hcju/isabelleps huggingface dataset:
state_rank_distinct: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and p :: "'d"   and q :: "'d"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "p \<noteq> q"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w p n = Some i"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some j" shows "i \<noteq> j"
measurable_envelopeI_countable: fixes M :: "'b measure"   and A :: "nat \<Rightarrow> 'b set"   and E :: "nat \<Rightarrow> 'b set" assumes "\<And>n. measurable_envelope M (A n) (E n)" shows "measurable_envelope M (\<Union> (range A)) (\<Union> (range E))"
divide_less_ennreal: fixes b :: "ennreal"   and a :: "ennreal"   and c :: "ennreal" assumes "b \<noteq> 0"   and "b < \<top>" shows "(a / b < c) = (a < c * b)"
dim_left_null_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.dim (left_null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A"
less_top: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and a :: "'c" assumes "class.order_top less_eq less top" shows "(a \<noteq> top) = less a top"
reachable_execute': fixes a :: "'c \<times> 'd"   and A :: "('c, 'd) nba"   and p :: "'d"   and q :: "'d" assumes "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p"   and "q \<in> reachable A (snd a)" shows "q \<in> reachable A p"
simps_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
indicator_simps_2: fixes x :: "'a"   and S :: "'a set" assumes "x \<notin> S" shows "indicator S x = (0::'b)"
lNact0: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" shows "dynamic_component.lNAct active c t 0 = 0"
assms_1: shows "add_mset s c = add_mset r' c'"
trnl\<^sub>\<epsilon>_def: fixes u :: "'a"   and \<nu> :: "'a" shows "k.trnl\<^sub>\<epsilon> u \<nu> \<equiv> \<l>[u] \<cdot> (k.\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[k, k\<^sup>*, u] \<cdot> (k \<star> \<nu>)"
components_eq_sing_iff: fixes s :: "'c set" shows "(components s = {s}) = (connected s \<and> s \<noteq> {})"
put_closure: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'a" assumes "weak_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v \<in> \<S>\<^bsub>x\<^esub>"
stable_ensures_Int: fixes F :: "'d program"   and C :: "'d set"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> stable C"   and "F \<in> A ensures B" shows "F \<in> C \<inter> A ensures C \<inter> B"
legendre_aux_eq_0: fixes x :: "real"   and p :: "nat" assumes "x < real p" shows "legendre_aux x p = 0"
generalizes_atm_def: fixes A :: "('a, 'b) Term.term"   and B :: "('a, 'b) Term.term" shows "generalizes_atm A B = (\<exists>\<sigma>. subst_atm_abbrev A \<sigma> = B)"
formulaEntailsFormula_def: fixes formula :: "Literal list list"   and formula' :: "Literal list list" shows "formulaEntailsFormula formula formula' \<equiv> \<forall>valuation. model valuation formula \<longrightarrow> model valuation formula'"
LeftDerivation_implies_append: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and c :: "symbol list" assumes "LeftDerivation a D b"   and "LeftDerivation b E c" shows "LeftDerivation a (D @ E) c"
idem_overrider_axioms_def: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "idem_overrider_axioms F \<equiv> \<forall>x. F x x = x"
inorder_baldL: fixes ba :: "bal"   and r :: "('b \<times> bal) tree"   and l :: "('b \<times> bal) tree alt"   and a :: "'b" assumes "ba = Rh \<longrightarrow> r \<noteq> \<langle>\<rangle>"   and "avl r" shows "Tree2.inorder (tree (baldL l a ba r)) = Tree2.inorder (tree l) @ a # Tree2.inorder r"
space_pair_measure: fixes A :: "'b measure"   and B :: "'c measure" shows "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
equalsffemptyI: fixes A :: "'c fset" assumes "\<And>y. y |\<in>| A \<Longrightarrow> False" shows "A = {||}"
simps_1: fixes v :: "'a"   and v' :: "'a" shows "G.is_path v [] v' = (v = v' \<and> v' \<in> G.V)"
strong_complement_character: fixes x :: "i" assumes "x \<noteq> u" shows "\<forall>v. P v (\<midarrow> x) = (\<not> O v x)"
