hcju/isabelleps huggingface dataset:
fits_Array: fixes G :: "prog"   and s :: "st"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "G,s\<turnstile>v\<Colon>\<preceq>T"   and "G\<turnstile>T'.[]\<preceq>T.[]"   and "G,s\<turnstile>v fits T'"   and "ws_prog G" shows "G,s\<turnstile>v\<Colon>\<preceq>T'"
lv_rel_comp_lv_rel_inverse: shows "lv_rel O lv_rel\<inverse> = {(x, y). x = y \<and> length x = DIM('a)}"
vsv_value: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" obtains b :: "V" where "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
nat_exists_least_iff: fixes P :: "nat \<Rightarrow> bool" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m<n. \<not> P m))"
mono_Mprefix0: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and Q :: "'b \<Rightarrow> 'b process" assumes "\<forall>x\<in>A. P x \<sqsubseteq> Q x" shows "Mprefix A P \<sqsubseteq> Mprefix A Q"
narrowid: fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.wn narrow widen less_eq less" shows "narrow x x = x"
OR4_analz_knows_Spy: fixes B :: "agent"   and X :: "msg"   and X' :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>X, Crypt (shrK B) X'\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "X \<in> analz (knows Spy evs)"
eval_prop_entailment_2: fixes \<phi> :: "'a ltln" shows "(eval \<phi> = No) = (\<not> UNIV \<Turnstile>\<^sub>P \<phi>)"
wt_elim_cases_14: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e0 :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (e0 ? e1 : e2)\<Colon>T" obtains T1 :: "ty" and T2 :: "ty" and Ta :: "ty" where "T = Inl Ta" and "E,dt\<Turnstile>e0\<Colon>-PrimT Boolean" and "E,dt\<Turnstile>e1\<Colon>-T1" and "E,dt\<Turnstile>e2\<Colon>-T2" and "prg E\<turnstile>T1\<preceq>T2 \<and> Ta = T2 \<or> prg E\<turnstile>T2\<preceq>T1 \<and> Ta = T1"
inj_onI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
simps_8: fixes uu :: "'b ltln set" shows "true\<^sub>n[uu]\<^sub>\<nu> = true\<^sub>n"
cases: fixes a :: "'a \<times> exp \<times> stack_elem list" assumes "boring_step a" obtains e :: "exp" and \<Gamma> :: "'a" and x :: "Vars.var" and S :: "stack_elem list" where "a = (\<Gamma>, e, Upd x # S)" and "isVal e"
assms_2: shows "exe_all_normalized_and_ex_tcsigs cs arss"
pos_bounded: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
dims1_def: shows "st.dims1 = nths (replicate (Suc (Suc k)) 2) {0..<Suc k}"
distinct_disc_1: fixes process :: "'a process" assumes "isAction process" shows "\<not> isChoice process"
times_mem_sum_mult: fixes A :: "'a set"   and B :: "'a set"   and a :: "'a"   and b :: "'a" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R"   and "a \<in> A"   and "b \<in> B" shows "a \<cdot>\<^sub>r b \<in> sum_mult R A B"
oriented_simplex_eq: fixes p :: "nat"   and l :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and l' :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "(oriented_simplex p l = oriented_simplex p l') = (\<forall>i\<le>p. l i = l' i)"
Lim_bounded: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a"   and M :: "nat"   and C :: "'a" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>M. f n \<le> C" shows "l \<le> C"
incm0: fixes n :: "nat" shows "f n - f 0 \<le>\<^sub>L f (Suc n) - f 0"
cases: fixes a1 :: "'b set"   and a2 :: "nat" assumes "R a1 a2" obtains A :: "'b set" where "a1 = Aa__" and "a2 = 0" and "finite Aa__"   | k :: "nat" where "a1 = {}" and "a2 = ka__"   | A :: "'b set" and x :: "'b" and k :: "nat" where "a1 = insert x A" and "a2 = Suc k" and "finite A" and "x \<notin> A" and "R A (Suc k)" and "R (insert x A) k"
