hcju/isabelleps huggingface dataset:
\<iota>\<^sub>1_in_hom: shows "\<guillemotleft>\<iota>\<^sub>1 : F \<I>\<^sub>C \<otimes>\<^sub>D F \<I>\<^sub>C \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright>"
iTL_defs_3: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iUntil I P Q \<equiv> \<diamond> t I. Q t \<and> (\<box> t' I \<down>< t. P t')"
subS: fixes n :: "nat" shows "C n \<subseteq> S"
is_zero_uminus: fixes a :: "('a, 'b) IEEE.float" shows "is_zero (- a) = is_zero a"
mset_filter_mono: fixes A :: "'a multiset"   and B :: "'a multiset"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A \<subseteq># B"   and "\<And>x. \<lbrakk>x \<in># A; P x\<rbrakk> \<Longrightarrow> Q x" shows "filter_mset P A \<subseteq># filter_mset Q B"
coplanar_perm_17: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C D B A"
fps_exp_deriv: fixes a :: "'a" shows "fps_deriv (fps_exp a) = fps_const a * fps_exp a"
IntD2: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
fail_SE_def: shows "fail\<^sub>S\<^sub>E = Map.empty"
run_tell_option: fixes tell :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and w :: "'c"   and m :: "('b, 'a) optionT" shows "run_option (tell_option tell w m) = tell w (run_option m)"
C: shows "sinks I D (D x) (c_ipurge I D u xs) = {}"
in_bitset_bound_weak: fixes b :: "nat"   and n :: "nat" assumes "b \<in> bitset n" shows "b \<le> n"
Bex_def: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
distinctPermClosed: fixes p :: "(name \<times> name) list"   and q :: "(name \<times> name) list" assumes "distinctPerm p" shows "distinctPerm (q \<bullet> p)"
mergesort_remdups_def: fixes xs :: "'b list" shows "mergesort_remdups xs = merge_list [] (map (\<lambda>x. [x]) xs)"
encode_operator_effect_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and op :: "'b strips_operator" shows "encode_operator_effect \<Pi> t op \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O> in \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> Atom (State (Suc t) (index vs v))) (add_effects_of op) @ map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ops op))) \<^bold>\<or> \<^bold>\<not> (Atom (State (Suc t) (index vs v)))) (delete_effects_of op)"
sum_lessThan_conv_atMost_nat: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum f {..<n} = sum f {..n} - f n"
div_to_mult_word_lt: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x \<le> y div z" shows "x * z \<le> y"
unit_simps_1: shows "D.seq (F g \<star>\<^sub>D F f) ((\<mu> \<star>\<^sub>D F f) \<cdot>\<^sub>D \<phi>)"
none_vertices: fixes c :: "'form" shows "((c, []) |\<in>| vertices) = (c \<in> set conclusions)"
M_def: shows "M \<equiv> median (map median (chop 5 xs))"
prems_2: shows "P \<turnstile> \<langle>Throw a,(h, l, sh')\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
factor_theorem': fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "a \<in> carrier R"   and "UP_cring.to_fun R f a = \<zero>\<^bsub>R\<^esub>" shows "f = X_poly_minus R a \<otimes>\<^bsub>UP R\<^esub> UP_cring.UP_root_div R f a"
polynate: fixes bs :: "'b list"   and p :: "poly" shows "\<lparr>polynate p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
div_neg_neg_trivial: fixes k :: "int"   and l :: "int" assumes "k \<le> 0"   and "l < k" shows "k div l = 0"
invc_baldR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc l"   and "invc2 r"   and "color l = Black" shows "invc (baldR l a r)"
a_star_redu_ImpR_elim: fixes x :: "name"   and a :: "coname"   and M :: "trm"   and y :: "coname"   and R :: "trm" assumes "ImpR x.<a>.M y \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = ImpR x.<a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
H_def: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. nil \<sqinter> (c::'a::refinement_lattice) ; x)"
fps_compose_linear: fixes f :: "'b fps"   and c :: "'b" shows "f oo fps_const c * fps_X = Abs_fps (\<lambda>n. c ^ n * f $ n)"
assms_1: shows "chain_subdiv_path \<gamma> one_chain"
Diff_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (rel_set A)) (-) (-)"
factorization: shows "is_factorization_of fctrs (lr_fps_denominator' cs)"
