hcju/isabelleps huggingface dataset:
whisker_can_left_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and b :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "horizontal_homs.obj V src b"   and "bicategorical_language.Trg trg t = bicategorical_language.Prim\<^sub>0 b" shows "H b (bicategory.can V H \<a> \<i> src trg u t) = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp (bicategorical_language.Prim\<^sub>0 b) u) (bicategorical_language.Hcomp (bicategorical_language.Prim\<^sub>0 b) t)"
fun_app_lifted_def: fixes f\<^sub>T :: "['d \<Rightarrow> ['c]]"   and x\<^sub>T :: "['d]" shows "f\<^sub>T . x\<^sub>T \<equiv> f\<^sub>T \<bind> (\<bind>) x\<^sub>T"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "normalization_euclidean_semiring_class.gcd (normalization_euclidean_semiring_class.gcd a b) c = normalization_euclidean_semiring_class.gcd a (normalization_euclidean_semiring_class.gcd b c)"
simps: fixes s :: "('c, 'd option) insert_message list"   and k :: "nat" shows "nth_visible s k = (let v = ext_ids (filter is_visible s) in if k < length v then Inr (v ! k) else Inl STR ''Argument k out of bounds.'')"
zeta_partial_sum_le_pos'': assumes "0 < 1 / 2"   and "1 / 2 \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>n. real n powr - (1 / 2)) x\<bar> \<le> c * x powr max 0 (1 - 1 / 2)"
d: shows "0 < stp \<and> crsp ly (abc_step_l (as, lm) (Some ins)) (steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) ires"
add_add_bot: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.aggregation_order plus bot less_eq less" shows "plus (plus x y) bot = plus x y"
assms_4: shows "put \<sigma> v = put \<rho> v"
finite_subst_img_if_finite_dom': fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "finite (subst_domain \<sigma>)" shows "finite (subst_range \<sigma>)"
\<pi>_def: shows "\<pi> \<equiv> path \<sigma>"
span_redundant: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and S :: "'e set" assumes "vector_space scale"   and "x \<in> module.span scale S" shows "module.span scale (insert x S) = module.span scale S"
forget: fixes x :: "name"   and t :: "trm"   and v :: "trm" assumes "x \<sharp> t" shows "t[x::=v] = t"
PiE_eq_empty_iff: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set" shows "(Pi\<^sub>E I F = {}) = (\<exists>i\<in>I. F i = {})"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "B $$ (i, j) \<le> A $$ (i, j)"
inv_can: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "local.inv (can u t) = can t u"
fds_subseries_def: fixes P :: "nat \<Rightarrow> bool"   and f :: "'a fds" shows "fds_subseries P f = fds (\<lambda>n. if P n then fds_nth f n else (0::'a))"
HInfinite_add_ge_zero: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HInfinite"   and "0 \<le> y"   and "0 \<le> x" shows "x + y \<in> HInfinite"
Int_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b tree list \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b tree list" shows "(case Fork x21 x22 of Node \<Rightarrow> f1 | Fork x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> Vset \<alpha>"
deriv_chain: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at x"   and "g field_differentiable at (f x)" shows "deriv (g \<circ> f) x = deriv g (f x) * deriv f x"
leastPD1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "leastP P x" shows "P x"
A: shows "geodesic_subsegment G x s t = geodesic_segment_param G x ` {s..t}"
meet_comm: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<inter> y = y \<inter> x"
rerename_noop: fixes x :: "'var"   and "from" :: "nat"   and V :: "'var set"   and to :: "nat"   and f :: "'var \<Rightarrow> 'var" assumes "x \<notin> freshenLC from ` V" shows "rerename V from to f x = f x"
left_adjoints_compose: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.is_left_adjoint f"   and "Span.is_left_adjoint f'"   and "Span.src f = Span.trg f'" shows "Span.is_left_adjoint (Span.hcomp f f')"
contour_integral_eq: fixes p :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x" shows "contour_integral p f = contour_integral p g"
empty_eq_affine_hull: fixes S :: "'b set" shows "({} = affine hull S) = (S = {})"
