hcju/isabelleps huggingface dataset:
clop_closure_set: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "range f = Fix f"
irreducible_reachable_imp_matchers_empty: fixes P :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" assumes "((P, \<sigma>), y) \<in> matchrel\<^sup>!"   and "fst y \<noteq> {#}" shows "matchers_map \<sigma> \<inter> matchers (set_mset P) = {}"
chainI: fixes C :: "'b set"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> P\<^sup>=\<^sup>= x y \<or> P\<^sup>=\<^sup>= y x" shows "chain_on P C A"
inv_simp: fixes f :: "'a \<times> 'a" assumes "VxV.iso f" shows "VxV.inv f = (local.inv (fst f), local.inv (snd f))"
mult_aform'_float_1: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "mult_aform' p x y = ((a, b), ba)" shows "a \<in> float"
simps_6: fixes x :: "'e" shows "phantom_nondetT (NondetT x) = {}"
coeff_C_eq: fixes x :: "'b"   and n :: "nat" shows "coeff (C x) n = (if n = 0 then x else (0::'b))"
l_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
rel_G_pos_distr: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and uu :: "('ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah) itself"   and L1 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and L2 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and L1' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and L2' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2')"
composition: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and e :: "b_e"   and t3s :: "t list" assumes "\<C> \<turnstile> es : t1s _> t2s"   and "\<C> \<turnstile> [e] : t2s _> t3s" shows "\<C> \<turnstile> es @ [e] : t1s _> t3s"
sup_inf_absorb: fixes x :: "'a upred"   and y :: "'a upred" shows "(x \<or> x \<and> y) = x"
vector_sup_closed: fixes x :: "'b"   and y :: "'b" assumes "comp_inf.vector x"   and "comp_inf.vector y" shows "comp_inf.vector (x \<squnion> y)"
setleI: fixes S :: "'a set"   and x :: "'a" assumes "\<forall>y\<in>S. y \<le> x" shows "S *<= x"
A_2: fixes a :: "'a" assumes "a \<in> A" shows "k a = \<Union> (k ` A)"
reachable_imp_identifiers_available_2: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s"   and "infinite UNIV" shows "\<exists>l. l \<notin> LID\<^sub>G s"
simps_1: fixes A :: "'a set" assumes "finite A" shows "esize A = enat (card A)"
d\<mu>_impl: shows "d\<mu>_impl fs = IArray.of_fun (\<lambda>i. IArray.of_fun (d\<mu> i) (Suc i)) (length fs)"
set_relD1: fixes s :: "'d set"   and s' :: "'e set"   and R :: "('d \<times> 'e) set"   and x :: "'d" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x \<in> s" shows "\<exists>x'\<in>s'. (x, x') \<in> R"
right_totalE: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and y :: "'g" assumes "right_total A" obtains x :: "'f" where "A x y"
inj_on_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and N :: "nat set" assumes "class.semiring_char_0 one times plus zero" shows "inj_on (semiring_1.of_nat one plus zero) N"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "new_shadow_root_ptr |\<in>| shadow_root_ptr_kinds h'"
INF_lower2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
arr_mset_append_alt: fixes l :: "int"   and r :: "int"   and arr :: "int \<Rightarrow> 'a"   and b :: "'a" assumes "l \<le> r" shows "arr_mset (arr(r := b)) l (r + 1) = arr_mset arr l r + {#b#}"
iso_def: fixes f :: "('a, 'a set) arr" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
wpt_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> '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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.wpt one times inf top x = x"
simps_8: shows "ccode.Eq \<noteq> ccode.Lt"
ground_aux_def: fixes t :: "trm"   and S :: "atom set" shows "ground_aux t S \<equiv> supp t \<subseteq> S"
add_set2: fixes x :: "'node"   and S :: "'node set"   and mrk :: "'node set"   and y :: "'node" assumes "x \<in> path S mrk"   and "x \<notin> path (insert y S) (insert y mrk)" shows "x = y"
cases: fixes a1 :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and a2 :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "Eq_RESUMPTION a1 a2" obtains a :: "'h" where "a1 = TERMINAL a" and "a2 = TERMINAL a"   | r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set" and b :: "'g" where "a1 = LINEAR b r" and "a2 = LINEAR b r'" and "Eq_RESUMPTION r r'"   | rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" and rs' :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" and c :: "'f" where "a1 = BRANCH c rs" and "a2 = BRANCH c rs'" and "\<And>d. Eq_RESUMPTION (rs d) (rs' d)"
