hcju/isabelleps huggingface dataset:
vle_trans: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<unlhd> y"   and "y \<unlhd> z" shows "x \<unlhd> z"
indecomposable_imp_Ord: fixes \<alpha> :: "V" assumes "indecomposable \<alpha>" shows "Ord \<alpha>"
assms_2: shows "\<forall>\<^sub>F x in at_top. f x \<le> g x"
last_prog_to_last_prog_same: fixes p' :: "'b"   and sb :: "'b memref list"   and p :: "'b" assumes "last_prog p' sb = p" shows "last_prog p sb = p"
coeff_mult_semiring_closed: fixes p :: "'a poly"   and q :: "'a poly"   and i :: "nat" assumes "\<And>i. coeff p i \<in> R"   and "\<And>i. coeff q i \<in> R" shows "coeff (p * q) i \<in> R"
consistent_Extend: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm" assumes "consistent S"   and "finite (params S)"   and "infinite UNIV" shows "consistent (Extend S f)"
defaultlabel_def: shows "defaultlabel = (defaultexpr, defaultguard, defaultaction)"
head_\<omega>_def: fixes M :: "hmultiset" shows "head_\<omega> M = (if M = 0 then 0 else \<omega>^ (Max_mset (hmsetmset M)))"
sin_i_times: fixes z :: "complex" shows "sin (\<i> * z) = \<i> * ((exp z - inverse (exp z)) / 2)"
MERGE: shows "\<Gamma>1' \<or>\<^sub>A \<Gamma>2' \<Longrightarrow>\<^sub>t \<Gamma>'"
arith_streamValue9aux: fixes t :: "nat"   and j :: "nat"   and d :: "nat"   and i :: "nat"   and k :: "nat" shows "Suc (t + (j + d) + (i + k)) = Suc (t + i + k + d + j)"
interchange_rows_code_nth: fixes A :: "(('b, 'c) vec, 'a) vec"   and a :: "'a"   and b :: "'a" shows "($) (interchange_rows A a b) = interchange_rows_row A a b"
ipurge_tr_aux_cons: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'a"   and xs :: "'a list" shows "ipurge_tr_aux I D U (x # xs) = (if \<exists>v\<in>U. (v, D x) \<in> I then ipurge_tr_aux I D (insert (D x) U) xs else x # ipurge_tr_aux I D U xs)"
elims: fixes x :: "nat"   and xa :: "real mpoly"   and xb :: "real mpoly"   and xc :: "atom fm"   and y :: "atom fm" assumes "linear_substitution_fm x xa xb xc = y" obtains var :: "nat" and b :: "real mpoly" and c :: "real mpoly" and F :: "atom fm" where "x = var" and "xa = b" and "xb = c" and "xc = F" and "y = linear_substitution_fm_helper var b c F 0"
15: shows "infs (gaccepting A) (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p)))"
r_smn_aux_def: fixes n :: "nat"   and m :: "nat" shows "r_smn_aux n m \<equiv> Cn (Suc m) (r_list_encode (m + n)) (map (\<lambda>i. Cn (Suc m) (r_code_constn n) [recf.Id (Suc m) i]) [0..<Suc m] @ map (\<lambda>i. r_constn m (code_id n i)) [0..<n])"
is_valid_problem_strips_operator_variable_sets_2: fixes \<Pi> :: "'c strips_problem"   and op :: "'c strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (add_effects_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
Vinv1_inductive_1: shows "init v_TS \<subseteq> Vinv1"
SUP_image: fixes Sup :: "'c set \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "Sup (g ` f ` A) = Sup ((g \<circ> f) ` A)"
closure_of_subset: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "S \<subseteq> X closure_of S"
start_of_suc_dec: fixes k :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and e :: "nat" assumes "k < length ap"   and "ap ! k = Dec n e" shows "start_of (layout_of ap) (Suc k) = start_of (layout_of ap) k + 2 * n + 16"
right_neutral: fixes a :: "'a" shows "a OR (0::'a) = a"
sup_derivable_fp: fixes p :: "'a" assumes "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"   and "extreme_bound A (\<sqsubseteq>) (Collect derivable) p" shows "f p = p"
