hcju/isabelleps huggingface dataset:
induces_hom_adjunction: shows "hom_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) S.comp \<phi>C \<phi>D F G \<Phi> \<Psi>"
B: shows "bn_count (Suc (Suc n) # ns) = Suc (Suc (foldl (+) 0 ns + n))"
myss'_def: fixes a :: "char list \<Longrightarrow> 'a" shows "myss' a \<equiv> vwb_lens a"
Act_eq_iff: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c, 'd) formula"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c, 'd) formula" shows "(Act f1 \<alpha>1 x1 = Act f2 \<alpha>2 x2) = (Act\<^sub>\<alpha> f1 \<alpha>1 (Rep_formula x1) = Act\<^sub>\<alpha> f2 \<alpha>2 (Rep_formula x2))"
fixed_point_in_\<omega>_limit_set_imp_\<omega>_limit_set_singleton_fixed_point: fixes K :: "'a set"   and x :: "'a"   and yfp :: "'a"   and z :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "yfp \<in> \<omega>_limit_set x"   and "f yfp = (0::'a)"   and "z \<in> \<omega>_limit_set x"   and "finite {y \<in> K. f y = (0::'a)}" shows "(\<exists>p1\<in>\<omega>_limit_set x. f p1 = (0::'a) \<and> \<omega>_limit_set z = {p1}) \<and> (\<exists>p2\<in>\<omega>_limit_set x. f p2 = (0::'a) \<and> \<alpha>_limit_set z = {p2})"
wf_subset: fixes r :: "('r \<times> 'r) set"   and p :: "('r \<times> 'r) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
cxt_eval: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>cxt_to_stmt [] c, mds, mem\<rangle> \<leadsto> \<langle>cxt_to_stmt [] c', mds', mem'\<rangle>" shows "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
main: fixes h :: "int poly" assumes "g =m f * h"   and "Mp h = h" shows "thesis"
subset_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
mset_concat: fixes xss :: "'a list list" shows "mset (concat xss) = sum_list (map mset xss)"
simps_4: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "hnf (a{b}.P) = True"
nth_prime_exists: fixes n :: "nat" shows "\<exists>p. prime p \<and> card {q. prime q \<and> q < p} = n"
le_rem_simps_1: fixes a :: "'a" shows "le_rem_s (LOC a) = a"
parts_pubkey_false_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "s \<turnstile> s'"   and "Crypt (SesK SK) (PubKey C) \<in> parts (used s')"   and "Crypt (SesK SK) (PubKey C) \<notin> parts (used s)"   and "\<forall>n. (Owner n, SesKey SK) \<notin> s'"   and "SesKey SK \<notin> spied s'" shows "False"
roofed_def: fixes G :: "('c, 'd) graph_scheme"   and B :: "'c set"   and S :: "'c set" shows "roofed_gen G B S = {x. \<forall>p. \<forall>y\<in>B. path G x p y \<longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S}"
intros_1: fixes d :: "nat"   and c :: "nat" assumes "0 \<le> d"   and "d \<le> c" shows "valid_intv c (Const d)"
assms_5: fixes g :: "'a" assumes "g \<in> G" shows "inverse g \<in> G"
SubstFormP_cong: fixes H :: "fm set"   and v :: "tm"   and v' :: "tm"   and i :: "tm"   and i' :: "tm"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> v EQ v'"   and "H \<turnstile> i EQ i'"   and "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> SubstFormP v i t u IFF SubstFormP v' i' t' u'"
open_segment_def: fixes a :: "'b"   and b :: "'b" shows "{a<--<b} \<equiv> {a--b} - {a, b}"
radical_sqrt_circle_circle_intersection: fixes A :: "point"   and B :: "point"   and C :: "point"   and D :: "point"   and E :: "point"   and F :: "point"   and X :: "point" assumes "abscissa A \<in> radical_sqrt"   and "ordinate A \<in> radical_sqrt"   and "abscissa B \<in> radical_sqrt"   and "ordinate B \<in> radical_sqrt"   and "abscissa C \<in> radical_sqrt"   and "ordinate C \<in> radical_sqrt"   and "abscissa D \<in> radical_sqrt"   and "ordinate D \<in> radical_sqrt"   and "abscissa E \<in> radical_sqrt"   and "ordinate E \<in> radical_sqrt"   and "abscissa F \<in> radical_sqrt"   and "ordinate F \<in> radical_sqrt"   and "distance A X = distance B C"   and "distance D X = distance E F"   and "\<not> (A = D \<and> distance B C = distance E F)" shows "abscissa X \<in> radical_sqrt \<and> ordinate X \<in> radical_sqrt"
