hcju/isabelleps huggingface dataset:
prime_dvd_prod_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and p :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "finite A"   and "normalization_semidom.prime zero times one normalize p" shows "dvd.dvd times p (comm_monoid_mult.prod times one f A) = (\<exists>x\<in>A. dvd.dvd times p (f x))"
inv_into_f_f: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
bind_def: shows "bind = COERCE(udom\<cdot>'d \<rightarrow> (udom \<rightarrow> udom\<cdot>'d) \<rightarrow> udom\<cdot>'d, 'c\<cdot>'d \<rightarrow> ('c \<rightarrow> 'f\<cdot>'d) \<rightarrow> 'f\<cdot>'d)\<cdot>bindU"
posint: shows "integrable M (expl_cond_expect M Y (\<lambda>w. max 0 (X w)))"
d0_def: shows "a.d0 = prod_list a.dims0"
simps: fixes a1 :: "('p, 'v, 's) configuration"   and a2 :: "'p set"   and a3 :: "('p, 'v, 's) configuration" shows "qReachable a1 a2 a3 = ((\<exists>cfg1 Q. a1 = cfg1 \<and> a2 = Q \<and> a3 = cfg1) \<or> (\<exists>cfg1 Q cfg2 msg cfg3. a1 = cfg1 \<and> a2 = Q \<and> a3 = cfg3 \<and> qReachable cfg1 Q cfg2 \<and> cfg2 \<turnstile> msg \<mapsto> cfg3 \<and> (\<exists>p\<in>Q. isReceiverOf p msg)))"
catch_assoc: fixes m :: "'m"   and m' :: "'m"   and m'' :: "'m" shows "catch (catch m m') m'' = catch m (catch m' m'')"
remove_from_eq_class_preserves_disjoint: fixes P :: "'b set set"   and X :: "'b set"   and elem :: "'b" assumes "is_non_overlapping P"   and "X \<in> P"   and "elem \<in> X" shows "X - {elem} \<notin> P"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "semiring_hom hom"   and "injective hom" shows "inj_semiring_hom hom"
less_mset_zmsetE: fixes M :: "'a zmultiset"   and N :: "'a zmultiset" assumes "M < N" obtains A :: "'a multiset" and B :: "'a multiset" and C :: "'a zmultiset" where "M = zmset_of A + C" and "N = zmset_of B + C" and "A < B"
integrable_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" shows "integrable (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) = integrable (lebesgue_on {a..b}) f"
multiplicity_one_nat: fixes p :: "nat" shows "multiplicity p (Suc 0) = 0"
unadjoin_unique: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c"   and y :: "'c"   and h :: "nat" assumes "finite_comm_group_adjoin G H a"   and "y \<in> H"   and "h < subgroup_indicator G H a" shows "finite_comm_group_adjoin.unadjoin G H a (y \<otimes>\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> h) = (y, h)"
is_up: fixes c :: "'a Word.word \<Rightarrow> 'b Word.word" shows "is_up c = (LENGTH('a) \<le> LENGTH('b))"
context_le_refl: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.context_le \<Gamma> \<Gamma>"
1: shows "Left_a.iso \<phi> \<and> Left_a.in_hom \<phi> (a \<star> f) (a \<star> g)"
map_bset_permute: fixes p :: "perm"   and B :: "'c set['d]" shows "p \<bullet> B = map_bset (permute p) B"
unit_simps_2: shows "local.dom \<eta>\<^sub>1 = src e\<^sub>1"
bex_eq_bex: fixes m :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>m. P x = Q x" shows "(\<exists>x\<in>m. P x) = (\<exists>x\<in>m. Q x)"
conSim: shows "weak_reduction_contrasimulation Rel (STCal Source Target)"
le_inf_iff: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "((y \<and> z) \<sqsubseteq> x) = (y \<sqsubseteq> x \<and> z \<sqsubseteq> x)"
inf_bot_right: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and x :: "'e" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
cases: fixes x :: "'ent atom formula" obtains a :: "predicate" and vs :: "'ent list" where "x = Atom (predAtm a vs)"   | va :: "'ent" and vb :: "'ent" where "x = Atom (Eq va__ vb)"   | "x = \<bottom>"   | v :: "'ent atom formula" where "x = \<^bold>\<not> v__"   | v :: "'ent atom formula" and va :: "'ent atom formula" where "x = vc__ \<^bold>\<and> vaa__"   | v :: "'ent atom formula" and va :: "'ent atom formula" where "x = vd__ \<^bold>\<or> vab__"   | v :: "'ent atom formula" and va :: "'ent atom formula" where "x = v \<^bold>\<rightarrow> va"