sifum_security_axioms: shows "sifum_security dma \<C>_vars eval\<^sub>w"
that: fixes n\<^sub>1 :: "'node"   and ns\<^sub>1 :: "'node list"   and n\<^sub>2 :: "'node"   and ns\<^sub>2 :: "'node list" assumes "var g r \<in> oldDefs g n\<^sub>1"   and "g \<turnstile> n\<^sub>1-ns\<^sub>1\<rightarrow>defNode g r"   and "var g r \<in> oldDefs g n\<^sub>2"   and "g \<turnstile> n\<^sub>2-ns\<^sub>2\<rightarrow>defNode g r"   and "set (butlast ns\<^sub>1) \<inter> set (butlast ns\<^sub>2) = {}" shows "thesis"
simps_2: fixes m :: "nat" shows "akiyama_tanigawa_step_aux m [] = []"
dickson_lessI: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and m :: "nat"   and t :: "'a" assumes "d s \<le> m"   and "d t \<le> m"   and "s \<prec> t" shows "dickson_less d m s t"
finfun_right_compose: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a" assumes "g \<in> finfun"   and "inj f" shows "g \<circ> f \<in> finfun"
hides_eq_sigD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G\<turnstile> new hides old" shows "msig old = msig new"
continuous_inverse_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "compact_space X"   and "Hausdorff_space Y"   and "continuous_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x"   and "S \<subseteq> f ` topspace X" shows "continuous_map (subtopology Y S) X g"
lucas_lehmer_mult_def: fixes m :: "nat" shows "lucas_lehmer_mult m = (\<lambda>(a, b) (c, d). ((a * c + 3 * b * d) mod m, (a * d + b * c) mod m))"
wf_tuple_append: fixes a :: "nat"   and A :: "nat set"   and xs :: "'a option list"   and b :: "nat"   and ys :: "'a option list" assumes "wf_tuple a {x \<in> A. x < a} xs"   and "wf_tuple b {x - a |x. x \<in> A \<and> a \<le> x} ys" shows "wf_tuple (a + b) A (xs @ ys)"
ptensor_mat_carrier: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "tensor_P m1 m2 \<in> carrier_mat ps2_P.d0 ps2_P.d0"
assms: shows "x \<in> \<real>\<^sub>\<ge>\<^sub>0"
term_brackets_dist: fixes vs :: "nat list"   and t :: "dB"   and t' :: "dB" assumes "term_brackets vs t = Some t'" shows "var_dist vs t' \<leftrightarrow> t"
ordIso_def: shows "ordIso = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
t: shows "g ^ totient (p ^ (k - 1)) = p ^ (k - 1) * t + 1"
par_comm: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
Left_equals_Right_coset_implies_normality: assumes "\<And>g. g \<in> G \<Longrightarrow> g \<cdot>| H = H |\<cdot> g" shows "normal_subgroup H G (\<cdot>) \<one>"
inf_Int_eq: fixes R :: "'a set"   and S :: "'a set" shows "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
elims_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and x :: "'b btree"   and xa :: "'b"   and y :: "bool" assumes "split split"   and "split.isin split x xa = y" obtains ya :: "'b" where "x = Leaf" and "xa = ya__" and "\<not> y"   | ts :: "('b btree \<times> 'b) list" and t :: "'b btree" and ya :: "'b" where "x = Node ts t" and "xa = ya" and "y = (case split ts ya of (x, []) \<Rightarrow> split.isin split t ya | (x, (sub, sep) # rs) \<Rightarrow> if ya = sep then True else split.isin split sub ya)"
int_or_not_bitORN_num: fixes n :: "num"   and m :: "num" shows "numeral n OR NOT (numeral m) = - numeral (bitORN_num n m)"
dim_def: fixes p :: "'a"   and V :: "(('a \<Rightarrow> real) \<Rightarrow> real) set" shows "tangent_space.dim p V = (if \<exists>b\<subseteq>tangent_space p. independent b \<and> span b = span V then card (SOME b. b \<subseteq> tangent_space p \<and> independent b \<and> span b = span V) else 0)"