A_incr: fixes bs :: "bool list" shows "T_incr bs + \<Phi> (incr bs) - \<Phi> bs = 2"
axioms_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "category A"
gc3_def: shows "gc3 = gcd pR3 pI3"
f: shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)"
FI_4: shows "M_lookup \<pi>i' = \<pi>' (Q_\<alpha> Qi) (M_lookup \<pi>i) u"
inf_map: fixes \<iota> :: "'f inference" assumes "\<iota> \<in> Inf_F"   and "\<G>_I \<iota> \<noteq> None" shows "the (\<G>_I \<iota>) \<subseteq> Red_I_G (\<G>_F (concl_of \<iota>))"
convex_standard_simplex: fixes x :: "nat \<Rightarrow> real"   and p :: "nat"   and y :: "nat \<Rightarrow> real"   and u :: "real" assumes "x \<in> standard_simplex p"   and "y \<in> standard_simplex p"   and "0 \<le> u"   and "u \<le> 1" shows "(\<lambda>i. (1 - u) * x i + u * y i) \<in> standard_simplex p"
quotient_element_nonempty: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "\<exists>x. x \<in> X"
compl_le_compl_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (uminus x) (uminus y) = less_eq y x"
scale_minus_right: fixes a :: "'a"   and x :: "'a" shows "a \<cdot> - x = - (a \<cdot> x)"
TS_inv_def: fixes c :: "('b list \<times> 'b list) pmf"   and x :: "'b"   and i :: "'b list" shows "TS_inv c x i \<equiv> (\<exists>hs. c = return_pmf (if x = hd i then i else rev i, [x, x] @ hs)) \<or> c = return_pmf (if x = hd i then i else rev i, [])"
right_neutral: fixes a :: "'b hrel" shows "a \<sqinter> false = a"
PostConjI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and R :: "'e set"   and B :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c R,B" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> B)"
expectation_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('b, 'c) \<I>"   and f :: "'a \<Rightarrow> ennreal" shows "expectation_gpv fail \<I> f \<equiv> lfp.fixp_fun (\<lambda>expectation_gpv gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None))"
pq: shows "[:f, e, 0, 1:] = [:- y1, 1:] * q"
integral_less_AE_space: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real" assumes "integrable M X"   and "integrable M Y"   and "AE x in M. X x < Y x"   and "emeasure M (space M) \<noteq> 0" shows "expectation X < expectation Y"
content_prod_eq_1_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(content (p * q) = (1::'a)) = (content p = (1::'a) \<and> content q = (1::'a))"
intro: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_monoid G"   and "abelian_group_axioms G" shows "abelian_group G"
lens_inv_def: fixes x :: "'d \<Longrightarrow> 'c" shows "inv\<^sub>L x = \<lparr>lens_get = create\<^bsub>x\<^esub>, lens_put = \<lambda>\<sigma>. get\<^bsub>x\<^esub>\<rparr>"
Seq_transT_invert: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" shows "\<not> (c1 ;; c2, s) \<rightarrow>t s'"
map_dual_func1: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "\<partial>\<^sub>F (f \<circ> g) = \<partial>\<^sub>F f \<circ> \<partial>\<^sub>F g"
distinct_9: fixes x1 :: "agent"   and x61 :: "msg"   and x62 :: "msg" shows "msg.Agent x1 \<noteq> \<lbrace>x61, x62\<rbrace>"
sset_range: fixes s :: "'b stream" shows "sset s = range ((!!) s)"
is_processT4_S1: fixes x :: "'c event list \<times> 'c event set"   and P :: "'c process"   and X :: "'c event set" assumes "x \<in> F P"   and "X \<subseteq> snd x" shows "(fst x, X) \<in> F P"
mat1: fixes nr :: "nat"   and ze :: "'c"   and on :: "'c" shows "mat nr nr (mat1I ze on nr)"
simps_1: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and x :: "'a"   and m :: "Mode"   and mds :: "Mode \<Rightarrow> 'a set" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang_no_dma.update_modes (x +=\<^sub>m m) mds = mds(m := insert x (mds m))"
module_prod_scale_eq_scaleC: shows "module_prod.scale (*\<^sub>C) (*\<^sub>C) = (*\<^sub>C)"