hyps_3: shows "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>1"
simps: fixes x :: "nat"   and y :: "nat" shows "lg x y = (if 1 < x \<and> 1 < y \<and> {u. lgR [x, y, u]} \<noteq> {} then Max {u. lgR [x, y, u]} else 0)"
measure_pmf_bind: fixes M :: "'b pmf"   and f :: "'b \<Rightarrow> 'a pmf" shows "measure_pmf (M \<bind> f) = measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x))"
H: shows "(p\<^sub>1 = Infinity \<or> p\<^sub>2 = Infinity \<or> p\<^sub>3 = Infinity) \<or> (p\<^sub>1 = opp p\<^sub>2 \<or> p\<^sub>2 = opp p\<^sub>3) \<or> opp p\<^sub>1 = local.add a p\<^sub>2 p\<^sub>3 \<or> opp p\<^sub>3 = local.add a p\<^sub>1 p\<^sub>2"
prod'_def: shows "prod' = (\<lambda>p. A $$ (p n, n) * signof p * (\<Prod>i = 0..<n. A $$ (p i, i)))"
2: shows "x ; 1 ; x\<^sup>T \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
E_of_succ_def: fixes succ :: "'a \<Rightarrow> 'b set" shows "E_of_succ succ \<equiv> {(u, v). v \<in> succ u}"
IH: shows "Suffixes (A ^^ n) = insert [] (Suffixes A @@ \<Union> ((^^) A ` {..<n}))"
parts_simp: fixes H :: "msg set" assumes "H \<subseteq> range Agent \<union> range Number \<union> range Nonce \<union> range Key \<union> range Hash \<union> range Passwd \<union> range Pri_AgrK \<union> range Pub_AgrK" shows "parts H = H"
maxsimp_vertices: fixes x :: "'a set" assumes "domain.chamber x" shows "order.greater_eq (\<Union> X) x"
int_diff_cases: fixes z :: "int" obtains m :: "nat" and n :: "nat" where "z = int m - int n"
alt_first_time_def: fixes n :: "nat"   and \<rho> :: "'b run"   and K :: "clock"   and \<tau> :: "'b tag_const" assumes "\<forall>m<n. time (Rep_run \<rho> m K) < \<tau>"   and "time (Rep_run \<rho> n K) = \<tau>" shows "first_time \<rho> K n \<tau>"
lens_plus_def: fixes X :: "'c \<Longrightarrow> 'e"   and Y :: "'d \<Longrightarrow> 'e" shows "X +\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. (get\<^bsub>X\<^esub> \<sigma>, get\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> (u, v). put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u\<rparr>"
singleton_eq_iff: fixes a :: "V"   and b :: "V" shows "(set {a} = set {b}) = (a = b)"
null_eqI: fixes n :: "'b" assumes "\<And>f. n \<cdot>\<^sub>D f = n \<and> f \<cdot>\<^sub>D n = n" shows "n = D.null"
lDown_tSngl: assumes "invar l \<and> invar r \<and> lv = Suc (lvl r) \<and> lv = Suc (Suc (lvl l))" shows "thesis"
elimation_rules_strictly_decreasing_meas': fixes \<S>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and \<S>\<^sub>2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" assumes "\<S>\<^sub>1 \<hookrightarrow>\<^sub>e \<S>\<^sub>2" shows "(\<S>\<^sub>2, \<S>\<^sub>1) \<in> measure \<mu>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
ab_1_2: fixes k :: "nat" shows "1 \<le> bb k"
eval\<o>_inject: fixes x :: "\<o>"   and y :: "\<o>" shows "(eval\<o> x = eval\<o> y) = (x = y)"
bij_rename_iff: fixes h :: "'c \<Rightarrow> 'd" shows "bij (rename h) = bij h"
sset_siterate: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "sset (siterate f x) = {(f ^^ n) x |n. True}"
dfa_steps_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and q :: "nat"   and w :: "bool list list" assumes "wf_dfa A n"   and "dfa_is_node A q"   and "list_all (is_alph n) w" shows "dfa_is_node A (dfa_steps A q w)"
typing_alt_simps_2: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "bool"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (BooleanLiteral c) : \<tau>) = (\<tau> = Boolean[1])"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(OK x2 = OK y2) = (x2 = y2)"
b3: fixes v :: "'a" shows "cblinfun_of_mat (M * N) *\<^sub>V v = basis_enum_of_vec (M * N *\<^sub>v vec_of_basis_enum v)"
Lambert_W_le_iff: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "- exp (- 1) \<le> y" shows "(Lambert_W x \<le> Lambert_W y) = (x \<le> y)"
that: fixes c :: "'a" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = c" shows "thesis"
iso_def: fixes f :: "'d" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
ab_2: shows "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t u"