assms_3: fixes s :: "real" assumes "t < s"   and "s \<le> t'" shows "flow0 x s \<noteq> flow0 x t"
subset_open_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a<--<b} \<subseteq> {c<--<d}) = (a = b \<or> a \<in> {c--d} \<and> b \<in> {c--d})"
exact3_comp_0: fixes L :: "('i, 'a, 'j) Module_scheme"   and M :: "('k, 'a, 'j) Module_scheme"   and N :: "('l, 'a, 'j) Module_scheme"   and f :: "'i \<Rightarrow> 'k"   and g :: "'k \<Rightarrow> 'l" assumes "R module L"   and "R module M"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "exact3 R L f M g N" shows "compos L g f = mzeromap L N"
append_Nil2: fixes xs :: "'g list" shows "xs @ [] = xs"
sfilter_shift_decomp: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and ys :: "'b list"   and zs :: "'b stream" assumes "sfilter P xs = ys @- zs"   and "alw (ev (holds P)) xs" shows "\<exists>ys' zs'. xs = ys' @- zs' \<and> filter P ys' = ys \<and> sfilter P zs' = zs"
assms_4: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) list"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "P (set ats) s"   and "index_valid (set ats) s" shows "P (insert a (set ats)) (assert_bound a s)"
axioms_3: 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 "functor A B F"
conj_simps_2: fixes P :: "fm" shows "Ferrack.conj P F = F"
matchT_M_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchT_M c d \<equiv> \<forall>s t s'. s \<approx> t \<and> (c, s) \<rightarrow>t s' \<longrightarrow> (\<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> discr d') \<or> (\<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t')"
evaln_nonstrict_Suc: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and n :: "nat"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w, s')" shows "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>Suc n\<rightarrow> (w, s')"
Ops_FvarsT: fixes f :: "'trm \<Rightarrow> 'trm"   and t :: "'trm" assumes "f \<in> Ops"   and "t \<in> trm" shows "FvarsT (f t) = FvarsT t"
less_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "less_fun f g = (\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k'))"
sFTrue: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and n :: "nat" assumes "pbval b ps"   and "vars b \<subseteq> dom ps" shows "wp\<^sub>3 (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
tl_of_assum_in_assum: fixes P :: "'b com option"   and s :: "'b"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list"   and pre :: "'b set"   and rely :: "('b \<times> 'b) set" assumes "(P, s) # (P, t) # xs \<in> assum (pre, rely)"   and "stable pre rely" shows "(P, t) # xs \<in> assum (pre, rely)"
T_def: shows "T = (\<lambda>n x. c n * inverse x ^ (j + n))"
finite_fourier_poly_def: fixes ws :: "complex list" shows "finite_fourier_poly ws = (let k = length ws in poly_of_list (map (\<lambda>n. complex_of_real (1 / real k) * (\<Sum>m<k. ws ! m * unity_root k (- int n * int m))) [0..<k]))"
derivative_component_fun_component: fixes f :: "real \<Rightarrow> 'a"   and x :: "real"   and i :: "'a" assumes "f differentiable at x" shows "vector_derivative f (at x) \<bullet> i = vector_derivative (\<lambda>x. f x \<bullet> i) (at x)"
nres_order_simps_21: fixes x :: "'a"   and Y :: "'a set" shows "(RETURN x \<le> RES Y) = (x \<in> Y)"
lookahead: fixes x :: "'a"   and p :: "'a" shows " |x\<^sup>\<star>] (d p \<cdot>  |x] d p) =  |x\<^sup>\<star>] d p"
project_Increasing: fixes F :: "'a program"   and G :: "'c program"   and func :: "'a \<Rightarrow> 'e" shows "(F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Increasing func) = (extend h F \<squnion> G \<in> Increasing (func \<circ> f))"
naturality: fixes f :: "'a" assumes "arr f" shows "\<rr>.map (cod f) \<cdot> R f = local.map f \<cdot> \<rr>.map (local.dom f)"
itrm_brackets_opaque: fixes vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_brackets vs x = Some x'" shows "set (opaque x') = set (opaque x) - set vs"
span_eq_real: shows "module_on.span (*\<^sub>R) = span"
assms: fixes t' :: "'a" assumes "Join t' \<in> set cas" shows "\<not> not_final_thread s t' \<and> t \<noteq> t'"
mulex_on_add_mset': fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'b"   and n :: "'b"   and A :: "'b set"   and M :: "'b multiset" assumes "P m n"   and "m \<in> A"   and "n \<in> A"   and "M \<in> multisets A" shows "mulex_on P A (add_mset m M) (add_mset n M)"