lens_comp_indep_cong_left: fixes Z :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'd"   and Y :: "'g \<Longrightarrow> 'd" assumes "mwb_lens Z"   and "X ;\<^sub>L Z \<bowtie> Y ;\<^sub>L Z" shows "X \<bowtie> Y"
set_pmf_iff: fixes x :: "'d"   and M :: "'d pmf" shows "(x \<in> set_pmf M) = (pmf M x \<noteq> 0)"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'd" assumes "i < nr"   and "j < nc" shows "Matrix.mat nr nc f $$ (i, j) = f (i, j)"
nn_integral_snd_count_space: fixes f :: "'b \<times> 'a \<Rightarrow> ennreal" shows "(\<Sum>\<^sup>+ y. \<Sum>\<^sup>+ x. f (x, y)) = integral\<^sup>N (count_space UNIV) f"
simps_2: fixes e :: "'a allegorical_term" shows "translation (A_Cnv e) = map_graph_fn (translation e) (\<lambda>x. if x < 2 then 1 - x else x)"
eq_iff_iszero_diff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = iszero (x - y)"
apply_outputs_unupdated: fixes ia :: "nat"   and r :: "nat"   and P :: "'a aexp list"   and j :: "'a \<Rightarrow> value option"   and v :: "'a aexp" assumes "ia \<noteq> r"   and "ia < length P" shows "apply_outputs P j ! ia = apply_outputs (P[r := v]) j ! ia"
contf: shows "continuous_map (subtopology X {x \<in> topspace X. p x \<le> q x}) Y f"
simps: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
Abs_resumption_inverse: fixes y :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "y \<in> resumption" shows "Rep_resumption (Abs_resumption y) = y"
assigneq_axiom_def: shows "assigneq_axiom \<equiv> \<langle> xid1 := Const fid1 \<rangle> pusall \<leftrightarrow> Exists xid1 (Equals (Var xid1) (Const fid1) && pusall)"
cwf_Link: fixes l :: "'link" shows "cwf (Link l) = cwf (pointsTo l)"
flow_continuous: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "isCont (flow0 x0) t"
deriv: fixes x :: "real" assumes "x \<in>\<^sub>r I ! 0" shows "isDERIV 0 f ((map real_of_float xs)[0 := x])"
s_prod_idr: fixes x :: "('a \<times> 'a set) set" shows "x \<cdot> 1\<^sub>\<sigma> = x"
disc_69: fixes x9 :: "'c ltl" shows "is_LTLFinal (F x9)"
paths_second_not_v1: fixes xs :: "'a list" assumes "xs \<in> paths" shows "hd (tl xs) \<noteq> v1"
sdrop_snth: fixes n :: "nat"   and s :: "'c stream"   and m :: "nat" shows "sdrop n s !! m = s !! (n + m)"
StableI: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> A Co A" shows "F \<in> Stable A"
access_mhead: fixes m :: "methd" shows "access (mhead m) = access m"
noFaultn_startD': fixes t :: "('d, 'e) xstate"   and f :: "'e"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and n :: "nat" assumes "t \<noteq> Fault f"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "s \<noteq> Fault f"
simps_2: shows "real_alg_2_list_hd_intern [] = of_rat_2 0"
FvarsT_substT: fixes s :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "s \<in> atrm"   and "t \<in> trm"   and "x \<in> var" shows "FvarsT (substT s t x) = FvarsT s - {x} \<union> (if x \<in> FvarsT s then FvarsT t else {})"
wf: shows "length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns"
initD_def: fixes s :: "'b" shows "Dijkstra.initD s \<equiv> (\<lambda>_. \<infinity>)(s := 0::'c)"
convex_hull_subset_affine_hull: fixes s :: "'c set" shows "convex hull s \<subseteq> affine hull s"
RPCNext_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCNext send rcv rst p \<equiv> RPCFwd send rcv rst p \<or> RPCReject send rcv rst p \<or> RPCFail send rcv rst p \<or> RPCReply send rcv rst p"
strict_monoI_Suc: fixes f :: "nat \<Rightarrow> 'b" assumes "\<And>n. f n < f (Suc n)" shows "strict_mono f"
shows_vec_def: fixes v :: "'b vec" shows "shows_vec v \<equiv> shows (list_of_vec v)"
circuit_inD_aux: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.circuit_in \<E> X" shows "indep_system.circuit \<E> (\<E>.indep_in \<E>) X"
simps_2: fixes bdd' :: "bool list bdd"   and bdds :: "bool list bdd list"   and b :: "bool"   and bs :: "bool list"   and bdd :: "bool list bdd" shows "subsetbdd (bdd' # bdds) (b # bs) bdd = (if b then subsetbdd bdds bs (bdd_binop bv_or bdd bdd') else subsetbdd bdds bs bdd)"
assms_2: shows "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d"
pnTimes_not_Zero_or_Plus: fixes x :: "'a rexp"   and r :: "'a rexp" assumes "\<not> is_Zero x"   and "\<not> is_Plus x" shows "pnTimes x r = Times x r"
