hcju/isabelleps huggingface dataset:
contractible_space_product_topology: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "contractible_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> (\<forall>i\<in>I. contractible_space (X i)))"
hull_insert: fixes P :: "'b set \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set" shows "P hull insert a S = P hull insert a (P hull S)"
WithoutQReach: shows "qReachable cfg (Proc - Q) cfg2"
concat_map_singleton: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "concat (map (\<lambda>x. [f x]) xs) = map f xs"
mono_neutral_cong_left: fixes T :: "'b set"   and S :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (0::'c)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
append_in_lists_conv: fixes xs :: "'a list"   and ys :: "'a list"   and A :: "'a set" shows "(xs @ ys \<in> lists A) = (xs \<in> lists A \<and> ys \<in> lists A)"
qc_imp_qc_trancl: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "quasi_commute r s" shows "quasi_commute r (s\<^sup>+)"
countable_web_axioms_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "countable_web_axioms \<Gamma> \<equiv> countable \<^bold>E\<^bsub>\<Gamma>\<^esub>"
length_filter_eq: fixes x :: "'b"   and xs :: "'b list" shows "length (filter ((=) x) xs) = count_list xs x"
submanifold_atlasI: fixes c :: "('a, 'b) chart" assumes "c \<in> atlas" shows "restrict_chart (domain \<psi>) c \<in> sub.atlas"
card_of_UNIV: fixes A :: "'a set" shows "|A| \<le>o |UNIV|"
assms_2: shows "character (G\<lparr>carrier := H\<rparr>) \<chi>"
wrm_sort: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<and> sortedTree h t \<longrightarrow> sortedTree h (wrm h t)"
let_to_bind_conv: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e nres" shows "Let x f = RETURN x \<bind> f"
Diagonalize_Cod: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (monoidal_language.Cod C t) = monoidal_language.Cod C (monoidal_language.Diagonalize C t)"
a_neigh_in: shows "a_neigh \<in> arcs G"
sup0: fixes a :: "'d" shows "a \<equiv> a \<squnion> \<bottom>"
sharing_consis: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent \<S> \<O> sb"
power_eq_imp_eq_base: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat"   and b :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times a n = power.power one times b n"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "a = b"
simps_1: fixes x :: "Preliminaries.var" shows "TE.wt (Var x) = True"
naturality: fixes f :: "'a \<times> 'a \<times> 'a" assumes "C.CCC.arr f" shows "C.\<alpha>' (C.CCC.cod f) \<cdot> C.T.ToCT f = C.T.ToTC f \<cdot> C.\<alpha>' (C.CCC.dom f)"
prod_lub_def: fixes Sup_a :: "'b set \<Rightarrow> 'b"   and Sup_b :: "'c set \<Rightarrow> 'c"   and Y :: "('b \<times> 'c) set" shows "prod_lub Sup_a Sup_b Y = (Sup_a (fst ` Y), Sup_b (snd ` Y))"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Src (t \<^bold>\<star> u) = Src u"
simps_6: fixes b :: "nat"   and I :: "\<I>"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Prev I \<phi>) = MFOTL.fvi b \<phi>"
simps_1: fixes n :: "nat" shows "val_size (VNat n) = 0"
frag_extend_minus: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and x :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend f (- x) = - frag_extend f x"
Scons_inject: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "Scons M N = Scons M' N'" obtains "M = M'" and "N = N'"
rbt_map_entry_inv2_1: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "inv2 (rbt_map_entry k f t) = inv2 t"
compseriesTr4: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "compseries G n f" shows "w_cmpser G n f"
P_1_lemma: fixes res :: "int \<times> int" assumes "0 \<le> fst res"   and "fst res < int p"   and "0 \<le> snd res"   and "snd res < int q" shows "\<exists>!x. P_1 res x"
