: fixes x :: "('a, 'b) pp" shows "abs_nat_pp (rep_nat_pp x) = x"
sep_conj_def: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "P \<and>* Q \<equiv> \<lambda>h. \<exists>x y. x ## y \<and> h = x + y \<and> P x \<and> Q y"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d \<Rightarrow> 'a"   and x :: "'d set['c]"   and y :: "'b set['c]" shows "rel_bset Sb (map_bset i x) y = rel_bset (\<lambda>x. Sb (i x)) x y"
mset_factors_imp_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and x :: "'a"   and f :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F x"   and "f \<in># F" shows "dvd.dvd times f x"
arg_min_if_finite_1: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "S \<noteq> {}" shows "arg_min_on f S \<in> S"
dehomogenize_id_iff: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "(dehomogenize x p = p) = (x \<notin> indets p)"
proj_stoch_proc_component': fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" assumes "k \<le> n" shows "f k x = proj_stoch_proc f n x !! k"
telescopic_base_dim_1: fixes K :: "'a set"   and F :: "'a set"   and E :: "'a set" assumes "subfield K R"   and "subfield F R"   and "finite_dimension K F"   and "finite_dimension F E" shows "finite_dimension K E"
assms_2: shows "are_all_operator_effects_consistent (a # ops)"
assms_2: shows "sig_of_pair p \<prec>\<^sub>t lt b"
axioms_2: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_pair_1_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 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 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 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" shows "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 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"
less_supI1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less x a" shows "less x (sup a b)"
I_asimp: fixes a :: "atom"   and xs :: "real list" shows "R.I (asimp a) xs = I\<^sub>R a xs"
suc_rw: shows "[0..<Suc k] = [0..<k] @ [k]"
leg0_composite: shows "gf.leg0 = f.leg0 \<cdot> gf.prj\<^sub>0"
quotient_map_compose: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f"   and "quotient_map X' X'' g" shows "quotient_map X X'' (g \<circ> f)"
inf_sup_distrib2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
supt_neqD: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<rhd> t" shows "s \<noteq> t"
surj_iff: fixes f :: "'c \<Rightarrow> 'b" shows "surj f = (f \<circ> inv f = id)"
prems_3: fixes c :: "'c" assumes "c \<in> set (x # xs)" shows "(\<lambda>a. case a of (a, b) \<Rightarrow> g a b c) \<in> A \<Otimes>\<^sub>M B \<rightarrow>\<^sub>M B"
comp_simp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1.seq (fst g) (fst f)"   and "A2.seq (snd g) (snd f)" shows "A1xA2.comp g f = (fst g \<cdot>\<^sub>A\<^sub>1 fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f)"
assms_8: shows "Smith_normal_form_wrt S' \<Q>"
exhaust: fixes y :: "gauss_int" obtains x1 :: "int" and x2 :: "int" where "y = Gauss_Int x1 x2"
decrease_ub_satisfied_inverse: fixes dir :: "('b, 'a) Direction"   and c :: "'a"   and s :: "('b, 'a) state"   and x :: "nat"   and v :: "nat \<Rightarrow> 'a"   and i :: "'b" assumes "\<lhd>\<^sub>u\<^sub>b (Direction.lt dir) c (Direction.UB dir s x)"   and "dir = Positive \<or> dir = Negative"   and "v \<Turnstile>\<^sub>b \<B> (update\<B>\<I> (Direction.UBI_upd dir) i x c s)" shows "v \<Turnstile>\<^sub>b \<B> s"
set_less_aux_finite_total: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "A \<sqsubset>' B \<or> A = B \<or> B \<sqsubset>' A"
prems_5: shows "hd (tl xs) = hd (tl P)"
A: shows "[Inl S] -G\<rightarrow>\<^sup>+ t"
neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "neg \<phi> \<in> fmla"
add_mset_remove_trivial_eq: fixes N :: "'a multiset"   and a :: "'a" shows "(N = a, N - (a, {#})) = (a \<in># N)"
semiring_norm_27: fixes n :: "num" shows "Num.BitM (num.Bit0 n) = num.Bit1 (Num.BitM n)"
cut_cs_cf: fixes \<Gamma> :: "'b formula multiset"   and A :: "'b formula"   and \<Delta> :: "'b formula multiset"   and \<Gamma>' :: "'b formula multiset"   and \<Delta>' :: "'b formula multiset" assumes "\<And>A \<Gamma> \<Delta>. \<lbrakk>\<Gamma> \<Rightarrow> A, \<Delta>; A, \<Gamma> \<Rightarrow> \<Delta>\<rbrakk> \<Longrightarrow> \<Gamma> \<Rightarrow> \<Delta>"   and "\<Gamma> \<Rightarrow> A, \<Delta>"   and "A, \<Gamma>' \<Rightarrow> \<Delta>'" shows "\<Gamma> + \<Gamma>' \<Rightarrow> \<Delta> + \<Delta>'"