trans_Step: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state"   and c'' :: "('test, 'atom) com"   and s'' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "(c', s') \<rightarrow>c (c'', s'')" shows "(c, s) \<rightarrow>*c (c'', s'')"
lunit'_simps_1: fixes f :: "'a" assumes "ide f" shows "arr \<l>\<^sup>-\<^sup>1[f]"
K_statefun_def: fixes c :: "'e"   and x :: "'f" shows "K_statefun c x \<equiv> c"
empty_map_add: fixes m :: "'c \<Rightarrow> 'd option" shows "Map.empty ++ m = m"
int_simps_7: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "\<turnstile> P \<noteq> Q = (P = (\<not> Q))"
Sup_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
bind_is_OK_E: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g" obtains x :: "'o" and h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
q65_upto_def: shows "q65 = (\<lambda>k. k\<^sup>2 mod 65) ` {..<65}"
soconf_sblank: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h :: "heap" assumes "P \<turnstile> C has_fields FDTs" shows "P,h,C \<turnstile>\<^sub>s sblank P C \<surd>"
collapse_1: fixes val :: "val" assumes "is_BoolVal val" shows "BoolVal (extract_bool val) = val"
UNIV_Plus_UNIV: shows "UNIV <+> UNIV = UNIV"
TSnopaid: fixes initH :: "nat list"   and "is" :: "nat list \<times> nat list"   and q :: "nat" shows "snd (fst (snd (rTS initH) is q)) = []"
A: fixes e :: "real" assumes "0 < e" shows "upper_asymptotic_density (B e) = 0"
tell_option_def: fixes w :: "'w"   and m :: "('a, 'm) optionT" shows "local.tell_option w m = OptionT (tell w (run_option m))"
empty_path: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" shows "local.path g [n]"
valid_fml_equiv: fixes \<phi> :: "fml"   and \<psi> :: "fml" shows "valid (\<phi> \<leftrightarrow> \<psi>) = fml_equiv \<phi> \<psi>"
ipcidr_union_cidr_split: fixes a :: "'b wordinterval" shows "ipcidr_union_set (set (cidr_split a)) = wordinterval_to_set a"
ln_powr_bound: fixes x :: "real"   and a :: "real" assumes "1 \<le> x"   and "0 < a" shows "ln x \<le> x powr a / a"
InrI: fixes b :: "'d"   and B :: "'d set"   and A :: "'e set" assumes "b \<in> B" shows "Inr b \<in> A <+> B"
simps: fixes a :: "trm" shows "SNa a = (\<exists>M. a = M \<and> (\<forall>x. M \<longrightarrow>\<^sub>a x \<longrightarrow> SNa x))"
p\<^sub>0_def: shows "p\<^sub>0 \<equiv> \<^bold>\<forall>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>E!,x\<^sup>P\<rparr>"
universal_affirmation_def: fixes A :: "'a set"   and B :: "'a set" shows "A Q B \<equiv> \<forall>b\<in>B. b \<in> A"
apartment_galleryD_chamber: fixes X :: "'a set set"   and xs :: "'a set list"   and x :: "'a set" assumes "X \<in> \<A>"   and "SimplicialComplex.maxsimpchain X xs"   and "x \<in> set xs" shows "SimplicialComplex.maxsimp X x"
assms_2: shows "P ^ 3 = a\<^sup>2 + 3 * b\<^sup>2"
count_roots_def: fixes p :: "real poly" shows "count_roots p = (if p = 0 then 0 else let ps = sturm_squarefree p in sign_changes_neg_inf ps - sign_changes_inf ps)"
Aheap_nonrec: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "nonrec \<Delta>" shows "\<A>\<^bsub>a\<^esub> e f|` domA \<Delta> \<sqsubseteq> Aheap \<Delta> e\<cdot>a"
map_undefined: fixes a :: "'a set" assumes "a \<notin> Partition" shows "induced a = undefined"
assms_4: shows "(c, c') \<in> R\<sigma> \<rightarrow> bool_rel"