lspasl_magicr: fixes Gamma :: "bool"   and h2 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and Delta :: "bool" assumes "(\<exists>h1 h0. Gamma \<and> (h1,h2\<triangleright>h0) \<and> A h1 \<and> \<not> B h0) \<longrightarrow> Delta" shows "Gamma \<longrightarrow> (A \<longrightarrow>* B) h2 \<or> Delta"
prj_simps_14: shows "C.dom f_src_src.Prj\<^sub>1\<^sub>0 = Chn (f \<star> src f \<star> src f)"
Cons: fixes a :: "'b"   and A :: "'b set"   and l :: "'b list" assumes "a \<in> A"   and "l \<in> lists A" shows "a # l \<in> lists A"
zero_smallest: fixes r :: "('d \<times> 'd) set"   and j :: "'d" assumes "wo_rel r"   and "j \<in> Field r" shows "(zero r, j) \<in> r"
ce_rel_lm_12: fixes x :: "nat"   and y :: "nat"   and r :: "(nat \<times> nat) set" shows "(c_pair x y \<in> ce_rel_to_set r) = ((x, y) \<in> r)"
transfer_WHILEIT: fixes f :: "'a \<Rightarrow> 'mc"   and F :: "'a \<Rightarrow> 'ma"   and I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x. \<alpha> (f x) \<le> F x" shows "\<alpha> (cWHILEIT I b f x) \<le> aWHILEIT I b F x"
bBnd: shows "b \<in> \<int>\<^sub>v \<inter> Bounded_vec Bnd"
semilattice_max_torder: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "torder_on A r" shows "semilattice (max_torder r)"
Ifun_congruent2: shows "congruent2 starrel starrel (\<lambda>F X. starrel `` {\<lambda>n. F n (X n)})"
dehomogenize_homogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "dehomogenize x (homogenize x p) = dehomogenize x p"
less_imp_not_less: fixes x :: "'a list"   and y :: "'a list" assumes "strict_prefix x y" shows "(\<not> strict_prefix y x) = True"
65110: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y \<squnion> (- y \<squnion> x)) = bot"
zero: shows "(\<Sum>i | i \<in> B \<and> i < j. 2 ^ i) div 2 ^ j = 0"
l_def: shows "l \<equiv> limsup (\<lambda>n. ereal (f n / f (Suc n)))"
composition_closed: fixes a :: "'b"   and b :: "'b" assumes "a \<in> A"   and "b \<in> A" shows "multA a b \<in> A"
g: shows "\<forall>i>M. g (Suc i) \<le>\<cdot> g i"
otr: shows "((\<sigma>, \<zeta>), a, \<sigma>', \<zeta>') \<in> automaton.trans (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg)"
set_mset_single: fixes b :: "'b" shows "set_mset {#b#} = {b}"
runProc_pState_id: fixes name :: "String.literal"   and args :: "expr list"   and prog :: "program"   and g :: "'b gState_scheme"   and p :: "pState" shows "snd (runProc name args prog g p) = p"
homeomorphic_subspaces: fixes S :: "'b set"   and T :: "'c set" assumes "subspace S"   and "subspace T"   and "dim S = dim T" shows "S homeomorphic T"
RID\<^sub>GI_2: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r' :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and r :: "'b" assumes "s r' = Some ls"   and "r \<in> RID\<^sub>L ls" shows "r \<in> RID\<^sub>G s"
prv_\<phi>L_eqv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (eqv (\<phi>L \<phi>) (imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>))"
AB_trms_disj: shows "\<forall>t\<in>set A. \<forall>s\<in>set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>). \<Gamma> t = \<Gamma> s \<and> mgu t s \<noteq> None \<longrightarrow> intruder_synth' public arity {} t \<and> intruder_synth' public arity {} s \<or> (\<exists>u\<in>Sec. is_wt_instance_of_cond \<Gamma> t u) \<and> (\<exists>u\<in>Sec. is_wt_instance_of_cond \<Gamma> s u)"
l3_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l3_step4 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xgnx, xgny, xsk} \<and> guessed_frame Rb xgnx = Some gnx \<and> implAuth A B \<langle>Number 1, gnx, Exp Gen (NonceF (Rb $ ny))\<rangle> \<in> ik s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp s\<rparr>}"
max_stepsize_ge_stepsize: fixes t :: "nat \<Rightarrow> real"   and j :: "nat"   and n :: "nat" assumes "grid t"   and "j \<le> n" shows "grid.stepsize t j \<le> grid.max_stepsize t n"
path_distinct: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list" assumes "graph weights"   and "graph.path weights l1 l2 xs" shows "\<exists>xs'. distinct xs' \<and> graph.path weights l1 l2 xs' \<and> map (\<lambda>(s, l, t). l) xs' \<preceq> map (\<lambda>(s, l, t). l) xs"
simps_1: fixes b :: "'a set \<times> 'a set" shows "attack \<N>\<^bsub>b\<^esub> = b"