CARD_1_def: shows "class.CARD_1 TYPE('b) \<equiv> CARD('b) = 1"
notsdom_notsdominate': fixes i :: "nat"   and j :: "nat" assumes "\<not> strict_dom i j"   and "j < length start" shows "\<not> strict_dominate i j"
convex_ball: fixes x :: "'b"   and e :: "real" shows "convex (ball x e)"
one_unique: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "x \<cdot> a = x" shows "a = \<one>"
max_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "Max x y = Max y x"
face_cycle_set_def: fixes a :: "'b" shows "G'.face_cycle_set a = orbit G'.face_cycle_succ a"
frequently_going_to_at_topE: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> real" assumes "frequently P (f going_to at_top)" obtains g :: "nat \<Rightarrow> 'a" where "\<And>n. P (g n)" and "LIM n sequentially. f (g n) :> at_top"
i1: shows "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
absorb1: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b" shows "linordered_monoid.min a b = a"
redT_updW_neq_Some_SomeD: fixes t' :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t :: "'b"   and w :: "'c" assumes "redT_updW t' ws wa ws'"   and "ws' t = \<lfloor>InWS w\<rfloor>"   and "ws t \<noteq> \<lfloor>InWS w\<rfloor>" shows "t = t' \<and> wa = Suspend w"
lifted_antisymmetric: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "f \<le>\<le> g"   and "g \<le>\<le> f" shows "f = g"
star_n_eq_iff: fixes X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'c" shows "(star_n X = star_n Y) = (\<forall>\<^sub>F n in \<U>. X n = Y n)"
2: shows "\<forall>i<length (t # T). P ((t # T) ! i) ((u # U) ! i)"
right_cancel: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a"   and z :: "'a \<times> 'a" assumes "x \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "z \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)" shows "(y \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> x = z \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> x) = (y = z)"
term_refl: fixes t :: "dB" shows "t \<leftrightarrow> t"
isOK_iff: fixes m :: "'c + 'd" shows "isOK m = (\<exists>x. m = Inr x)"
onelem: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<cdot> S \<parallel> Multirelations.U \<subseteq> R \<otimes> (S \<parallel> Multirelations.U)"
design_blocks_nempty: shows "\<B>\<^sup>C \<noteq> {#}"
hd_multc: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and c :: "val" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "val_type c = REAL"   and "c \<noteq> RealVal 0" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Mult $$ <e ,  Val c> \<Rightarrow> (\<lambda>\<rho> x. f \<rho> (op_sem Mult <|x, op_sem Inverse c|>) * ennreal (inverse \<bar>extract_real c\<bar>))"
subst_rename: fixes c :: "name"   and t1 :: "lam"   and a :: "name"   and t2 :: "lam" assumes "c \<sharp> t1" shows "t1[a::=t2] = ([(c, a)] \<bullet> t1)[c::=t2]"
in_set_update_listI: fixes y :: "nat \<times> 'd"   and n :: "nat"   and x :: "'d"   and ys :: "(nat \<times> 'd) list" assumes "y = (n, x) \<or> fst y \<noteq> n \<and> y \<in> set ys" shows "y \<in> set (update_list n x ys)"
length_partition_by_median_7: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "1 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "0 < length r"
domintros_1: fixes uu :: "lit" shows "is_value_dom (Lit uu)"
elims: fixes x :: "nat option"   and y :: "enat" assumes "Dijkstra_Abstract.enat_of_option x = y" obtains "x = None" and "y = \<infinity>"   | n :: "nat" where "x = Some n" and "y = enat n"
mult_sv_mult_sv: fixes k1 :: "complex"   and k2 :: "complex"   and v :: "complex \<times> complex" shows "k1 *\<^sub>s\<^sub>v (k2 *\<^sub>s\<^sub>v v) = (k1 * k2) *\<^sub>s\<^sub>v v"