smult_l_minus: fixes a :: "'a"   and x :: "'c" assumes "a \<in> carrier K"   and "x \<in> carrier V" shows "\<ominus>\<^bsub>K\<^esub> a \<odot>\<^bsub>V\<^esub> x = \<ominus>\<^bsub>V\<^esub> (a \<odot>\<^bsub>V\<^esub> x)"
inject: fixes x1 :: "'m"   and x2 :: "'n"   and y1 :: "'m"   and y2 :: "'n" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
lookup_eq_order_irrelevant: fixes pairs :: "('a \<times> 'b) list"   and pairs' :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst pairs)"   and "distinct (map fst pairs')"   and "set pairs = set pairs'" shows "lookup (\<lambda>x. x = k) pairs = lookup (\<lambda>x. x = k) pairs'"
eq_of_exps: fixes a1 :: "nat"   and b1 :: "nat"   and i :: "nat"   and a2 :: "nat"   and b2 :: "nat"   and j :: "nat" shows "(fst (rep_nat_term (Term_Order.of_exps a1 b1 i)) = fst (rep_nat_term (Term_Order.of_exps a2 b2 j))) = (a1 = a2 \<and> b1 = b2)"
root: fixes x :: "name"   and t :: "Lambda_Z.term"   and s :: "Lambda_Z.term" assumes "atom x \<sharp> t" shows "App (Abs x s) t \<rightarrow>\<^sub>\<beta> subst x t s"
320: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> x = bot"
a_d_iff: fixes x :: "'a" shows "a x = (1::'a) \<sqinter> - (x \<cdot> \<top>)"
presheaf_of_rings_def: 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" shows "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> topological_space S is_open \<and> presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<T>e.map f) = Cop.arr f"
assms_1: shows "(f has_field_derivative D) (at x within A)"
pred_subset_eq2: fixes R :: "('c \<times> 'd) set"   and S :: "('c \<times> 'd) set" shows "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)"
empty_def: fixes ko :: "'o" shows "local.empty ko = oalist_of_list ([], ko)"
prime_product_def: fixes n :: "nat" shows "prime_product n = (\<forall>p. prime p \<longrightarrow> multiplicity p n \<le> 1)"
leq_refl: fixes x :: "'b spmf" shows "ord_spmf (=) x x"
n\<^sub>1n\<^sub>2: shows "n\<^sub>1 \<le> n\<^sub>2"
simps_2: fixes xs :: "'b list"   and k :: "nat"   and n :: "nat" shows "list_slice_aux xs k (Suc n) = xs \<down> k # list_slice_aux (xs \<up> k) k n"
singlPart_surj: fixes f :: "'d \<Rightarrow> 'c"   and I :: "'d set"   and J :: "'c set" shows "((`) f ` singlPart I = singlPart J) = (f ` I = J)"
solves_ode_subset: fixes x :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set"   and S :: "real set" assumes "(x solves_ode f) T X"   and "S \<subseteq> T" shows "(x solves_ode f) S X"
filter_append: fixes P :: "'b \<rightarrow> tr"   and xs :: "['b]"   and ys :: "['b]" shows "filter\<cdot>P\<cdot>(xs ++ ys) = filter\<cdot>P\<cdot>xs ++ filter\<cdot>P\<cdot>ys"
closedin_singleton: fixes U :: "'a set"   and a :: "'a" shows "closedin (top_of_set U) {a} = (a \<in> U)"
ins_type_1: fixes t :: "'b bro"   and h :: "nat"   and x :: "'b" assumes "t \<in> B h" shows "ins x t \<in> Bp h"
f''_imp_f': fixes C :: "real set"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and f'' :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "convex C"   and "\<And>x. x \<in> C \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x \<in> C \<Longrightarrow> (f' has_real_derivative f'' x) (at x)"   and "\<And>x. x \<in> C \<Longrightarrow> 0 \<le> f'' x"   and "x \<in> C"   and "y \<in> C" shows "f' x * (y - x) \<le> f y - f x"