group_eq_aux: fixes a :: "'c"   and b :: "'c" shows "a + (b - a) = b"
mcont_const: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'d llist" shows "mcont luba orda lSup lprefix (\<lambda>x. c)"
baby_Sard: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "CARD('a) \<le> CARD('b)"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> rank (matrix (f' x)) < CARD('b)" shows "negligible (f ` S)"
cases: fixes a :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set" assumes "a \<in> fake bad IK chan" obtains M :: "chan" where "a = M__" and "M__ \<in> chan"   | M :: "msg" and c :: "tag" and A :: "agent" and B :: "agent" where "a = Chan c A B M" and "M \<in> IK" and "c = insec \<or> c = confid \<or> A \<in> bad \<or> B \<in> bad"
coprime_of_int_polyI: fixes p :: "int poly"   and q :: "int poly" assumes "algebraic_semidom_class.coprime p q" shows "algebraic_semidom_class.coprime (Ring_Hom_Poly.of_int_poly p) (Ring_Hom_Poly.of_int_poly q)"
scalar7_triple2: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "x \<bullet> y \<times>\<^sub>7 z = z \<bullet> x \<times>\<^sub>7 y"
lcm_eucl: shows "normalization_euclidean_semiring_class.lcm = lcm"
genideal_one: shows "Idl {\<one>} = carrier R"
single_zero: fixes k :: "'e" shows "monomial (0::'f) k = 0"
extremum_unique: fixes a :: "'g multiset" shows "(a \<subseteq># {#}) = (a = {#})"
irreducibleD': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.irreducible times one zero a"   and "dvd.dvd times b a" shows "dvd.dvd times a b \<or> dvd.dvd times b one"
v1: shows "\<v> = int (\<Sum>x\<in>#image_mset card (mset_set \<G>). x)"
specializedtoprimes1a: fixes p :: "nat"   and q :: "nat"   and a :: "nat"   and b :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> q"   and "a mod p = b mod p"   and "a mod q = b mod q"   and "b < p * q" shows "a mod (p * q) = b"
leftmost_notword: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and i :: "nat"   and a :: "symbol list"   and j :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftmost \<NN> \<TT> i a"   and "i < j" shows "\<not> CFG.is_word \<TT> (take j a)"
bind_lunit: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c option" shows "Some x \<bind> f = f x"
non_empty_class: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<noteq> {}"
prpl_of_lasso_def: fixes L :: "('a, 'b) lasso_scheme" shows "prpl_of_lasso L \<equiv> (lasso_reach L, lasso_va L # lasso_cysfx L)"
act_emp_var: fixes x :: "'a" shows "bot_pres (\<alpha> x)"
invar_l: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" assumes "invar (Node k v l r)" shows "invar l"
simps: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "rquicksort' R xs = (if xs = [] then return_pmf ([], 0) else pmf_of_set {..<length xs} \<bind> (\<lambda>i. let x = xs ! i in case threeway_partition x R (delete_index i xs) of (ls, es, rs) \<Rightarrow> rquicksort' R ls \<bind> (\<lambda>(ls, n1). rquicksort' R rs \<bind> (\<lambda>(rs, n2). return_pmf (ls @ x # es @ rs, length xs - 1 + n1 + n2)))))"
eval_Inj_elim: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')" shows "case t of In1 (Inl e) \<Rightarrow> \<exists>v. w = In1 v | In1 (Inr c) \<Rightarrow> w = \<diamondsuit> | In2 e \<Rightarrow> \<exists>v. w = In2 v | In3 e \<Rightarrow> \<exists>v. w = In3 v"
emeasure_density_distr_interval: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> ennreal"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "a \<le> b"   and "f \<in> borel_measurable borel"   and "g \<in> borel_measurable borel"   and "g' \<in> borel_measurable borel"   and "h \<in> borel_measurable borel"   and "subprob_space (density lborel f)"   and "\<And>x. 0 \<le> f x"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "continuous_on {a..b} g'"   and "strict_mono_on g {a..b}"   and "\<And>x. h x \<in> {a..b} \<Longrightarrow> g (h x) = x"   and "{a..b} \<subseteq> range h" shows "emeasure (distr (density lborel f) lborel h) {a..b} = emeasure (density lborel (\<lambda>x. f (g x) * ennreal (g' x))) {a..b}"