S_def: fixes h :: "'pub_input"   and e :: "'challenge" shows "S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S_raw h e)"
simps_2: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and used :: "'c set" shows "witness_list (p # ps) used = (case proper_dia p of None \<Rightarrow> witness_list ps used | Some q \<Rightarrow> let i = SOME i. i \<notin> used in (\<^bold>@ i q) # (\<^bold>\<diamond> Nom i) # witness_list ps ({i} \<union> used))"
vconst_on_eq_vtimes: fixes A :: "V"   and c :: "V" shows "vconst_on A c = A \<times>\<^sub>\<circ> set {c}"
sep_dom2_aux2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and z :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "x \<in> dom (Cp (list2FWpolicy (separate (y # z)))) \<longrightarrow> x \<in> dom (Cp (list2FWpolicy (y # z)))"   and "x \<in> dom (Cp (list2FWpolicy (a # separate (y # z))))" shows "x \<in> dom (Cp (list2FWpolicy (a \<oplus> y # z)))"
assms_2: shows "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"
Compl_atLeast: fixes k :: "'b" shows "- {k..} = {..<k}"
K: fixes k' :: "stack" assumes " k \<mapsto> k' "   and "r = t \<star> k'" shows "P"
Perp_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D \<or> B A Perp C D \<or> A B Perp D C \<or> B A Perp D C \<or> C D Perp A B \<or> C D Perp B A \<or> D C Perp A B \<or> D C Perp B A" shows "A B Perp C D"
gprod_2_1_def: fixes R :: "('a \<times> 'a) set" shows "gprod_2_1 R \<equiv> {((a, b), c, d). a = d \<and> (b, c) \<in> R \<or> b = c \<and> (a, d) \<in> R}"
v: shows "\<forall>n\<in>V. n \<noteq> v \<longrightarrow> adjacent v n"
exhaust_sel: fixes prefix_match :: "'b prefix_match" obtains "prefix_match = PrefixMatch (pfxm_prefix prefix_match) (pfxm_length prefix_match)"
starset_Int: fixes A :: "'b set"   and B :: "'b set" shows "*s* (A \<inter> B) = *s* A \<inter> *s* B"
Idl_subset_eq_dvd: fixes k :: "int"   and l :: "int" shows "(Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) = (l dvd k)"
assms: shows "LeftDerivationFix [A] i D j \<gamma>"
dist_execution_axioms_def: fixes events :: "'a \<Rightarrow> ('a, 'b) event list" shows "dist_execution_axioms events \<equiv> ((\<forall>i s m. dist_execution_preliminary.event_at events i (Receive s m) \<longrightarrow> dist_execution_preliminary.event_at events s (Send m)) \<and> (\<forall>i j s m. dist_execution_preliminary.event_at events i (Receive s m) \<longrightarrow> dist_execution_preliminary.event_at events j (Receive s m) \<longrightarrow> fst i = fst j \<longrightarrow> i = j)) \<and> acyclicP (dist_execution_preliminary.happened_immediately_before events) \<and> (\<forall>m s i j i'. dist_execution_preliminary.event_at events (i, j) (Receive s m) \<longrightarrow> i' \<in> deps m \<longrightarrow> (\<exists>s' j' m'. dist_execution_preliminary.event_at events (i, j') (Receive s' (Insert m')) \<and> j' < j \<and> I m' = i')) \<and> (\<forall>m i. dist_execution_preliminary.event_at events i (Send m) \<longrightarrow> (\<exists>n \<sigma>. Inr m = dist_execution_preliminary.state events i \<bind> (\<lambda>s. create_insert s n \<sigma> i)) \<or> (\<exists>n. Inr m = dist_execution_preliminary.state events i \<bind> (\<lambda>s. create_delete s n)))"
sgn2_eq_1_sub_arg: shows "sgn2 = (-) 1"
simps_6: fixes bs :: "'a list"   and a :: "poly"   and b :: "poly" shows "\<lparr>Mul a b\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>a\<rparr>\<^sub>p\<^bsup>bs\<^esup> * \<lparr>b\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
dg_digraph_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "digraph \<beta> \<AA>"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> mat nr nc (\<lambda>ij. 0::'c)"
compactin_absolute: fixes X :: "'c topology"   and S :: "'c set" shows "compactin (subtopology X S) S = compactin X S"
exact_seq_cons_iff: fixes G :: "'b monoid"   and K :: "'b monoid"   and H :: "'b monoid"   and l :: "'b monoid list"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and q :: "('b \<Rightarrow> 'b) list" shows "exact_seq (G # K # H # l, g # h # q) = (exact_seq (K # H # l, h # q) \<and> Group.group G \<and> g \<in> hom K G \<and> kernel K G g = h ` carrier H)"