orthogonal_to_subspace_exists_gen: fixes S :: "'a set"   and T :: "'a set" assumes "span S \<subset> span T" obtains x :: "'a" where "x \<noteq> (0::'a)" and "x \<in> span T" and "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
get_ancestors_locs_def: shows "get_ancestors_locs \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs"
mult_left_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)" shows "(c * a = c * b) = (a = b)"
is_functor: shows "functor J (\<cdot>\<^sub>A) \<chi>.A.map"
dg_dghm_id_is_iso_dghm: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "dghm_id \<CC> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<CC>"
Ig_generate_I: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "Vr K v \<diamondsuit>\<^sub>p Ig K v I = I"
projection_subset_elim: fixes Y :: "'b set"   and X :: "'b set"   and l :: "'b list" assumes "Y \<subseteq> X" shows "l \<upharpoonleft> X \<upharpoonleft> Y = l \<upharpoonleft> Y"
vertices_restrict: fixes G :: "('f, 'e) labeled_graph" shows "vertices (restrict G) = vertices G"
assms: shows "fst args \<in> keys (rep_list (snd args))"
12: shows "path s\<^sub>3 (target s\<^sub>1 q)"
Nom_7: shows "set ((ps, a) # branch) \<subseteq> set branch'"
ideal_generated_in: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.ring plus zero minus uminus times"   and "a \<in> A" shows "a \<in> ring.ideal_generated plus zero uminus times A"
bisim1AAcc2: fixes P :: "'a prog"   and h :: "'heap"   and i :: "'addr expr1"   and i' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and a :: "'addr expr1"   and v :: "'addr val" assumes "P,i,h \<turnstile> (i', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<lfloor>i\<rceil>,h \<turnstile> (Val v\<lfloor>i'\<rceil>, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 a) + pc, xcp)"
borel_measurable_nn_transfer_operator_iterates: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal"   and n :: "nat" assumes "qmpt M T"   and "f \<in> borel_measurable M" shows "(qmpt.nn_transfer_operator M T ^^ n) f \<in> borel_measurable M"
obsne: shows "obs \<sigma> k \<noteq> obs \<sigma>' k"
jkbpTn_jkbpCn_represents: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.jkbpTn jkbp envInit envAction envTrans envVal jview n (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview) = PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n"
wf_fmla_atom_alt: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and \<phi> :: "'a atom formula" shows "wf_fmla_atom ty_ent \<phi> = (is_predAtom \<phi> \<and> wf_fmla ty_ent \<phi>)"
fv_less_nfv: fixes x :: "nat" assumes "x \<in> fv" shows "x < nfv"
PO_l3_inv1: shows "reach l3 \<subseteq> l3_inv1"
par_assoc: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<parallel> (b::'a::refinement_lattice) \<parallel> (c::'a::refinement_lattice) = a \<parallel> (b \<parallel> c)"
Tag_Enc: fixes X :: "tag"   and X' :: "msg"   and Y' :: "msg" shows "Tag X \<noteq> Enc X' Y'"
fg: fixes y :: "'b" assumes "y \<in> T" shows "f (g y) = y"
lfp_greatest: 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"   and A :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>u. less_eq (f u) u \<Longrightarrow> less_eq A u" shows "less_eq A (complete_lattice.lfp Inf less_eq f)"
vec_elements_def: fixes v :: "'a Matrix.vec" shows "vec_elements v = set (map (($v) v) [0..<dim_vec v])"
d_OUT_0: fixes x :: "'a" shows "d_OUT (\<lambda>_. 0) x = 0"