rQuot_conc: fixes c :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "rQuot c (A @@ B) = A @@ rQuot c B \<union> (if [] \<in> B then rQuot c A else {})"
simps_11: fixes v :: "v"   and vb :: "v"   and vd :: "v"   and ve :: "v list" shows "do_opapp (v # vb # vd # ve) = None"
not_may_lock_conv: fixes l :: "('a \<times> nat) option"   and t :: "'a" shows "(\<not> may_lock l t) = (\<exists>t'. t' \<noteq> t \<and> has_lock l t')"
Sbis_finished_discr_R: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "finished c"   and "c \<approx>s d" shows "discr d"
induced_play_lprefix_non_positions: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a" assumes "2 * N \<le> enat (length p)" shows "induced_play \<sigma> p = ltake (2 * N) (llist_of p)"
negligible_plus: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "negligible f"   and "negligible g" shows "negligible (\<lambda>\<eta>. f \<eta> + g \<eta>)"
has_subprob_density_imp_has_density: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "has_density M N f"
step: shows "pn.eq_m C (A * B) \<and> monic A \<and> Mp D =m A \<and> Mp H =m B \<and> pn.Mp A = A \<and> pn.Mp B = B"
lappend_llist_of_stream_conv_shift: fixes xs :: "'b list"   and ys :: "'b stream" shows "lappend (llist_of xs) (llist_of_stream ys) = llist_of_stream (xs @- ys)"
span_base: fixes Sa :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and a :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "Sa \<subseteq> sub.tangent_space p"   and "a \<in> Sa" shows "a \<in> span Sa"
path_split_Cons: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n as n'"   and "as \<noteq> []" obtains a' :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" and as' :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list" where "as = a' # as'" and "n = src a'" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a'" and "CFGExit_wf.path (trg a') as' n'"
a_get_host_locs_def: shows "get_host_locs \<equiv> \<Union> (range get_shadow_root_locs) \<union> (\<Union>ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
class_type_of'_ty_of_htype: fixes hT :: "htype" shows "class_type_of' (ty_of_htype hT) = \<lfloor>class_type_of hT\<rfloor>"
prefix_of_appendD: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and ys :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs @ ys prefix of i" shows "xs prefix of i"
A: shows "((s, c), w, s', c') \<in> trcl (ntrs fg)"
obs2: shows "Y\<^sub>2 \<in> Sols f (\<lambda>s. {t\<^sub>0--t}) S t\<^sub>0 s"
word_unat_Rep_inject1: fixes x :: "'b word" shows "(unat x = unat 1) = (x = 1)"
right_hom_with_unit_axioms: shows "right_hom_with_unit (\<cdot>) (\<star>) \<a> \<iota> a"
HM_def: shows "HM = \<lparr>edge_rev = perm_restrict (edge_rev M) (arcs G - {a, a'}), edge_succ = perm_rem a (perm_rem a' (edge_succ M))\<rparr>"
tagged_partial_division_ofD_1: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_partial_division_of i" shows "finite s"
linorder_not_le: fixes x :: "'e"   and y :: "'e" shows "(\<not> x \<le> y) = (y < x)"
pick_middlep: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'c"   and c :: "'e" assumes "(P OO Q) a c" shows "P a (BNF_Def.pick_middlep P Q a c) \<and> Q (BNF_Def.pick_middlep P Q a c) c"
grid_estimate: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "d < length b"   and "p \<in> grid b ds" shows "ix p d < (ix b d + 1) * 2 ^ (lv p d - lv b d) \<and> (ix b d - 1) * 2 ^ (lv p d - lv b d) < ix p d"
small_fpairs: fixes r :: "V" shows "small {x. x \<in>\<^sub>\<circ> r \<and> (\<exists>a b. x = [a, b]\<^sub>\<circ>)}"
abs_inverse: fixes n :: "nat" shows "rep_nat (abs_nat n) = n"
lossless_S1_14: fixes m :: "bool \<times> bool \<times> bool \<times> bool"   and out :: "unit" shows "lossless_spmf (S1_14 m out)"
finite_field_ops_integer: fixes pp :: "integer" assumes "p = int_of_integer pp" shows "field_ops (finite_field_ops_integer pp) mod_ring_rel_integer"
digit_to_char_simps_2: shows "digit_to_char (Suc 0) = CHR ''1''"
compact_arc_image: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "compact (path_image g)"
dims0_def: shows "ps2_P.dims0 = nths dims ps2_P.vars0"