OUTfromChCorrect_data16: shows "OUTfromChCorrect data16"
pred_is_dep: fixes m :: "('a, 'b) insert_message"   and i :: "'a" shows "P m = \<lbrakk>i\<rbrakk> \<longrightarrow> i \<in> deps (Insert m)"
runs1a1_def: fixes h :: "role_t \<Rightarrow> atom list \<Rightarrow> atom list"   and runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and rid :: "fid_t" shows "map_runs h runz rid \<equiv> case runz rid of None \<Rightarrow> None | Some (rol, agts, al) \<Rightarrow> Some (rol, agts, h rol al)"
inverts_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "inverts_mat A B \<equiv> A * B = 1\<^sub>m (dim_row A)"
init_item_rhs: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_rhs (init_item r k) = snd r"
bi_unique_alt_def2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_unique R = (R ===> R ===> (=)) (=) (=)"
rtrancl_mono: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s" shows "r\<^sup>* \<subseteq> s\<^sup>*"
Fail_HInv3_R: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "Fail s s' t"   and "HInv3_R s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_R s' p q d"
hom_power: fixes x :: "'a"   and n :: "nat" shows "hom (x ^ n) = hom x ^ n"
A_intro': fixes A :: "'a set"   and y :: "'a"   and z :: "'a" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "y \<in> A"   and "z \<in> X"   and "(y, z) \<in> r"   and "(z, y) \<in> r" shows "z \<in> A"
simps_6: fixes f :: "nat \<Rightarrow> nat"   and e :: "cexpr"   and t :: "pdf_type" shows "map_vars f (\<integral>\<^sub>c e \<partial>t) = \<integral>\<^sub>c map_vars (case_nat 0 (\<lambda>x. Suc (f x))) e \<partial>t"
countable_set_spmf: fixes p :: "'c spmf" shows "countable (set_spmf p)"
of_nat_in_Nats: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n \<in> semiring_1.Nats one plus zero"
l_comb_sc1: fixes A :: "'b set"   and H :: "'a set"   and r :: "'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "r \<in> carrier R"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> n} \<rightarrow> H" shows "r \<cdot>\<^sub>s \<Sigma>\<^sub>e M (\<lambda>k. s k \<cdot>\<^sub>s g k) n = \<Sigma>\<^sub>e M (\<lambda>k. (r \<cdot>\<^sub>r\<^bsub>R\<^esub> s k) \<cdot>\<^sub>s g k) n"
Abs_norm: fixes x :: "'b" shows "Abs (norm x) = Abs x"
elims_2: fixes x :: "'a set \<times> (nat \<Rightarrow> 'b)"   and xa :: "('a, 'b) state" assumes "x \<Turnstile>\<^sub>i\<^sub>s\<^sub>e xa" obtains I :: "'a set" and v :: "nat \<Rightarrow> 'b" and s :: "('a, 'b) state" where "x = (I, v)" and "xa = s" and "v \<Turnstile>\<^sub>t \<T> s \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>b\<^sub>e \<B>\<I> s"
nonce_notin_analz_initState: fixes N :: "nat"   and A :: "agent" shows "Nonce N \<notin> analz (initState A)"
ds_monotone: fixes S :: "'b set"   and T :: "'b set"   and r :: "('b \<times> 'b) set" assumes "S \<subseteq> T" shows "r \<down>s S \<subseteq> r \<down>s T"
subst_comp_notin_dom_eq: fixes x :: "'a"   and \<theta>1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta>2 :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "x \<notin> subst_domain \<theta>1" shows "(\<theta>1 \<circ>\<^sub>s \<theta>2) x = \<theta>2 x"
bi_unique_mod_ring_rel32_2: shows "left_unique mod_ring_rel32"
of_int_div_aux: fixes d :: "int"   and x :: "int" assumes "d \<noteq> 0" shows "of_int x / of_int d = of_int (x div d) + of_int (x mod d) / of_int d"
pw_eq_on_sym: fixes S :: "'b set"   and w :: "'c \<Rightarrow> 'b set"   and w' :: "'c \<Rightarrow> 'b set" assumes "pw_eq_on S w w'" shows "pw_eq_on S w' w"
models_in_range: fixes P :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> P" shows "in_range h"
list_wset_rel_internal_def: fixes R :: "('a \<times> 'b) set" shows "list_wset_rel R = br set top O \<langle>R\<rangle>set_rel"
Un_least: fixes A :: "'c set"   and C :: "'c set"   and B :: "'c set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
cinsert_is_cUn: fixes a :: "'f"   and A :: "'f cset" shows "cinsert a A = cUn (csingle a) A"
islimpt_finite: fixes s :: "'b set"   and x :: "'b" assumes "finite s" shows "\<not> x islimpt s"
at_within_Icc_at: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and x :: "'c"   and b :: "'c" assumes "class.order_topology less_eq less open"   and "less a x"   and "less x b" shows "topological_space.at_within open x (ord.atLeastAtMost less_eq a b) = topological_space.at_within open x UNIV"