loose_linkable_bounded: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "bounded_countable_web \<Gamma>"   and "loose \<Gamma>" shows "linkable \<Gamma>"
a_in_s_set: fixes a :: "'e"   and A :: "'e set"   and R :: "('f, 'g) Ring_scheme"   and s :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "a \<in> A" shows "a \<in> s_set R s A"
sgn_0: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn zero = zero"
P5: shows "Coplanar A B C Y \<and> Bet Q Y S"
assms: fixes x :: "'a" assumes "x \<in> set (coeffs p)" shows "f x = x"
f1: shows "\<forall>e. e \<notin> set pre \<or> e \<in> set (history k)"
AE_T_iff_n: fixes P :: "'s stream \<Rightarrow> bool"   and n :: "nat"   and x :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P"   and "0 < n" shows "almost_everywhere (T x) P = (\<forall>xs y. Graph_Defs.steps (\<lambda>a b. b \<in> set_pmf (K a)) (x # xs @ [y]) \<and> length xs + 1 = n \<longrightarrow> (AE \<omega> in T y. P (xs @- y ## \<omega>)))"
partition_on_split1: fixes A :: "'b set"   and P :: "'b set set"   and Q :: "'b set set" assumes "partition_on A (P \<union> Q)" shows "partition_on (\<Union> P) P"
coboundedI1: fixes c :: "'a Word.word"   and a :: "'a Word.word"   and b :: "'a Word.word" assumes "c \<le>s a" shows "c \<le>s signed.max a b"
finite_fvi: fixes b :: "nat"   and \<phi> :: "formula" shows "finite (fvi b \<phi>)"
set_cases: fixes e :: "'c"   and a :: "'c llist" assumes "e \<in> lset a" obtains z2 :: "'c llist" where "a = LCons e z2__"   | z1 :: "'c" and z2 :: "'c llist" where "a = LCons z1 z2" and "e \<in> lset z2"
dist_les_4: fixes cfun' :: "ValD \<rightarrow> ValD" shows "ValTT \<notsqsubseteq> ValF\<cdot>cfun'"
hyps: shows "1 + int n gchoose n = 1 + int n"
Smith_rw: shows "Smith_1xn_aux (Suc i) A (S, Q) = Smith_1xn_aux i A (S * make_mat (dim_col A) (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then S $$ (0, 0) else S $$ (0, Suc i))))), Q * make_mat (dim_col A) (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then S $$ (0, 0) else S $$ (0, Suc i))))))"
feasible_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "feasible t = (\<forall>P b. bounded_by b P \<and> nneg P \<longrightarrow> bounded_by b (t P) \<and> nneg (t P))"
pd_intro: fixes l :: "'n"   and k :: "'n"   and \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "l pd\<rightarrow> k"   and "is_path \<pi>"   and "\<pi> 0 = k"   and "\<pi> n = local.return" obtains i :: "nat" where "i \<le> n" and "\<pi> i = l"
f: fixes b :: "'a" assumes "b \<in> set bl" shows "0 < f b"
surjective_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "surjective x"   and "surjective y" shows "surjective (x * y)"
matrix_vector_right_distrib_diff: fixes M :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "('a, 'c) Finite_Cartesian_Product.vec"   and w :: "('a, 'c) Finite_Cartesian_Product.vec" shows "M *v (v - w) = M *v v - M *v w"
representation_extend: fixes basis :: "'b set"   and v :: "'b"   and basis' :: "'b set" assumes "vs1.independent basis"   and "v \<in> vs1.span basis'"   and "basis' \<subseteq> basis" shows "vs1.representation basis v = vs1.representation basis' v"
deleteMin_def: fixes q :: "('c, 'd) BinomialHeap" shows "deleteMin q \<equiv> if q = local.empty then local.empty else Abs_BinomialHeap (BinomialHeapStruc.deleteMin (Rep_BinomialHeap q))"
Red_F_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a inference set"   and Red_F_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "intersection_calculus Bot Inf Q entails_q Red_I_q Red_F_q" shows "intersection_calculus.Red_F Q Red_F_q N = (\<Inter>q\<in>Q. Red_F_q q N)"
L2_set_infinite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "infinite A" shows "L2_set f A = 0"