det_mult: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "Determinant.det (A * B) = Determinant.det A * Determinant.det B"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Inl x1 = Inl y1) = (x1 = y1)"
gcd_0_left: fixes a :: "'a" shows "gcd (0::'a) a = normalize a"
distinct_relpow_impl: fixes L :: "('b \<times> 'b) list"   and new :: "'b list"   and "have" :: "'b list"   and M :: "nat" assumes "distinct L"   and "distinct new"   and "distinct have"   and "distinct (new @ have)" shows "distinct (relpow_impl (\<lambda>as. remdups (map snd (filter (\<lambda>(a, b). a \<in> set as) L))) (\<lambda>xs ys. filter (\<lambda>x. x \<notin> set ys) xs @ ys) (\<lambda>x xs. x \<in> set xs) new have M)"
has_vector_derivative_transform_within: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and S :: "real set"   and d :: "real"   and g :: "real \<Rightarrow> 'b" assumes "(f has_vector_derivative f') (at x within S)"   and "0 < d"   and "x \<in> S"   and "\<And>x'. \<lbrakk>x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g has_vector_derivative f') (at x within S)"
elts_subset_ON: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "elts \<alpha> \<subseteq> ON"
subtyping_append: fixes CT :: "nat \<Rightarrow> classDef option"   and Cs :: "nat list"   and Ds :: "nat list"   and C :: "nat"   and D :: "nat" assumes "CT \<turnstile>+ Cs <: Ds"   and "CT \<turnstile> C <: D" shows "CT \<turnstile>+ (Cs @ [C]) <: (Ds @ [D])"
PROVE_\<Theta>_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and f\<^sub>0 :: "'b"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<Theta> :: "('b \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "PROVE_\<Theta> \<pi> f\<^sub>0 s\<^sub>0 \<Theta> \<equiv> \<forall>P c Q. (f\<^sub>0, P, c, Q) \<in> \<Theta> \<and> P s\<^sub>0 \<longrightarrow> wp \<pi> (c s\<^sub>0) (Q s\<^sub>0) s\<^sub>0"
sub_gpvs_def: fixes \<I> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" shows "sub_gpvs \<I> gpv \<equiv> {x. sub_gpvsp \<I> x gpv}"
IH: fixes E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : T"   and "P,h \<turnstile> l(V := None) (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
cases: fixes a1 :: "nat \<Rightarrow> 'a"   and a2 :: "nat"   and prob :: "('a, 'b, 'c, 'd) ifc_problem" assumes "(a1, a2) \<in> IFC_def.scop prob" obtains \<pi> :: "nat \<Rightarrow> 'a" and n :: "nat" where "a1 = \<pi>" and "a2 = n" and "(\<pi>, n) \<in> IFC_def.scp prob" and "\<pi> n \<in> dom (IFC_def.att prob)"
wall_crossings_are_walls: fixes H :: "'a set set set"   and Cs :: "'a set list" assumes "H \<in> set (wall_crossings Cs)"   and "H \<noteq> {}" shows "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}})"
simps_2: fixes e :: "event"   and s :: "event list"   and r :: "room" shows "isin (e # s) r = (let G = isin s r in case e of Check_in g r c \<Rightarrow> G | Enter g r' c \<Rightarrow> if r' = r then {g} \<union> G else G | Exit g r' \<Rightarrow> if r' = r then G - {g} else G)"
drop_Cons': fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
transpose_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $ j $ i)"
openInjectAuxAux: fixes xvec1 :: "name list"   and xvec2 :: "name list"   and yvec :: "name list" assumes "length (xvec1 @ xvec2) = length yvec" shows "\<exists>yvec1 yvec2. yvec = yvec1 @ yvec2 \<and> length xvec1 = length yvec1 \<and> length xvec2 = length yvec2"
\<I>'_3: shows "subst_domain \<I>' = {x \<in> vars\<^sub>s\<^sub>t S. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X}"
lm123: fixes pre :: "'b \<Rightarrow> bool"   and list :: "'b list" shows "set (filterpositions2 pre list) \<subseteq> {0..<length list}"
skolemize1: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)" shows "\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x)"
cil_set_induct_fix\<alpha>: fixes w :: "'a list"   and w1 :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and w2 :: "'a list"   and P :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b set \<times> 'b set) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2"   and "P [] \<alpha> [] []"   and "\<And>e w' w1' w2. \<lbrakk>w' \<in> w1' \<otimes>\<^bsub>\<alpha>\<^esub> w2; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w2) = {}; P w' \<alpha> w1' w2\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> (e # w1') w2"   and "\<And>e w' w2' w1. \<lbrakk>w' \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2'; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w1) = {}; P w' \<alpha> w1 w2'\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> w1 (e # w2')" shows "P w \<alpha> w1 w2"