bangInduct: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Prop :: "'a \<Rightarrow> ccs \<Rightarrow> act \<Rightarrow> ccs \<Rightarrow> bool"   and \<C> :: "'a" assumes "!P \<longmapsto>\<alpha> \<prec> P'"   and "\<And>\<alpha> P' \<C>. P \<longmapsto>\<alpha> \<prec> P' \<Longrightarrow> Prop \<C> (P \<parallel> !P) \<alpha> (P' \<parallel> !P)"   and "\<And>\<alpha> P' \<C>. \<lbrakk>!P \<longmapsto>\<alpha> \<prec> P'; \<And>\<C>. Prop \<C> !P \<alpha> P'\<rbrakk> \<Longrightarrow> Prop \<C> (P \<parallel> !P) \<alpha> (P \<parallel> P')"   and "\<And>a P' P'' \<C>. \<lbrakk>P \<longmapsto>a \<prec> P'; !P \<longmapsto>coAction a \<prec> P''; \<And>\<C>. Prop \<C> !P (coAction a) P''; a \<noteq> \<tau>\<rbrakk> \<Longrightarrow> Prop \<C> (P \<parallel> !P) (\<tau>) (P' \<parallel> P'')"   and "\<And>\<alpha> P' \<C>. \<lbrakk>P \<parallel> !P \<longmapsto>\<alpha> \<prec> P'; \<And>\<C>. Prop \<C> (P \<parallel> !P) \<alpha> P'\<rbrakk> \<Longrightarrow> Prop \<C> !P \<alpha> P'" shows "Prop \<C> !P \<alpha> P'"
star_unfoldl: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
notinset_notinEdge2: fixes y :: "nat"   and xs :: "nat list"   and x :: "nat" assumes "y \<notin> set xs" shows "(x, y) \<notin> Edges xs"
assms_3: shows "orient_lit_inst L1 t s pos \<sigma>"
factors_wfactors: fixes as :: "'a list"   and a :: "'a" assumes "factors G as a"   and "set as \<subseteq> carrier G" shows "wfactors G as a"
ctr_transfer_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_Tree R E E"
push_def: fixes k :: "'c"   and p :: "'d"   and q :: "('c, 'd) pq" shows "push k p q = Abs_pq (if k \<notin> set |q| then insort_key snd (k, p) (alist_of q) else alist_of q)"
test_dvd_def: fixes u :: "int poly"   and ws :: "int poly multiset" shows "test_dvd u ws = (\<forall>v l. v dvd u \<longrightarrow> 0 < degree v \<longrightarrow> degree v < degree u \<longrightarrow> Mp v \<noteq> Mp (Polynomial.smult l (\<Prod>\<^sub># ws)))"
class_semiring_def: fixes uu :: "'c itself"   and b :: "'d" shows "class_semiring uu b \<equiv> \<lparr>carrier = UNIV, mult = (*), one = 1::'c, zero = 0::'c, add = (+), \<dots> = b\<rparr>"
nonregular_event_induces_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and c :: "('d, 'e, 'f) configuration"   and p :: "'d"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "\<not> ps c p \<noteq> None"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev = p"   and "\<not> regular_event ev" shows "\<not> regular_event ev \<longrightarrow> ps c' p \<noteq> None"
beta_reduction_left_pairE: fixes A :: "'a trm"   and B :: "'a trm"   and M' :: "'a trm" assumes "SimplyTyped.Pair A B \<rightarrow>\<beta> M'" shows "(\<exists>A'. A \<rightarrow>\<beta> A' \<and> M' = SimplyTyped.Pair A' B) \<or> (\<exists>B'. B \<rightarrow>\<beta> B' \<and> M' = SimplyTyped.Pair A B')"
vdomain_iff: fixes a :: "V"   and r :: "V" shows "(a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r) = (\<exists>y. \<langle>a, y\<rangle> \<in>\<^sub>\<circ> r)"
dvdm_smult: fixes f :: "int poly"   and g :: "int poly"   and c :: "int" assumes "f dvdm g" shows "f dvdm smult c g"
types_preserved_br_if: fixes n :: "i32"   and i :: "nat"   and e :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[$C ConstInt32 n, $Br_if i]\<rparr> \<leadsto> \<lparr>e\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 n, $Br_if i] : ts _> ts'"   and "e = [$Br i] \<or> e = []" shows "\<S>\<bullet>\<C> \<turnstile> e : ts _> ts'"
Lambda_app: fixes x :: "ZF"   and A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" assumes "x |\<in>| A" shows "Lambda A f \<acute> x = f x"