HPhase1or2ReadElse_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d r"   and "HInv4d s q" shows "HInv4d s' q"
check_cs_unsat: 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"   and I :: "'a list" assumes "invariant_simplex' cs J s"   and "check_simplex' s = Inl I" shows "set I \<subseteq> J \<and> minimal_unsat_core (set I) cs"
rep_eq: fixes x :: "'e['f, 'g]"   and xa :: "'e['h, 'g]" shows "(x \<cong>\<^sub>Q xa) = (fromQ x = fromQ xa)"
backward_finite_path_root_move_root: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "backward_finite_path_root r x"   and "point q"   and "q \<le> x\<^sup>\<star> ; r" shows "backward_finite_path_root q x"
distinct_3: fixes x31 :: "com"   and x32 :: "com" shows "SKIP \<noteq> x31;; x32"
fref_def: fixes P :: "'c \<Rightarrow> bool"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
clo: fixes n :: "nat" shows "closed (A n) \<and> \<phi> (A n)"
unfold_while: fixes s :: "state"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and u :: "state" shows "(s -WHILE b DO c\<rightarrow> u) = (s -IF b THEN c; WHILE b DO c ELSE Do (\<lambda>s. {s})\<rightarrow> u)"
Disj_def: fixes xset :: "('b, 'c, 'd) formula set['b]" shows "Disj xset = Formula.Not (Conj (map_bset Formula.Not xset))"
lexord_cons_cons: fixes a :: "'g"   and x :: "'g list"   and b :: "'g"   and y :: "'g list"   and r :: "('g \<times> 'g) set" shows "((a # x, b # y) \<in> lexord r) = ((a, b) \<in> r \<or> a = b \<and> (x, y) \<in> lexord r)"
foldmatrix_transpose: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "\<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d)" shows "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
prefix_suffix: fixes x :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "x = prefix n x \<frown> suffix n x"
Exit_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "\<exists>as. CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as Exit"
distinct_11: fixes x11 :: "'a"   and x12 :: "'b"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "x11 \<leftarrow> x12 \<noteq> Await x71 x72"
intro: fixes \<eta> :: "'d \<Rightarrow> 'e"   and M :: "'d set"   and M' :: "'e set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d"   and composition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and unit' :: "'e" assumes "Set_Theory.map \<eta> M M'"   and "Group_Theory.monoid M composition unit"   and "Group_Theory.monoid M' composition' unit'"   and "monoid_homomorphism_axioms \<eta> M composition unit composition' unit'" shows "monoid_homomorphism \<eta> M composition unit M' composition' unit'"
le_numeral_extra_4: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one one"
simps_13: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and env :: "taylor_model list" shows "approx_tm prec ord I a Pi env = Some (tm_pi prec)"
dvd_add: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times a b"   and "dvd.dvd times a c" shows "dvd.dvd times a (plus b c)"
his_determ_the: fixes n' :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "(n', nid') \<in> his t n nid" shows "(THE nid'. (n', nid') \<in> his t n nid) = nid'"
bij_finite_const_subst_exists: fixes S :: "'a set"   and T :: "'b set"   and U :: "'b set" assumes "finite S"   and "finite T"   and "infinite U" shows "\<exists>\<sigma>. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> (\<lambda>c. Fun c []) ` (U - T)"
pwd_owner_v_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "let (s, S, A, B, C, D) = pwd_owner_v n; SK = (Some S, {A, B}, {C, D}) in pwd_owner_v_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s)) \<and> {SesKey SK} \<in> key_sets (Pwd n) (crypts (Log -` spied s))"
monom_span: fixes p :: "nat" shows "poly_vs.span {monom (1::'b) x |x. x \<le> p} = {x. degree x \<le> p}"