g: shows "calls_chain [''FORWARD'' \<mapsto> [Rule m1 Log, Rule m2 (Call ''foo''), Rule m3 Accept, Rule m' (Call ''baz'')], ''foo'' \<mapsto> [Rule m4 Log, Rule m5 Return, Rule m6 (Call ''bar'')], ''bar'' \<mapsto> [], ''baz'' \<mapsto> []] = {(''FORWARD'', ''foo''), (''FORWARD'', ''baz''), (''foo'', ''bar'')}"
path_connected_space_iff_path_component: fixes X :: "'b topology" shows "path_connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. path_component_of X x y)"
index_add_mat_3: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_col (A + B) = dim_col B"
bound: shows "\<forall>i. i \<in> obs_ids \<pi> \<inter> {..<j} \<longrightarrow> i \<le> j"
mgu_gives_wellformed_MGU: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "mgu s t = Some \<theta>" shows "wf\<^sub>M\<^sub>G\<^sub>U \<theta> s t"
f1im1_sum: shows "map of_int_hom.vec_hom fs ! (i - 1) = gs.M.sumlist (map (\<lambda>j. fs.gs.\<mu> (i - 1) j \<cdot>\<^sub>v fs.gs.gso j) [0..<i - 1]) + fs.gs.gso (i - 1)"
UN_singleton: fixes A :: "'b set" shows "(\<Union>x\<in>A. {x}) = A"
symmetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "Q \<sim> P"
ipurge_tr_rev_t_form_def: fixes X :: "('c, 'd) ipurge_rec" shows "ipurge_tr_rev_t_form X \<equiv> In X = []"
in_aug_pm_set: fixes x :: "'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "(x \<in> aug_pm_set z i A) = (x = z \<or> x \<in> A \<or> x \<in> minus_set i A)"
ranked: fixes q :: "'a"   and f :: "'b"   and qs :: "'a list" assumes "q \<rightarrow> f qs \<in> ta1.\<delta>" shows "A f = Some (length qs)"
ok_SKIP2: fixes F :: "'c program" shows "F ok \<bottom>"
rep_eq: fixes x :: "fbint set" shows "open x = (\<forall>xa\<in>int_of_fbint ` x. \<exists>b>0. arith_prog xa b \<subseteq> int_of_fbint ` x)"
abs_eq: fixes xa :: "'b ltl set"   and x :: "'b ltl" shows "xa \<up>\<Turnstile>\<^sub>P Abs x = xa \<Turnstile>\<^sub>P x"
IH: assumes "ps \<in> snd ` set pth"   and "ps \<in> snd ` set pth" shows "\<exists>pth1 e pth2. pth = pth1 @ [e] @ pth2 \<and> snd e = ps \<and> ps \<notin> snd ` set pth1"
extend_basis_superset: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "B \<subseteq> vector_space.extend_basis scale B"
Hermite_Lindemann_aux3: fixes X :: "complex set"   and \<beta> :: "complex \<Rightarrow> rat" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> \<beta> x \<noteq> 0"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic x"   and "(\<Sum>x\<in>X. of_rat (\<beta> x) * exp x) = 0" shows "X = {}"
l\<^sub>2inc\<^sub>2: shows "l\<^sub>2 \<in> C\<^sub>2"
rem_condless_act_pair_2: fixes s :: "('c, 'd) fmap"   and pfx_a :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "rem_condless_act s pfx_a [] = pfx_a"
assms: shows "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
Suc_leD: fixes m :: "nat"   and n :: "nat" assumes "Suc m \<le> n" shows "m \<le> n"
min: shows "min_rvar_incdec_eq dir s eq = Inl is"
gronwall_general_segment: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and K :: "real"   and t :: "real" assumes "\<And>t. t \<in> closed_segment a b \<Longrightarrow> g t \<le> C + K * integral (closed_segment a t) g"   and "continuous_on (closed_segment a b) g"   and "\<And>t. t \<in> closed_segment a b \<Longrightarrow> 0 \<le> g t"   and "0 < C"   and "0 < K"   and "t \<in> closed_segment a b" shows "g t \<le> C * exp (K * \<bar>t - a\<bar>)"