divide_nonneg_pos: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) < y" shows "(0::'a) \<le> x / y"
pell_valuation_power_nat: fixes D :: "nat"   and z :: "nat \<times> nat"   and n :: "nat" assumes "pell D" shows "pell.pell_valuation D (case pell.pell_power D z n of (a, b) \<Rightarrow> (int a, int b)) = pell.pell_valuation D (case z of (x, y) \<Rightarrow> (int x, int y)) ^ n"
fds_exp_add: fixes f :: "'a fds"   and g :: "'a fds" shows "fds_exp (f + g) = fds_exp f * fds_exp g"
Out_def: fixes x3 :: "'d"   and x2 :: "name list"   and x1 :: "'d" shows "x3\<lparr>\<nu>*x2\<rparr>\<langle>x1\<rangle> \<equiv> Abs_action (Out_Rep x3 x2 x1)"
hyps_1: fixes l :: "nat" assumes "lk = l - k"   and "k \<le> l"   and "\<forall>j\<in>{k..<l}. v \<notin> local.writes (path \<sigma> j)" shows "(\<sigma>\<^bsup>l\<^esup>) v = (\<sigma>\<^bsup>k\<^esup>) v"
ps2_P_dims0: shows "ps2_P.dims0 = dims"
D_def: shows "D = Utrans_P vars1 mat_O;; hadamard_n n;; Utrans_P vars1 mat_Ph;; hadamard_n n;; Utrans_P vars2 (mat_incr K)"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus> y \<in> carrier G"
padic_add_assoc0: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "prime p" shows "padic_add p (padic_add p x y) z = padic_add p x (padic_add p y z)"
assms_3: fixes t1 :: "('a, unit) RBT_Impl.rbt"   and t2 :: "('a, unit) RBT_Impl.rbt"   and color :: "color"   and a :: "'a"   and b :: "unit" assumes "P t1"   and "P t2"   and "t2 \<noteq> rbt.Empty" shows "P (Branch color t1 a b t2)"
assms: shows "dX X d \<subseteq> Field (Pd d)"
extend_map_lemma: fixes \<F> :: "'a set set"   and \<G> :: "'a set set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "\<G> \<subseteq> \<F>"   and "convex T"   and "bounded T"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"   and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> S \<inter> T face_of S"   and "continuous_on (\<Union> \<G>) f"   and "f ` \<Union> \<G> \<subseteq> rel_frontier T" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on (\<Union> \<F>) g" and "g ` \<Union> \<F> \<subseteq> rel_frontier T" and "\<And>x. x \<in> \<Union> \<G> \<Longrightarrow> g x = f x"
part_sum_identity: fixes y :: "i"   and x :: "i" assumes "P y x" shows "x \<oplus> y = x"
shift_shift: fixes r :: "mregex"   and k :: "nat"   and j :: "nat" shows "Monitor.shift (Monitor.shift r k) j = Monitor.shift r (k + j)"
le0: shows "le_vec 0 (sr *s y - A *v y)"
assms_1: shows "AEeq M (\<lambda>w. c) (\<lambda>w. d)"
gamma2_differentiable_2: shows "\<forall>x\<in>{0..1} - s' \<inter> {0..1}. \<gamma>2 differentiable at x"
bis_defs_2: shows "ZObis \<equiv> gfp ZOretr"
b_in_bs: fixes i :: "nat" assumes "i < k" shows "bs ! i \<in> set bs"
Q3: shows "Quotient R3 Abs3 Rep3 T3"
intro: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and B :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and c :: "'a list"   and ins :: "'b list" assumes "lbv A r f T B"   and "lbvs_axioms A T B step c ins" shows "lbvs A r f T B step c ins"
dvd_mult_div_cancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times a (divide b a) = b"
diff_conv_add_uminus: fixes a :: "'b"   and b :: "'b" shows "a - b = a + - b"
spick_pseudo_proj: fixes n :: "nat"   and w :: "bool stream"   and v :: "bool" shows "spick (pseudo_proj_True (Suc n) w) n v = spick w n v"
comparator_linordered_group_def: shows "comparator_linordered_group = default"
face_cycle_set_self: fixes a :: "'b" shows "a \<in> G.face_cycle_set a"
add_One_commute: fixes n :: "num" shows "num.One + n = n + num.One"