not_zero_m: shows "\<not> is_zero_row_upt_k m k A"
1: shows "length (M1 \<otimes> M2 \<otimes> M3) = length M1 * length M2 * length M3"
compatible_iff_seq: fixes g :: "'a"   and f :: "'a" shows "(domains g \<inter> codomains f \<noteq> {}) = seq g f"
eq: shows "lappend (ltake (llength ys) xs) (lappend (ldrop (llength ys) xs) xs') = lappend ys zs"
lt_plus_lessI': fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p + q \<noteq> 0"   and "lt q = lt p"   and "lc p + lc q = (0::'c)" shows "lt (p + q) \<prec>\<^sub>t lt p"
set_Macaulay_red: fixes fs :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "set (punit.Macaulay_red (punit.Keys_to_list fs) fs) = set (punit.Macaulay_list fs) - {p. punit.lt p \<in> punit.lt_set (set fs)}"
bounded_linear_inner_right: fixes a :: "'a" shows "bounded_linear ((\<bullet>) a)"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "x0 \<in> X"
iso_tuple_fst_update_def: fixes isom :: "('a, 'b, 'c) tuple_isomorphism"   and f :: "'b \<Rightarrow> 'b" shows "iso_tuple_fst_update isom f = abst isom \<circ> apfst f \<circ> repr isom"
set_object_value_invariant: fixes s :: "state_t"   and ob :: "obj_id_t"   and va :: "obj_t" shows "atomic_step_invariant s = atomic_step_invariant (set_object_value ob va s)"
list_all2_map1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and as :: "'d list"   and bs :: "'c list" shows "list_all2 P (map f as) bs = list_all2 (\<lambda>x. P (f x)) as bs"
induct: fixes x1 :: "'b \<Rightarrow> ('c, 'd) trm"   and x2 :: "('c + 'b, 'd) trm"   and P :: "('b \<Rightarrow> ('c, 'd) trm) \<Rightarrow> ('c + 'b, 'd) trm \<Rightarrow> bool" assumes "TadmitFFO x1 x2"   and "\<And>\<sigma> \<theta>. \<lbrakk>TadmitFFO \<sigma> \<theta>; P \<sigma> \<theta>; NTUadmit \<sigma> \<theta> UNIV\<rbrakk> \<Longrightarrow> P \<sigma> (Differential \<theta>)"   and "\<And>\<sigma> args f. \<lbrakk>\<And>i. TadmitFFO \<sigma> (args i); \<And>i. P \<sigma> (args i)\<rbrakk> \<Longrightarrow> P \<sigma> ($f (Inl f) args)"   and "\<And>\<sigma> args f. \<lbrakk>\<And>i. TadmitFFO \<sigma> (args i); \<And>i. P \<sigma> (args i); dfree (\<sigma> f)\<rbrakk> \<Longrightarrow> P \<sigma> ($f (Inr f) args)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitFFO \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitFFO \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Plus \<theta>1 \<theta>2)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitFFO \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitFFO \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Times \<theta>1 \<theta>2)"   and "\<And>\<sigma> x. P \<sigma> (Var x)"   and "\<And>\<sigma> r. P \<sigma> (Const r)" shows "P x1 x2"
comm_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "multA a b = multA b a"
space_measure_spmf: fixes p :: "'a spmf" shows "space (measure_spmf p) = UNIV"
reduced_ifex: fixes b :: "'d bool_expr" shows "reduced (ifex_of b) {}"
rwof_init: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" assumes "nofail (m0 \<bind> WHILE cond step)" shows "m0 \<le> SPEC (rwof m0 cond step)"
IH2: assumes "\<Gamma> \<turnstile> e2 : Low"   and "x \<in> fv e2" shows "\<Gamma> x = Some Low"
mult_scalar_monomial: fixes t :: "'a"   and c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "MPoly_Type_Class.punit.mult_scalar (monomial c t) p = MPoly_Type_Class.punit.monom_mult c t p"
mbr_method_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "mbr (method sig m) = mdecl (sig, mthd m)"
density_states_def: fixes dims :: "nat list" shows "state_sig.density_states dims = {\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims). partial_density_operator \<rho>}"
simple_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "simple_group G \<equiv> Group.group G \<and> simple_group_axioms G"