G_ctor_o_rec: fixes s :: "('e, ('e G \<times> 'f) F) G0 \<Rightarrow> 'f" shows "ctor_rec_G s \<circ> ctor_G = s \<circ> map_pre_G id <id, ctor_rec_G s>"
d_1: shows "D \<in> N_of_state (lnth Sts i)"
dis_rews_6: fixes a :: "'b"   and list :: "['b]" shows "null\<cdot>(a : list) = FF"
main: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "(adj_mat A * A) $$ (i, j) = (det A \<cdot>\<^sub>m 1\<^sub>m n) $$ (i, j)"
inv_chart_inverse: fixes x :: "'a"   and c :: "('a, 'b) chart" assumes "x \<in> domain c" shows "inv_chart c (apply_chart c x) = x"
tt_in_keys: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "punit.tt p \<in> keys p"
inverse_mult_distrib: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (times a b) = times (inverse a) (inverse b)"
d_IN_def: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a" shows "d_IN g y = (\<Sum>\<^sup>+ x. g (x, y))"
invert_side_of_triangle_2: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "B.arr h"   and "f \<cdot>\<^sub>B g = h"   and "B.iso g" shows "B.seq h (B.inv g) \<and> f = h \<cdot>\<^sub>B B.inv g"
simps: fixes rx :: "real interval"   and ix :: "real interval"   and ry :: "real interval"   and iy :: "real interval" shows "Complex_Interval rx ix * Complex_Interval ry iy = Complex_Interval (rx * ry - ix * iy) (rx * iy + ix * ry)"
real_n: fixes s :: "'s" assumes "s \<in> S" shows "ennreal (enn2real (n s)) = n s"
P_Rcd: fixes fps :: "(char list \<times> pat) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta> :: "binding list" assumes "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>" shows "\<turnstile> PRcd fps : RcdT fTs \<Rightarrow> \<Delta>"
oInv_eq_0: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "normal F"   and "a \<le> F 0" shows "oInv F a = 0"
vector_derivative_at_within_ivl_diff: fixes s :: "real set"   and a :: "real"   and b :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "finite s \<and> a < b \<and> x \<in> {a..b} - s \<and> (f has_vector_derivative f') (at x)" shows "vector_derivative f (at x within {a..b} - s) = f'"
rep_eq: fixes x :: "'d pmf" shows "pmf x = (\<lambda>xa. measure_pmf.prob x {xa})"
dist_midpoint_1: fixes a :: "'b"   and b :: "'b" shows "dist a (midpoint a b) = dist a b / 2"
map_append: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'d list" shows "map f (xs \<cdot> ys) = map f xs \<cdot> map f ys"
simps: fixes rs :: "(name \<times> sterm) list"   and a1 :: "sterm"   and a2 :: "sterm" shows "(rs \<turnstile>\<^sub>s a1 \<longrightarrow> a2) = ((\<exists>name rhs. a1 = Sconst name \<and> a2 = rhs \<and> srewrite_step rs name rhs) \<or> (\<exists>cs t t'. a1 = Sabs cs $\<^sub>s t \<and> a2 = t' \<and> rewrite_first cs t t') \<or> (\<exists>t t' u. a1 = t $\<^sub>s u \<and> a2 = t' $\<^sub>s u \<and> rs \<turnstile>\<^sub>s t \<longrightarrow> t') \<or> (\<exists>u u' t. a1 = t $\<^sub>s u \<and> a2 = t $\<^sub>s u' \<and> rs \<turnstile>\<^sub>s u \<longrightarrow> u'))"
SN_on_Image_conv: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "SN_on r (r `` A) = SN_on r A"
assms_1: fixes n :: "nat" shows "0 \<le> u n"
rn_price_ind_def: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "rn_price_ind N der matur n w = rn_rev_price N der matur (matur - n) w"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and err :: "'d err" shows "P (case err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = ((err = Err \<longrightarrow> P f1) \<and> (\<forall>x2. err = OK x2 \<longrightarrow> P (f2 x2)))"