expands_to_cong': fixes f :: "real \<Rightarrow> real"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g' :: "real \<Rightarrow> real" assumes "(f expands_to MS xs g) basis"   and "\<forall>\<^sub>F x in at_top. g x = g' x" shows "(f expands_to MS xs g') basis"
lossless_map_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf" shows "lossless_spmf (map_spmf f p) = lossless_spmf p"
genPrefix_imp_nth: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "i < length xs"   and "(xs, ys) \<in> genPrefix r" shows "(xs ! i, ys ! i) \<in> r"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_resumption return bind pause" shows "monad_resumption_axioms bind pause"
idempotent_def: fixes L :: "('d, 'e) gorder_scheme"   and f :: "'d \<Rightarrow> 'd" shows "Idem\<^bsub>L\<^esub> f \<equiv> \<forall>x\<in>carrier L. f (f x) .=\<^bsub>L\<^esub> f x"
tm_wf_t_twice_compile: shows "tm_wf0 t_twice_compile"
model_to_dimacs_pos_nin: fixes n_vars :: "nat"   and x :: "nat"   and M :: "nat \<Rightarrow> bool"   and a :: "nat" assumes "n_vars \<le> x" shows "- int x \<notin> set (model_to_dimacs_model M [a..<n_vars])"
prefix_imp_pfixGe: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<le> ys" shows "xs pfixGe ys"
empty_nz: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m Empty # rs \<Rightarrow>\<^sub>z t"
fmran_of_list: fixes m :: "('d \<times> 'c) list" shows "fmran (fmap_of_list m) = snd |`| fset_of_list (AList.clearjunk m)"
F_def: shows "F = make\<Pi>\<^sub>1 (\<lambda>u s w. u = \<nu>\<upsilon> a)"
span_li_not_depend_1: fixes S :: "'e set"   and N :: "'e set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N W" shows "LinearCombinations.module.span K (W.vs N) S = W.module.span S"
wt_app_eff_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list" shows "wt_app_eff r app step \<tau>s = (\<forall>p<length \<tau>s. app p (\<tau>s ! p) \<and> (\<forall>(q, \<tau>)\<in>set (step p (\<tau>s ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q))"
weak_le_iff_join: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq> y) = (x .= x \<sqinter> y)"
mult_lcm_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "c * lcm a b = unit_factor c * lcm (c * a) (c * b)"
chamber_in_image: fixes y :: "'b set" assumes "y \<in> f \<turnstile> X"   and "codomain.chamber y" shows "SimplicialComplex.maxsimp (f \<turnstile> X) y"
lunit_coherence1: fixes f :: "'c" assumes "C.ide f" shows "\<l>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (unit (trg\<^sub>C f) \<star>\<^sub>D F f) = lF f"
add_4: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys"   and "add_mset x X = mset xs"   and "add_mset y Y = mset ys"   and "\<forall>i<length ys. (xs ! i, ys ! i) \<in> ns" shows "thesis"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_new_shadow_root_get_shadow_root_axioms get_shadow_root_locs" shows "l_new_shadow_root_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
dim_le_card: fixes V :: "'c set"   and W :: "'c set" assumes "V \<subseteq> vs2.span W"   and "finite W" shows "vs2.dim V \<le> card W"
totient_gcd: fixes a :: "nat"   and b :: "nat" shows "totient (a * b) * totient (gcd a b) = totient a * totient b * gcd a b"
4: fixes x :: "'a"   and h :: "'a" assumes "x \<in> cball (0::'a) \<delta>"   and "x + h \<in> cball (0::'a) \<delta>" shows "norm (f (x + h) - f x - h) \<le> 1 / 2 * norm h"
x2: shows "flow0 x t2 \<in> {flow0 x t1<--<b}"
dist_metric_completion_limit: fixes x :: "'a metric_completion"   and y :: "'a metric_completion" shows "(\<lambda>n. dist (rep_metric_completion x n) (rep_metric_completion y n)) \<longlonglongrightarrow> dist x y"
relpowp_imp_rtranclp: fixes n :: "nat"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "(P ^^ n) x y" shows "P\<^sup>*\<^sup>* x y"
less_nat_zero_code: fixes n :: "nat" shows "order.greater 0 n = False"
simps_9: fixes \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Until \<phi> I \<psi>) = max (MFOTL.future_reach \<phi>) (MFOTL.future_reach \<psi>) + right I + 1"