arr_iff_in_hom: fixes f :: "'b" shows "D.arr f = \<guillemotleft>f : D.dom f \<Rightarrow>\<^sub>D D.cod f\<guillemotright>"
assms_1: shows "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U"
assms_3: assumes "x \<notin> lvars (\<T> s)"   and "lt dir c (\<langle>\<V> s\<rangle> x)"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)" shows "P (update x c (update\<B>\<I> (UBI_upd dir) i x c s))"
path_split_single: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c"   and t :: "'d list"   and e :: "'d"   and sn :: "'c" assumes "s1 (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> sn" shows "\<exists>s'. s1 t\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' e\<longrightarrow>\<^bsub>SES\<^esub> sn"
ncc_precond_def: shows "ncc_precond TYPE('c) = (\<forall>(Xi, X)\<in>appr_rel. ncc X)"
ideD_1: fixes a :: "'c" assumes "Fop.Bop.ide a" shows "Fop.Bop.arr a"
low_mds_eq_dma: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2" shows "dma\<^sub>C mem\<^sub>1 = dma\<^sub>C mem\<^sub>2"
fst_quot_of_fract_eq_0_iff: fixes x :: "'a fract" shows "(fst (quot_of_fract x) = (0::'a)) = (x = 0)"
dims1_def: shows "st.dims1 = nths [2, 2] {0}"
simps_2: fixes m :: "nat"   and ms :: "nat list" shows "hd0 (m # ms) = m"
bij_betw_axis_sphere: shows "bij_betw (\<lambda>x. (\<i> - x) / (\<i> + x)) {x. Im x = 0} (sphere 0 1 - {- 1})"
\<alpha>_limit_set_def: fixes x :: "'a" shows "rev.\<alpha>_limit_set x = {p. rev.\<alpha>_limit_point x p}"
Rep_cfun_inject: fixes x :: "'c \<rightarrow> 'd"   and y :: "'c \<rightarrow> 'd" shows "(Rep_cfun x = Rep_cfun y) = (x = y)"
sig_of_p_less: fixes j :: "nat" assumes "Inr j \<in> set ps" shows "sig_of_pair p \<prec>\<^sub>t term_of_pair (0::'a, j)"
B_Issues_A: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says B A (Crypt servK (Number T3)) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "B Issues A with Crypt servK (Number T3) on evs"
simps_2: fixes x :: "('a, 'b) Combinators"   and v :: "('a, 'b) Combinators"   and va :: "('a, 'b) Combinators list" shows "list2policyR (x # v # va) = list2policyR (v # va) \<oplus> x"
reachable_steps_elem: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "Graph_Start_Defs.reachable E s\<^sub>0 x"   and "Graph_Defs.steps E xs"   and "y \<in> set xs"   and "hd xs = x" shows "Graph_Start_Defs.reachable E s\<^sub>0 y"
subtype_sound: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set" assumes "t \<le>:\<^sub>P t'" shows "\<forall>mem. pred P mem \<longrightarrow> type_max t mem \<le> type_max t' mem"
r_dovetail_def: shows "r_dovetail \<equiv> Cn 2 r_not [Cn 2 r_result1 [Cn 2 r_pdec2 [recf.Id 2 1], recf.Id 2 0, Cn 2 r_pdec1 [recf.Id 2 1]]]"
valid_write_sops: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>write_sops sb. valid_sop sop"
concat_take_Suc_app_nth: fixes x :: "nat"   and l :: "'a list list" assumes "x < length l" shows "concat (take (Suc x) l) = concat (take x l) @ l ! x"
word_log2_maximum: fixes w :: "'b word"   and n :: "nat" assumes "bit w n" shows "n \<le> word_log2 w"
abs_leq_squared: fixes z :: "int" shows "\<bar>z\<bar> \<le> z\<^sup>2"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "Late_Semantics.residual" assumes "a<b>.P \<longmapsto>\<^sub>l Rs" obtains x :: "name" and aa :: "name" and Pa :: "pi" where "a = aa \<and> [b].P = [x].Pa" and "Rs = aa<x> \<prec>\<^sub>l Pa" and "x \<noteq> aa"
surjective: fixes r :: "('e, 'f) TS_scheme" shows "r = \<lparr>init = init r, trans = TS.trans r, \<dots> = TS.more r\<rparr>"
simps_4: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalQuadratic var a b c d (Neq p) = neg (allZero p var)"