steph: fixes xs :: "real list"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))"
distributive_def: fixes fmul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "distributive fmul fadd \<equiv> l_distributive fmul fadd \<and> r_distributive fmul fadd"
bound_plus_of_nat: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and default :: "'a"   and bound :: "'a \<Rightarrow> nat"   and a :: "'a"   and n :: "nat" assumes "complexity_one_mono_ordered_semiring_1 gt default bound"   and "(0::'a) \<le> a" shows "bound (a + of_nat n) = bound a + bound (of_nat n)"
icsSp_Append_sum: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" shows "n ics-ns @ [n'']\<rightarrow>\<^sub>d* n'"
M_in_subprob: shows "M2 \<in> space (subprob_algebra M2)"
retraction_is_epi: fixes g :: "('a S.E.term, 'a) S.arr" assumes "S.retraction g" shows "S.epi g"
additive_implies_homogenous: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (0::'d) = (0::'e)"
n: fixes j :: "nat" assumes "n < j" shows "b j = 0"
zip_eq_conv: fixes xs :: "'d list"   and ys :: "'e list"   and zs :: "('d \<times> 'e) list" assumes "length xs = length ys" shows "(xs || ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
that: fixes d :: "real" assumes "0 < d"   and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e" shows "thesis"
Ch_XH_largest_Field: fixes x :: "'a"   and h :: "'c" assumes "x \<in> Ch h XH_largest" shows "x \<in> Field (Pd (Xd x))"
cong: fixes Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "NoCardinalityAnalysis.cHeap Aheap \<equiv> NoCardinalityAnalysis.cHeap Aheap"
inv_op_closed2: fixes x :: "'a"   and h :: "'a" assumes "x \<in> carrier G"   and "h \<in> H" shows "x \<otimes> h \<otimes> inv x \<in> H"
pairwise_orthogonal_independent: fixes S :: "'a set" assumes "pairwise orthogonal S"   and "(0::'a) \<notin> S" shows "independent S"
j: shows "j < length (filter (\<lambda>ys. i < length ys) xs)"
assms_2: shows "open0_Var (x, T) u ::: U"
clock_trans_agent: fixes t :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a :: "'a" assumes "t \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a t" shows "toSet (local.clock_trans (fst ec) (snd ec)) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = s \<and> t' \<leadsto> s \<in> Clock.jkbpC \<and> clock_jview a t' = clock_jview a t)"
periodic_simps'_3: fixes nc :: "nat"   and n :: "int" shows "pbernpoly nc (real_of_int n) = pbernpoly nc 0"
measure_pmf_single: fixes M :: "'a pmf"   and x :: "'a" shows "measure_pmf.prob M {x} = pmf M x"
PDG_path_Append: fixes n'' :: "'node"   and n' :: "'node"   and n :: "'node" assumes "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n'' n'"   and "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n n''" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n n'"
vlrestriction_app: fixes a :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)"   and "a \<in>\<^sub>\<circ> A" shows "(VLambda Aa f \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = VLambda Aa f\<lparr>a\<rparr>"
mergesort_3: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "length (mergesort f xs) = length xs"
le_take_same: fixes i :: "nat"   and j :: "nat"   and a :: "'b list"   and b :: "'b list" assumes "i \<le> j"   and "take j a = take j b" shows "take i a = take i b"
case_eq_if: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and list :: "'d list" shows "(case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (if list = [] then f1 else f2 (hd list) (tl list))"