CH_def: fixes X :: "'a set" shows "Bossy.CH X = (\<Union>h. BCh h X)"
contour_integral_subcontour_integral: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<le> v" shows "contour_integral (subpath u v g) f = integral {u..v} (\<lambda>x. f (g x) * vector_derivative g (at x))"
predicate2I: fixes P :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
ereal_minus_eq_minus_iff: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" shows "(a - b = a - c) = (b = c \<or> a = \<infinity> \<or> a = - \<infinity> \<and> b \<noteq> - \<infinity> \<and> c \<noteq> - \<infinity>)"
slice1_param_in_slice1: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice1 n"   and "n s-p:V\<rightarrow>\<^bsub>in\<^esub> n'" shows "nx \<in> sum_SDG_slice1 n'"
domain_zero_sum_2: fixes x :: "brick list"   and y :: "brick list" assumes "domain_block x + domain_block y = 0" shows "domain_block y = 0"
endofunctor_axioms: shows "endofunctor (\<cdot>) trg"
cinner_zero_right: fixes x :: "'a" shows "cinner x (0::'a) = 0"
eqOn_FV_sem: fixes \<phi> :: "'aprop dfmla"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<pi>l1 :: "('state \<times> 'aprop set) stream list"   and env :: "pvar \<Rightarrow> nat"   and env1 :: "pvar \<Rightarrow> nat" assumes "wff \<phi>"   and "pointerOf \<pi>l = pointerOf \<pi>l1"   and "cpt (FV \<phi>) env \<pi>l"   and "cpt (FV \<phi>) env1 \<pi>l1"   and "local.eqOn (FV \<phi>) env \<pi>l env1 \<pi>l1" shows "sem \<phi> env \<pi>l = sem \<phi> env1 \<pi>l1"
Map_ide_1: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "Map a = (\<lambda>x\<in>Dom a. x)"
intro: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<subseteq> carrier G"   and "\<And>x y. \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H"   and "\<one>\<^bsub>G\<^esub> \<in> H" shows "submonoid H G"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>last ns'"   and "last ns' \<in> set (predecessors g m)"   and "distinct ns'"   and "set ns' \<subseteq> set ns"   and "m \<notin> set ns'" shows "thesis"
coinduct_strong: fixes R :: "'a process \<Rightarrow> 'a process \<Rightarrow> bool"   and process :: "'a process"   and process' :: "'a process" assumes "R process process'"   and "\<And>process process'. R process process' \<Longrightarrow> isAction process = isAction process' \<and> (isAction process \<longrightarrow> isAction process' \<longrightarrow> prefOf process = prefOf process' \<and> (R (contOf process) (contOf process') \<or> contOf process = contOf process')) \<and> (isChoice process \<longrightarrow> isChoice process' \<longrightarrow> (R (ch1Of process) (ch1Of process') \<or> ch1Of process = ch1Of process') \<and> (R (ch2Of process) (ch2Of process') \<or> ch2Of process = ch2Of process'))" shows "process = process'"
inputs1_def: shows "inputs1 = take (length (input_sizes m1)) inputs"
Rf_mono_on_iia_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "Basis.mono_on (Pow A) (Rf f) = iia_on A f"
int_add_floor: fixes z :: "int"   and x :: "'a" shows "z + \<lfloor>x\<rfloor> = \<lfloor>of_int z + x\<rfloor>"
fwi_innermost_id: fixes i' :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and j' :: "nat"   and j :: "nat" assumes "i' < i" shows "fwi m n k i' j' i j = m i j"
add_cancel_right_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
poly_of_vars_simps_2: fixes \<phi> :: "char list \<Rightarrow> nat"   and xs :: "char list multiset"   and ys :: "char list multiset" assumes "poly_embed \<phi>" shows "poly_embed.poly_of_vars \<phi> (xs + ys) = poly_embed.poly_of_vars \<phi> xs * poly_embed.poly_of_vars \<phi> ys"
borel_UNIV: fixes n :: "nat" shows "space (M n) = UNIV"
ignore_mono: fixes \<Delta> :: "'state \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'b"   and vl1 :: "'c"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'a"   and \<Delta>' :: "'state \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ignore \<Delta> s s1 vl1 a ou s' vl'"   and "\<And>s vl s1 vl1. \<Delta> s vl s1 vl1 \<Longrightarrow> \<Delta>' s vl s1 vl1" shows "ignore \<Delta>' s s1 vl1 a ou s' vl'"