expands_to_drop_zero: fixes C :: "'a"   and f :: "real \<Rightarrow> real"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "\<forall>\<^sub>F x in at_top. eval C x = 0"   and "(f expands_to MS (MSLCons (C, e) xs) g) basis" shows "(f expands_to MS xs g) basis"
summable_sums: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f" shows "f sums suminf f"
tape_of_nl_rev: fixes lm :: "nat list" shows "rev (<lm>) = <rev lm>"
bar1: fixes ws :: "letter list list" assumes "good ws" shows "bar ws"
has_derivative_compose: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x))" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
sum_nonneg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "less_eq zero (comm_monoid_add.sum plus zero f A)"
I_or: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "int list" shows "Z.I (or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = Z.I (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
g_2: fixes t :: "real" assumes "t \<in> A" shows "g t = f t"
a_eq_g: fixes m :: "nat" shows "1 / of_nat k * (\<Sum>ma<k. s f g k ma * unity_root k (- int m * int ma)) = a k m"
neg_ci: fixes l :: "'a list"   and P :: "pol" assumes "in_carrier l" shows "Ipol l (neg P) = \<ominus> Ipol l P"
mult_p'_irred: shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G (monom_mult (lc q / lc p) (lp q - lp p) p')"
simps_1: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "Nielson_VCG.pre SKIP Q = Q"
mset_isort: fixes xs :: "'b list" shows "mset (isort xs) = mset xs"
fac_acc_body3_def: shows "fac_acc_body3 \<equiv> \<Lambda> r n. if n = 0 then n2a\<cdot>1 else n2a\<cdot>n >=> r\<cdot>(n -\<^sub># 1)"
cmax2: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" assumes "Card_order r"   and "Card_order s"   and "r \<le>o s" shows "cmax r s =o s"
PseudoHA_HARoot: fixes SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "HARoot (PseudoHA SA D) = SA"
lindemann_weierstrass_integral: fixes u :: "complex"   and f :: "complex poly" shows "((\<lambda>t. exp (u - t) * poly f t) has_contour_integral exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) - (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)) (linepath 0 u)"
assms_5: fixes p :: "int poly" assumes "p \<in> P" shows "0 < Polynomial.degree p"
set_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>set_rel \<equiv> {(A, B). (\<forall>x\<in>A. \<exists>y\<in>B. (x, y) \<in> R) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. (x, y) \<in> R)}"
is_root_insert: fixes h :: "'b tree"   and x :: "'b" assumes "is_root h" shows "is_root (Pairing_Heap_Tree.insert x h)"
greatest_def: fixes P :: "'b \<Rightarrow> bool"   and po :: "'b potype" shows "greatest P po \<equiv> SOME x. x \<in> pset po \<and> P x \<and> (\<forall>y\<in>pset po. P y \<longrightarrow> (y, x) \<in> order po)"
disj_def: fixes p :: "fm"   and q :: "fm" shows "Parametric_Ferrante_Rackoff.disj p q \<equiv> if p = T \<or> q = T then T else if p = F then q else if q = F then p else if p = q then p else Or p q"
list_slice_nth_nth: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat"   and n :: "nat" assumes "m < length xs div k"   and "n < k" shows "list_slice xs k ! m ! n = xs ! (m * k + n)"
l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs create_element ShadowRootClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptr"
ecut_def: fixes f :: "hf \<Rightarrow> 'a"   and x :: "hf" shows "ecut f x \<equiv> \<lambda>y. if y \<^bold>\<in> eclose x then f y else undefined"
dual_cases: fixes x :: "'a dual" obtains y :: "'a" where "x = dual y" and "y \<in> UNIV"
characteriseSeq: fixes C :: "'b sequent" shows "\<exists>A B. C = ( A \<Rightarrow>* B)"
simps: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and s :: "nat"   and l :: "cell list"   and r :: "cell list" shows "P holds_for (s, l, r) = P (l, r)"
assms: shows "\<xi>' \<in> is_rreq \<xi>"
add_sign_intros_4: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"