assms_4: shows "h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h'"
S_sound_ntrp: fixes u :: "'n"   and eel :: "('p, 'ba) label list el_step"   and sh :: "'n list"   and ch :: "'n list multiset" assumes "(([u], {#}), eel, sh, ch) \<in> ntrp fg" obtains p :: "'p" and u' :: "'n" and v :: "'n" and w :: "('p, 'ba) label list" where "eel = LOC (LCall p # w)" and "(u, Call p, u') \<in> edges fg" and "sh = [v, u']" and "proc_of fg v = p" and "mon_c fg ch = {}" and "\<And>s. s \<in># ch \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p" and "\<And>P. {#[entry fg p]. p \<in># P#} \<subseteq># ch \<Longrightarrow> (v, mon_w fg w, P) \<in> S_cs fg (size P)"
assms_2: fixes k :: "int" assumes "k \<le> n" shows "f $$ k = (0::'a)"
le: shows "(x = 0) = (int_of_uint32 x = 0)"
less_trans: fixes x :: "char"   and y :: "char"   and z :: "char" assumes "less_char x y"   and "less_char y z" shows "less_char x z"
right_hom_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b" shows "right_hom V H \<omega> \<equiv> weak_composition V H \<and> subcategory V (weak_composition.right V H \<omega>) \<and> right_hom_axioms V \<omega>"
tt: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and rest1 :: "'a list" assumes "a \<in> {x, y}" shows "OPT2 rest1 (step [x, y] a (hd (OPT2 (a # rest1) [x, y]))) = tl (OPT2 (a # rest1) [x, y])"
check': shows "check_OR (x0, x1) (a0, a1) s' ((e0', z0'), e1', z1')"
Derivs_alt_def: fixes w :: "'a list"   and L :: "'a list set" shows "Derivs w L = fold Deriv w L"
noFault_c1': shows "\<Gamma>\<turnstile> \<langle>c1',Normal s\<rangle> \<Rightarrow>\<notin>range Fault"
heap_wellformed_induct_si: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>parent. \<lbrakk>\<And>children child. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children; child \<in> set children\<rbrakk> \<Longrightarrow> P (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child); \<And>shadow_root host. \<lbrakk>parent = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host; h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root\<rbrakk> \<Longrightarrow> P (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root); \<And>owner_document disc_nodes node_ptr. \<lbrakk>parent = cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document; h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes; node_ptr \<in> set disc_nodes\<rbrakk> \<Longrightarrow> P (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)\<rbrakk> \<Longrightarrow> P parent" shows "P ptr"
infinite_imp_asc_chain: fixes S :: "nat set" assumes "infinite S" shows "\<forall>m\<in>S. \<exists>n\<in>S. m < n"
inDomConc: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<notin> dom (Cp a)"   and "x \<notin> dom (Cp (list2FWpolicy p))" shows "x \<notin> dom (Cp (list2FWpolicy (a # p)))"
assms_2: shows "from_nat a \<noteq> from_nat b"
free_eta: fixes s :: "Core.term"   and t :: "Core.term"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "loose_bvar1 t i = loose_bvar1 s i"
except_plus: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "except (p + q) S = except p S + except q S"
ladder_last_j_bound: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder \<alpha> D L \<gamma>" shows "ladder_last_j L < length \<gamma>"
simple_function_mult: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "simple_function M f"   and "simple_function M g" shows "simple_function M (\<lambda>x. f x * g x)"
cs_select_id: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<pi> k \<noteq> IFC_def.return prob" shows "IFC_def.cs_select prob \<pi> (IFC_def.cs prob \<pi> k) = k"
d\<mu>_def: fixes i :: "nat"   and j :: "nat" shows "fs.d\<mu> i j = int_of_rat (rat_of_int (fs.d fs (Suc j)) * fs.gs.\<mu> i j)"
item_dot_is_\<alpha>_length: fixes x :: "item" assumes "wellformed_item x" shows "item_dot x = length (item_\<alpha> x)"
continuous_imp_open_vimage: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "continuous_on s f"   and "open s"   and "open B"   and "f -` B \<subseteq> s" shows "open (f -` B)"
h2: shows "(\<Gamma> @ \<Delta>)\<turnstile>T1<:S1"
der_g: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S) \<and> invertible (matrix (g' x))"
assms_1: shows "comm_monoid_mult_class.coprime (h - [:c1:]) (h - [:c2:])"
red_preserves_defass: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "\<D> e \<lfloor>dom l\<rfloor>" shows "\<D> e' \<lfloor>dom l'\<rfloor>"