inject_5: fixes x51 :: "expr"   and x52 :: "expr"   and y51 :: "expr"   and y52 :: "expr" shows "(x51 \<cdot> x52 = y51 \<cdot> y52) = (x51 = y51 \<and> x52 = y52)"
card_of_Times3: fixes A :: "'a set" shows "|A| \<le>o |A \<times> A|"
arc_eq_1: fixes x :: "'a" assumes "arc x" shows "x = x * x\<^sup>T * x"
prod_cases4: fixes y :: "'b \<times> 'c \<times> 'd \<times> 'e" obtains a :: "'b" and b :: "'c" and c :: "'d" and d :: "'e" where "y = (a, b, c, d)"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> xs = vcard xs"
invar_enq: fixes empty :: "'b"   and enq :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and first :: "'b \<Rightarrow> 'c"   and deq :: "'b \<Rightarrow> 'b"   and is_empty :: "'b \<Rightarrow> bool"   and list :: "'b \<Rightarrow> 'c list"   and invar :: "'b \<Rightarrow> bool"   and q :: "'b"   and x :: "'c" assumes "Queue empty enq first deq is_empty list invar"   and "invar q" shows "invar (enq x q)"
le_imp_less_or_eq: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (a \<prec>\<^bsub>D\<^esub> b \<or> a = b)"
assms_2: shows "HT_mods \<pi> mods P c Q"
conjunction_sup: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice) \<sqsubseteq> c \<squnion> d"
member_remove: fixes x :: "'g"   and y :: "'g"   and A :: "'g set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"
t_find_closest_mono: fixes \<delta>' :: "real"   and \<delta> :: "real"   and p :: "int \<times> int"   and ps :: "(int \<times> int) list" assumes "\<delta>' \<le> \<delta>" shows "t_find_closest p \<delta>' ps \<le> t_find_closest p \<delta> ps"
counit_def: fixes a :: "'a" shows "counit a = (SOME \<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (map\<^sub>0 a) (unit a) \<epsilon>)"
ring_closed_def: fixes A :: "'b set" shows "ring_closed A \<equiv> ((0::'b) \<in> A \<and> (1::'b) \<in> A) \<and> (\<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x + y \<in> A) \<and> (\<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x * y \<in> A) \<and> (\<forall>x. x \<in> A \<longrightarrow> - x \<in> A)"
flip_flip: fixes tup :: "'a \<times> 'b" shows "flip (flip tup) = tup"
assms_2: shows "A \<Delta> T --` A \<in> null_sets M"
neqE: fixes x :: "'t"   and y :: "'t" assumes "x \<noteq> y" obtains "x \<prec>\<^sub>t y"   | "y \<prec>\<^sub>t x"
SSpec_derivable_empty: fixes c :: "OBJ" shows "{} \<rhd> c : SSpec c"
emeasure_count_space: fixes X :: "'d set"   and A :: "'d set" assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
rep_eq: fixes x :: "'c Filters.filter"   and xa :: "'c Filters.filter" shows "Filters.filter.Rep_filter (x \<sqinter> xa) = Filters.filter.Rep_filter x \<inter> Filters.filter.Rep_filter xa"
assms: shows "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)"
rel_interior_of_polyhedron: fixes S :: "'b set" assumes "polyhedron S" shows "rel_interior S = S - \<Union> {F. F facet_of S}"
continuous_on_mult': fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. f x * g x)"
card_of_Func_UNIV: fixes B :: "'b set" shows "|Func UNIV B| =o |{f. range f \<subseteq> B}|"
Col_def: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec" shows "real_euclid.Col a b c \<equiv> B\<^sub>\<real> a b c \<or> B\<^sub>\<real> b c a \<or> B\<^sub>\<real> c a b"
assms_2: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> f x \<bullet> k"
snd_sym_preproc_aux_dgrad_set_le: fixes d :: "'a \<Rightarrow> nat"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list" assumes "dickson_grading d"   and "set vs \<subseteq> Keys (set fs)" shows "dgrad_set_le d (pp_of_term ` Keys (set (snd (sym_preproc_aux gs ks (vs, fs))))) (pp_of_term ` Keys (set gs \<union> set fs))"
actions_in_execution_def: fixes exec :: "'a list list" shows "actions_in_execution exec \<equiv> {a. \<exists>aseq\<in>set exec. a \<in> set aseq}"
new_document_is_l_new_document: shows "l_new_document DocumentClass.type_wf"
x12_1: shows "x1 \<equiv> Variable (Suc (Suc 0))"
meet_equality: fixes x :: "'b"   and y :: "'b"   and inf :: "'b" assumes "is_inf x y inf" shows "x \<sqinter> y = inf"