tendsto_cmult_ereal: fixes c :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'b filter" assumes "\<bar>c\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
disjointp_def: fixes xs :: "('c \<Rightarrow> bool) list" shows "disjointp xs = disjoint_family_on (\<lambda>n. {x. (xs ! n) x}) {0..<length xs}"
l5_5_2: 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 "\<exists>X. Bet A B X \<and> Cong A X C D" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D"
var_eq_def: shows "var_eq \<equiv> \<lambda>ns var. \<forall>no1\<in>set ns. \<forall>no2\<in>set ns. var no1 = var no2"
cycle_free_antimono: fixes g :: "'a ugraph"   and g' :: "'a ugraph" assumes "edges g \<subseteq> edges g'"   and "cycle_free g'" shows "cycle_free g"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (\<sigma>\<tau>.map f) = A.arr f"
monic_imp_nonzero: fixes x :: "'b poly" assumes "monic x" shows "x \<noteq> 0"
lookup_def: fixes xs :: "'z" shows "oaz.lookup xs = lookup_raw (list_of_oalistz xs)"
l_2: shows "(u \<circ> a) \<longlonglongrightarrow> liminf (u \<circ> r)"
assms_3: shows "group_hom (add_monoid G) (add_monoid H) h"
inj_imp_surj_inv: fixes f :: "'d \<Rightarrow> 'e" assumes "inj f" shows "surj (inv f)"
two: fixes x :: "'a" shows "2 \<times> x = x \<squnion> x"
extremum_strict: fixes a :: "'b list" shows "\<not> strict_prefix a []"
Int_commute: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B = B \<inter> A"
r_one: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> = x"
scale_sum_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" assumes "module scale" shows "scale (sum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
column_Gram_Schmidt_column_k_eq_efficient: fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "nat"   and b :: "'a" assumes "Gram_Schmidt_upt_k A k = foldl Gram_Schmidt_column_k_efficient A [0..<Suc k]"   and "Suc k < ncols A" shows "column b (Gram_Schmidt_column_k (Gram_Schmidt_upt_k A k) (Suc k)) = column b (Gram_Schmidt_column_k_efficient (Gram_Schmidt_upt_k A k) (Suc k))"
FSbImorph_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('b, 'c, 'd, 'e, 'f, 'k, 'h, 'j, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.FSbImorph Delta h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> FixSyn.ipresIGCons Delta h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> FixSyn.ipresIGSubstAll Delta h hA MOD MOD'"
real: fixes k :: "nat" shows "\<chi> k \<in> \<real>"
simps: fixes var :: "nat"   and p :: "real mpoly" shows "convertDerivative var p = (if MPoly_Type.degree p var = 0 then fm.Atom (Less p) else Or (fm.Atom (Less p)) (And (fm.Atom (Eq p)) (convertDerivative var (derivative var p))))"
sets_PiM_cong: fixes I :: "'a set"   and J :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and N :: "'a \<Rightarrow> 'b measure" assumes "I = J"   and "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (Pi\<^sub>M I M) = sets (Pi\<^sub>M J N)"
dd_def: shows "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
blinding_of_on_T: fixes A :: "'e set"   and rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "blinding_of_on A rh bo" shows "blinding_of_on {x. set1_T\<^sub>m x \<subseteq> A} (root_hash_T rh) (blinding_of_T rh bo)"
tau_steps_not_self: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula"   and n :: "nat" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "(x = indexed_weak_nominal_ts.tau_steps \<tau> x n) = (n = 0)"