absolutely_integrable_on_combine: fixes a :: "real"   and c :: "real"   and f :: "real \<Rightarrow> 'c"   and b :: "real" assumes "f absolutely_integrable_on {a..c}"   and "f absolutely_integrable_on {c..b}"   and "a \<le> c"   and "c \<le> b" shows "f absolutely_integrable_on {a..b}"
elims_2: fixes x :: "'b poly" assumes "isconstant x" obtains v :: "'b" where "x = (v__)\<^sub>p"   | v :: "nat" where "x = Bound va__"   | v :: "'b poly" and va :: "'b poly" where "x = Add vb__ vaa__"   | v :: "'b poly" and va :: "'b poly" where "x = Sub vc__ vab__"   | v :: "'b poly" and va :: "'b poly" where "x = Mul vd__ vac__"   | v :: "'b poly" where "x = Neg ve__"   | v :: "'b poly" and va :: "nat" where "x = Pw vf__ va"   | v :: "'b poly" and vc :: "nat" and vb :: "'b poly" where "x = CN v (Suc vc) vb"
concurrent_ops_commute_def: fixes xs :: "'a list" shows "concurrent_ops_commute xs \<equiv> \<forall>x y. {x, y} \<subseteq> set xs \<longrightarrow> x \<parallel> y \<longrightarrow> \<langle>x\<rangle> \<rhd> \<langle>y\<rangle> = \<langle>y\<rangle> \<rhd> \<langle>x\<rangle>"
h_pos: fixes t :: "real" assumes "t < 2" shows "0 < h t"
Smith_nx1_def: fixes A :: "'a mat" shows "Smith_nx1 A = (let (S, P) = Smith_1xn_aux (dim_row A - 1) A\<^sup>T (A\<^sup>T, 1\<^sub>m (dim_row A)) in (P\<^sup>T, S\<^sup>T))"
locality: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and I :: "real set"   and V :: "real \<Rightarrow> 'c set"   and s :: "'d" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "open_cover_of_open_subset S is_open U I V"   and "\<And>i. i \<in> I \<Longrightarrow> V i \<subseteq> U"   and "s \<in> \<FF> U"   and "\<And>i. i \<in> I \<Longrightarrow> \<rho> U (V i) s = zero_str (V i)" shows "s = zero_str U"
zero_less_mult_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (times a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
dup_dmdD: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<diamond>\<diamond>F" shows "sigma \<Turnstile> \<diamond>F"
RedNew: fixes h :: "heap"   and a :: "nat"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h' :: "heap"   and l :: "sfields"   and sh :: "sheap" assumes "new_Addr h = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h' = h(a \<mapsto> blank P C)" shows "P \<turnstile> \<langle>new C,(h, l, sh),True\<rangle> \<rightarrow> \<langle>addr a,(h', l, sh),False\<rangle>"
new_shadow_root_children: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "h' \<turnstile> get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t new_shadow_root_ptr RShadowRoot.child_nodes \<rightarrow>\<^sub>r []"
Let_perm_simps: fixes p :: "perm"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "p \<bullet> Let \<Gamma> e = Let (p \<bullet> \<Gamma>) (p \<bullet> e)"
tllist_all2_TNil1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and b :: "'i"   and ts :: "('h, 'j) tllist" shows "tllist_all2 P Q (TNil b) ts = (\<exists>b'. ts = TNil b' \<and> Q b b')"
strip_all_vars_mk_all: fixes ty :: "typ"   and s :: "variable"   and t :: "Core.term" shows "strip_all_vars (mk_all s ty t) = ty # strip_all_vars t"
ltl_StrongRelease_Until_con: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> M\<^sub>c \<psi> = \<xi> \<Turnstile>\<^sub>c \<psi> U\<^sub>c (\<phi> and\<^sub>c \<psi>)"
split_spec_param_def: fixes n :: "nat"   and X :: "'b set" shows "split_spec_param n X = SPEC (\<lambda>(A, B). X \<subseteq> A \<union> B)"
sfirst_eq_0: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(sfirst P \<omega> = 0) = P \<omega>"
set_list_R1_list_to_list_R1: fixes xs :: "'b list" shows "set_list_R1 (list_to_list_R1 xs) = set xs"
induct: fixes x1 :: "nat \<Rightarrow> Syntax.type"   and x2 :: "nat \<Rightarrow> Syntax.type"   and x3 :: "ctxt"   and x4 :: "Syntax.type"   and x5 :: "Syntax.type"   and P :: "(nat \<Rightarrow> Syntax.type) \<Rightarrow> (nat \<Rightarrow> Syntax.type) \<Rightarrow> ctxt \<Rightarrow> Syntax.type \<Rightarrow> Syntax.type \<Rightarrow> bool" assumes "x1 , x2 \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t x3 : x4 \<Leftarrow> x5"   and "\<And>\<Gamma> \<Delta> T. P \<Gamma> \<Delta> \<diamond> T T"   and "\<And>\<Gamma> \<Delta> E T1 T2 U t. \<lbrakk>\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T1 \<rightarrow> T2 \<Leftarrow> U; P \<Gamma> \<Delta> E (T1 \<rightarrow> T2) U; \<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1\<rbrakk> \<Longrightarrow> P \<Gamma> \<Delta> (E \<^sup>\<bullet> t) T2 U" shows "P x1 x2 x3 x4 x5"