seq_char: fixes g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "Cop_S.seq g f = (Cop_S.arr f \<and> Cop_S.arr g \<and> Cop_S.Dom g = Cop_S.Cod f)"
\<alpha>n_correct: fixes g :: "'g" assumes "invar g" shows "getFrom ` \<alpha>e g \<union> getTo ` \<alpha>e g \<subseteq> set (\<alpha>n g)"
assms_1: shows "y * f $$ fls_subdegree f = (1::'a)"
healthy_intros_6: fixes t :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy (wlp t)"   and "nearly_healthy (wlp u)" shows "nearly_healthy (wlp (t ;; u))"
valid_no_mon: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and c :: "'b list multiset" assumes "mon_c fg c = {}" shows "valid fg c"
concat_length_le: fixes xs :: "nat list"   and f :: "nat \<Rightarrow> 'a list"   and g :: "nat \<Rightarrow> real" assumes "\<forall>x\<in>set xs. real (length (f x)) \<le> g x" shows "real (length (concat (map f xs))) \<le> sum_list (map g xs)"
iNextWeak_False: fixes t0 :: "nat"   and I :: "nat set" shows "(\<not> (\<circle>\<^sub>W t t0 I. False)) = (I \<down>> t0 \<noteq> {} \<and> t0 \<in> I)"
Act_eq_iff_perm: fixes \<alpha>1 :: "'g"   and x1 :: "('e, 'f, 'g) formula"   and \<alpha>2 :: "'g"   and x2 :: "('e, 'f, 'g) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2)"
action_proj_inter: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs1 :: "'d set"   and vs2 :: "'d set" shows "action_proj (action_proj a vs1) vs2 = action_proj a (vs1 \<inter> vs2)"
LtK_LtK: fixes X :: "ltkey"   and X' :: "ltkey" shows "(LtK X = LtK X') = (X = X')"
expectation_gpv_le_exec_gpv: fixes \<I> :: "('b, 'c) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and gpv :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ennreal"   and s :: "'a" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (callee s x)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s. \<I> \<turnstile>c callee s \<surd>" shows "expectation_gpv 0 \<I> f gpv \<le> \<integral>\<^sup>+ (x, s). f x \<partial>measure_spmf (exec_gpv callee gpv s)"
continuous_on_Shleg: fixes A :: "real set" shows "continuous_on A local.Shleg"
cinsertCI: fixes a :: "'f"   and B :: "'f cset"   and b :: "'f" assumes "\<not> cin a B \<Longrightarrow> a = b" shows "cin a (cinsert b B)"
foldli_cong: fixes l :: "'d list"   and l' :: "'d list"   and \<sigma> :: "'e"   and \<sigma>' :: "'e"   and c :: "'e \<Rightarrow> bool"   and c' :: "'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "l = l'"   and "\<sigma> = \<sigma>'"   and "c = c'"   and "\<And>\<sigma> x. \<lbrakk>x \<in> set l'; c' \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> = f' x \<sigma>" shows "foldli l c f \<sigma> = foldli l' c' f' \<sigma>'"
rel_flip: fixes R1 :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'e + 'g"   and b :: "'f + 'h" shows "rel_sum R1\<inverse>\<inverse> R2\<inverse>\<inverse> a b = rel_sum R1 R2 b a"
assms_2: shows "P \<Longrightarrow>\<^sub>t P'"
simps_2: fixes v :: "('b \<times> nat) tree"   and va :: "'b \<times> nat"   and vb :: "('b \<times> nat) tree" shows "combine \<langle>\<rangle> \<langle>v, va, vb\<rangle> = \<langle>v, va, vb\<rangle>"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case loop of init \<Rightarrow> f1 | loop \<Rightarrow> f2 | final \<Rightarrow> f3) = f2"
replicate_Cons_eq: fixes n :: "nat"   and x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "(replicate n x = y # ys) = (\<exists>n'. n = Suc n' \<and> x = y \<and> replicate n' x = ys)"
lemma4: fixes a :: "'a"   and a1 :: "'a"   and d :: "'a"   and b :: "'a"   and b1 :: "'a"   and d' :: "'a" assumes "a = a1 * d"   and "b = b1 * d"   and "ideal_generated {a1, b1} = ideal_generated {1::'a}"   and "ideal_generated {a, b} = ideal_generated {d'}" shows "\<exists>a1' b1'. a = a1' * d' \<and> b = b1' * d' \<and> ideal_generated {a1', b1'} = ideal_generated {1::'a}"
unhalt: shows "{\<lambda>tp. tp = (Bk \<up> i, <args>)} p \<up>"
directed: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "connex A less_eq" shows "directed A less_eq"
span_zero: fixes S :: "'b set" shows "(0::'b) \<in> cspan S"
Dickson_fun: fixes V :: "'a set" assumes "finite V" shows "almost_full_on (adds) {x. supp_fun x \<subseteq> V}"