hcomp_in_hhom: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>"
well_def: fixes j :: "nat" assumes "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))" shows "j < length (cols (M_mat signs subsets))"
prems_2: shows "\<Psi> \<rhd> P \<sim> P''"
AKcryptSK_not_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "AKcryptSK authK servK evs"   and "evs \<in> kerbV" shows "\<not> AKcryptSK servK K evs"
assms_1: shows "\<forall>(a, b)\<in>A. \<forall>(c, d)\<in>A. b = c \<and> a \<noteq> d \<longrightarrow> (a, d) \<in> A"
dg_dghm_id_is_dghm: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "dghm_id \<CC> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
is_REL_state_def: shows "is_REL_state \<equiv> \<lambda>ts. \<exists>t<N. \<exists>tk. ts t = REL tk"
CastRedsNull: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>null,s',b'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow>* \<langle>null,s',b'\<rangle>"
continuous_at_ball: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f x = (\<forall>e>0. \<exists>d>0. f ` ball x d \<subseteq> ball (f x) e)"
ex_sum_in_interval: fixes la :: "'b"   and ua :: "'b"   and lb :: "'b"   and ub :: "'b"   and x :: "'b" assumes "la \<le> ua"   and "lb \<le> ub"   and "la + lb \<le> x"   and "x \<le> ua + ub" shows "\<exists>xa\<ge>la. xa \<le> ua \<and> (\<exists>xb\<ge>lb. xb \<le> ub \<and> x = xa + xb)"
SHORun_eq: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and HOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set"   and SHOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" shows "SHORun A rho HOs SHOs = (HOinitConfig A (rho 0) \<and> (\<forall>r. SHOnextConfig A r (rho r) (HOs r) (SHOs r) (rho (Suc r))))"
ltrm_is_UP_monom: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "is_UP_monom (monom P (p (deg R p)) (deg R p))"
tauBangI: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<parallel> P \<longmapsto> \<tau> \<prec> P'"   and "guarded P" obtains Q :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> !P \<longmapsto> \<tau> \<prec> Q" and "Q \<sim> P' \<parallel> !P"
fv_set_mono: fixes M :: "('a, 'b) Term.term set"   and N :: "('a, 'b) Term.term set" assumes "M \<subseteq> N" shows "fv\<^sub>s\<^sub>e\<^sub>t M \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t N"
simps_2: fixes n :: "nat"   and S :: "'b formula set" shows "extender (Suc n) S = (let r = extender n S; rt = from_nat n \<triangleright> r; rf = \<^bold>\<not> (from_nat n) \<triangleright> r in if fin_sat rf then rf else rt)"
minus_poly_mapping_def: shows "(-) \<equiv> map_fun lookup (map_fun lookup Abs_poly_mapping) (\<lambda>f1 f2 k. f1 k - f2 k)"
simps_2: fixes x31 :: "com"   and x32 :: "com"   and y31 :: "com"   and y32 :: "com" shows "(x31;; x32 = y31;; y32) = (x31 = y31 \<and> x32 = y32)"
nths_def: fixes xs :: "'b list"   and A :: "nat set" shows "nths xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<length xs]))"
simps_4: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and P :: "'b"   and f :: "'e"   and \<alpha> :: "'d"   and t :: "('f, 'c, 'd, 'e) Tree" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P (tAct f \<alpha> t) = (\<exists>\<alpha>'' t' P'. tAct f \<alpha> t =\<^sub>\<alpha> tAct f \<alpha>'' t' \<and> transition (effect_apply f P) \<langle>\<alpha>'',P'\<rangle> \<and> effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P' t')"