nn_transfer_operator_sum: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_transfer_operator (\<lambda>x. f x + g x) x = nn_transfer_operator f x + nn_transfer_operator g x"
bt_augment_alist_subset_1: fixes q :: "('c, 'd) pq"   and t :: "('d, 'c) bintree" shows "set (pq.alist_of q) \<subseteq> set (pq.alist_of (bt_augment t q))"
ind_nonzero: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "0 < n" shows "ind P n = (if P n then 1::'b else (0::'b))"
Replace_cong: fixes A :: "hf"   and B :: "hf"   and P :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and Q :: "hf \<Rightarrow> hf \<Rightarrow> bool" assumes "A = B"   and "\<And>x y. x \<^bold>\<in> B \<Longrightarrow> P x y = Q x y" shows "Replace A P = Replace B Q"
pequiv_refl: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<simeq> a"
parts_union: fixes H :: "msg set"   and H' :: "msg set" shows "parts (H \<union> H') = parts H \<union> parts H'"
infinite: fixes A :: "'b set" assumes "infinite A" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A = (0::'b)"
emptyLeftProperty: fixes A :: "'a event set"   and t :: "'a event list" shows "\<forall>s. s setinterleaves (([], t), A) \<longrightarrow> s = t"
le_bool_def: fixes P :: "bool"   and Q :: "bool" shows "(P \<le> Q) = (P \<longrightarrow> Q)"
ccpo_mono_simp: fixes f :: "'g \<Rightarrow> 'h" shows "monotone (\<le>) (\<le>) f = mono f"
InvariantEquivalentAfterLearn: fixes F0 :: "Literal list list"   and F :: "Literal list list"   and C :: "Literal list"   and F' :: "Literal list list" assumes "InvariantEquivalent F0 F"   and "formulaEntailsClause F C"   and "F' = F @ [C]" shows "InvariantEquivalent F0 F'"
simps_11: fixes v :: "'b acom"   and va :: "'b acom"   and vb :: "char list"   and vc :: "aexp"   and vd :: "'b" shows "(v;; va \<sqsubseteq> vb ::= vc {vd}) = False"
rem_0_coeff: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and x :: "char list multiset"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p (p, q, add_mset (x, 0) r) (p, q, r)"
fresh_psubst_I1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "fresh zs z X"   and "freshEnv zs z rho" shows "fresh zs z X #[rho]"
trap_2: shows "Lfilled 0 es [Trap] lholed"
subseq_diagonal_rest: fixes k :: "nat" shows "strict_mono (\<lambda>x. fold_reduce k x (k + x))"
remove_all_subs: fixes x :: "'b"   and xs :: "'b list" shows "set (remove_all x xs) \<subseteq> set xs"
order_less_imp_not_eq: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "(x = y) = False"
M_alt: fixes \<pi> :: "'a prefix" shows "M \<pi> = {(i, v). i < progress \<pi> \<and> wf_tuple nfv fv v \<and> (\<exists>\<sigma>. prefix_of \<pi> \<sigma> \<and> sat \<sigma> (map the v) i)}"
simps: fixes Q :: "'d \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "tailrec.fun Q g h x = (if Q x then g x else tailrec.fun Q g h (h x))"
simps: fixes a1 :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and a2 :: "('a, 'b) form"   and a3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" shows "rightPrincipal a1 a2 a3 = (\<exists>C A Ps R. a1 = (Ps, C) \<and> a2 = A \<and> a3 = R \<and> C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<and> (Ps, C) \<in> R)"
med12s_def: fixes t :: "l2_state" shows "med12s t \<equiv> \<lparr>ik = ik t, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t, signalsInit2 = signalsInit2 t, signalsResp2 = signalsResp2 t\<rparr>"
inversion_def: shows "inversion = conjugate \<circ> reciprocal"
finsum_reindex: fixes f :: "'b \<Rightarrow> 'a vec"   and h :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "f \<in> h ` A \<rightarrow> carrier_vec n"   and "inj_on h A" shows "finsum V f (h ` A) = (\<Oplus>\<^bsub>V\<^esub>x\<in>A. f (h x))"