pfinite_INF_le_expectation_gpv: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and f :: "'j \<Rightarrow> ennreal"   and fail :: "ennreal" assumes "pfinite_gpv \<I> gpv" shows "min (\<Sqinter> (f ` results_gpv \<I> gpv)) fail \<le> expectation_gpv fail \<I> f gpv"
less_def: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<squnion> y = y \<and> y \<squnion> x \<noteq> x)"
inject_3: fixes x31 :: "int poly"   and x32 :: "nat"   and y31 :: "int poly"   and y32 :: "nat" shows "(Real_Alg_Info x31 x32 = Real_Alg_Info y31 y32) = (x31 = y31 \<and> x32 = y32)"
pderiv_no_occurrence: fixes x :: "'b"   and r :: "'b rexp" assumes "x \<notin> atoms r" shows "pderiv x r = {}"
simps: fixes a :: "('test, 'atom) com" shows "discr0 a = (\<exists>c. a = c \<and> (\<forall>x xa xb. mustT c x \<longrightarrow> (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> x \<approx> xb \<and> discr0 xa) \<and> (\<forall>x xa. mustT c x \<longrightarrow> (c, x) \<rightarrow>t xa \<longrightarrow> x \<approx> xa))"
wtl_stable: fixes pc :: "nat" assumes "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"   and "s\<^sub>0 \<in> A"   and "pc < length ins" shows "stable r step \<tau>s pc"
assms_1: shows "cpt (P - {p}) env \<pi>l"
components_are_iso: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.iso (B.\<alpha>' a)"
path_component_of_eq_empty: fixes X :: "'c topology"   and x :: "'c" shows "(path_component_of_set X x = {}) = (x \<notin> topspace X)"
entails_pconj_assumption: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real" assumes "feasible t"   and "\<And>s. P s \<Longrightarrow> Q s \<le> t R s"   and "unitary Q"   and "unitary R" shows "\<guillemotleft> P \<guillemotright> && Q \<tturnstile> t R"
P_inf_eq_E_inf: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred St P" shows "P_inf s P = E_inf s (indicator {x \<in> space St. P x})"
inDed_mono: fixes x :: "formula list"   and y :: "formula list" assumes "x \<in> deductions CutFreePC"   and "set x \<subseteq> set y" shows "y \<in> deductions CutFreePC"
2: shows "atoms (tseytin_toatom F F) \<union> \<Union> (atoms ` set (tseytin_tran1 (tseytin_toatom F) F)) \<subseteq> atoms F \<union> (\<Union>I\<in>set (subformulae F). atoms (tseytin_toatom F I))"
awalkI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "set (awalk_verts u p) \<subseteq> verts G"   and "set p \<subseteq> arcs G"   and "cas u p v" shows "awalk u p v"
ldrop_inf_iffT: fixes t :: "'b llist"   and i :: "nat" shows "(t \<up> i \<in> UNIV\<^sup>\<omega>) = (t \<in> UNIV\<^sup>\<omega>)"
dist_trans: fixes u :: "nat"   and d1 :: "nat"   and w :: "nat"   and d2 :: "nat"   and v :: "nat" assumes "dist u d1 w"   and "dist w d2 v" shows "dist u (d1 + d2) v"
HSem_def': fixes \<Gamma> :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" shows "\<lbrace> \<Gamma> \<rbrace>\<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
pos_oriented_cmat_mult_positive': fixes H1 :: "complex \<times> complex \<times> complex \<times> complex"   and H2 :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H1 \<and> H1 \<noteq> mat_zero"   and "hermitean H2 \<and> H2 \<noteq> mat_zero"   and "\<exists>k>0. H2 = cor k *\<^sub>s\<^sub>m H1"   and "pos_oriented_cmat H1" shows "pos_oriented_cmat H2"
dominates_def: fixes g :: "'g"   and n :: "'node"   and m :: "'node" shows "old.dominates g n m \<equiv> m \<in> set (\<alpha>n g) \<and> (\<forall>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<longrightarrow> n \<in> set ns)"
fls_prpart_eq0_iff: fixes f :: "'a fls" shows "(fls_prpart f = 0) = (0 \<le> fls_subdegree f)"
emb_trans: fixes P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "\<And>f g h. \<lbrakk>f \<in> F; g \<in> F; h \<in> F; P f g; P g h\<rbrakk> \<Longrightarrow> P f h"   and "emb P s t"   and "emb P t u" shows "emb P s u"
countable_insert: fixes A :: "'d set"   and a :: "'d" assumes "countable A" shows "countable (insert a A)"
positively_invariant_def: fixes M :: "'a set" shows "positively_invariant M = (\<forall>x\<in>M. trapped_forward x M)"
carrier_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "carrier_mat nr nc = {m. dim_row m = nr \<and> dim_col m = nc}"
sum_mono2: fixes B :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> {#} \<subseteq># f b" shows "comm_monoid_add.sum (+) {#} f A \<subseteq># comm_monoid_add.sum (+) {#} f B"
ed_ou_witness_UNIV: shows "UNIV = set [ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d, e]"
assms_3: shows "Case_Labeling.CTXT (Suc OC1) ((''else'', Suc OC1, []) # CT') OC (ovalidNF R r Q)"
in_set_impl_in_set_zip2: fixes xs :: "'b list"   and ys :: "'c list"   and y :: "'c" assumes "length xs = length ys"   and "y \<in> set ys" obtains x :: "'b" where "(x, y) \<in> set (zip xs ys)"
next_always: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> \<circle>\<box>F"
unit_simps_4: fixes a :: "'b" assumes "B.obj a" shows "C.dom (H.unit a) = H.map\<^sub>0 a"
normalize_match_preserves_nodisc: fixes disc :: "'b \<Rightarrow> bool"   and m :: "'b match_expr"   and m' :: "'b match_expr" assumes "\<not> has_disc disc m"   and "m' \<in> set (normalize_match m)" shows "\<not> has_disc disc m'"
graph_union_iff: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'e) labeled_graph" shows "(graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2) = (edges G\<^sub>1 \<subseteq> edges G\<^sub>2 \<and> vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2)"
LeftDerivationFix_splits_at_derives: fixes a :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and b :: "symbol list" assumes "LeftDerivationFix a i D j b" shows "\<exists>U a1 a2 b1 b2. splits_at a i a1 U a2 \<and> splits_at b j b1 U b2 \<and> derives a1 b1 \<and> derives a2 b2"
proj2_def: shows "\<pi>\<^sub>2 \<equiv> \<lambda>(dsn, uu_, uu_, uu_, uu_). dsn"
has_vderiv_on_diff: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and T :: "real set"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a" assumes "(f has_vderiv_on f') T"   and "(g has_vderiv_on g') T" shows "((\<lambda>x. f x - g x) has_vderiv_on (\<lambda>x. f' x - g' x)) T"
asBinp_equal_imp_alphaBinp: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and qbinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp"   and "asBinp qbinp = asBinp qbinp'" shows "(qbinp, qbinp') \<in> alphaBinp"
map_iterator_linord_rule_P: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'a \<Rightarrow> 'c option"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<times> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_linord less_eq it (map_to_set m)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<And>k'. k' \<in> it \<Longrightarrow> less_eq k k'; \<And>k'. k' \<in> dom m - it \<Longrightarrow> less_eq k' k\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<And>k k'. \<lbrakk>k \<in> dom m - it; k' \<in> it\<rbrakk> \<Longrightarrow> less_eq k k'\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
s_2: shows "\<phi> C1_differentiable_on {0..1} - s"
rotate_downpos_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "rotate_downpos w1 w2" shows "kauff_mat w1 = kauff_mat w2"
map_G_comp: fixes l1 :: "'ac \<Rightarrow> 'w"   and l2 :: "'ad \<Rightarrow> 'x"   and co1 :: "'ae \<Rightarrow> 'y"   and co2 :: "'af \<Rightarrow> 'z"   and contra1 :: "'aa \<Rightarrow> 'ag"   and contra2 :: "'ab \<Rightarrow> 'ah"   and l1' :: "'p \<Rightarrow> 'ac"   and l2' :: "'q \<Rightarrow> 'ad"   and co1' :: "'r \<Rightarrow> 'ae"   and co2' :: "'s \<Rightarrow> 'af"   and contra1' :: "'ag \<Rightarrow> 't"   and contra2' :: "'ah \<Rightarrow> 'u" shows "map_G l1 l2 co1 co2 contra1 contra2 \<circ> map_G l1' l2' co1' co2' contra1' contra2' = map_G (l1 \<circ> l1') (l2 \<circ> l2') (co1 \<circ> co1') (co2 \<circ> co2') (contra1' \<circ> contra1) (contra2' \<circ> contra2)"
left_diff_distrib: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "(x - y) \<times>\<^sub>7 z = x \<times>\<^sub>7 z - y \<times>\<^sub>7 z"
binary_functor_axioms: shows "binary_functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F"
integral_const: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c"   and "a \<le> b" shows "integral (\<lambda>_. c) a b = (b - a) * c"
less_irrefl: fixes x :: "'b Word.word" shows "\<not> x <s x"