AddSA_CompFun_ran3: fixes SA1 :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and SA2 :: "('d, 'e, 'f) seqauto"   and SA3 :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and T :: "'d" assumes "States SA1 \<inter> HAStates A = {}"   and "States SA2 \<inter> (HAStates A \<union> States SA1) = {}"   and "States SA3 \<inter> (HAStates A \<union> States SA1 \<union> States SA2) = {}"   and "S \<in> HAStates A"   and "T \<in> States SA1" shows "ran (CompFun (A [++] (S, SA1) [++] (T, SA2) [++] (T, SA3))) = insert {} (insert {SA3, SA2} (ran (CompFun (A [++] (S, SA1)))))"
AngFlat_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "AngFlat a \<equiv> QCongA a \<and> (\<forall>A B C. a A B C \<longrightarrow> Bet A B C)"
WhileT: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and c :: "char list exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>3 :: "char list exp"   and s\<^sub>3 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P \<turnstile> \<langle>while (e) c,s\<^sub>2\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>"
IH: fixes x2aa :: "fterm" assumes "x2aa \<in> set ts"   and "\<forall>x\<in>vars\<^sub>t x2aa. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "x2aa \<cdot>\<^sub>t \<sigma>\<^sub>1 = x2aa \<cdot>\<^sub>t \<sigma>\<^sub>2"
subset_Compl_self_eq: fixes A :: "'b set" shows "(A \<subseteq> - A) = (A = {})"
components_are_iso: fixes a :: "'a \<times> 'a" assumes "C.VV.ide a" shows "C.iso (fst a \<star>\<^sub>C snd a)"
sorted_list_of_set_alt_def: fixes A :: "'b set" shows "sorted_list_of_set_alt A = (if A = {} then [] else THE xs. is_sorted_list_of_set A xs)"
Proc_CFG_Entry_no_targetnode: fixes prog :: "cmd"   and n :: "label"   and et :: "p_edge_kind" assumes "prog \<turnstile> n -et\<rightarrow>\<^sub>p Entry" shows "False"
GF_advice_inter: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "X \<inter> subformulas\<^sub>\<mu> \<phi> \<subseteq> S" shows "\<phi>[X \<inter> S]\<^sub>\<nu> = \<phi>[X]\<^sub>\<nu>"
k_3: fixes p :: "nat"   and X :: "'a topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "singular_chain (Suc p) X (k p c)"
\<S>: fixes S' :: "('fun, 'var) strand_step list" assumes "S' \<in> update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)" shows "S' = S \<or> S' \<in> \<S>"
eThin: fixes A' :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "vassn"   and A :: "triple set" assumes "A' \<turnstile>\<^sub>e {P} e {Q}"   and "A' \<subseteq> A" shows "A \<turnstile>\<^sub>e {P} e {Q}"
AX_mono: fixes p :: "'b set"   and q :: "'b set" assumes "p \<subseteq> q" shows "\<^bold>A\<^bold>X p \<subseteq> \<^bold>A\<^bold>X q"
inserted_ids_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "inserted_ids ops \<equiv> List.map_filter (\<lambda>oper. case oper of (oid, Insert ref val) \<Rightarrow> Some oid | (oid, Delete x) \<Rightarrow> Map.empty x) ops"
assms: shows "(\<lambda>n. f (Suc n)) sums l"
WT_sub_gpvsD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and gpv' :: "('f, 'd, 'e) gpv" assumes "\<I> \<turnstile>g gpv \<surd>"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "\<I> \<turnstile>g gpv' \<surd>"
univ_poly_is_principal: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subfield K M" shows "principal_domain (K [X]\<^bsub>M\<^esub>)"
simple_function_Pair: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "simple_function M f"   and "simple_function M g" shows "simple_function M (\<lambda>x. (f x, g x))"
map_component_term_of_pair: fixes f :: "'k \<Rightarrow> 'k"   and t :: "'a"   and k :: "'k" shows "map_component f (term_of_pair (t, k)) = term_of_pair (t, f k)"
dominates_arcsD: fixes v :: "'a"   and w :: "'a"   and u :: "'b" assumes "v \<rightarrow>\<^bsub>H.del_arc u\<^esub> w" shows "v \<rightarrow>\<^bsub>del_arc u\<^esub> w"
finite: fixes A :: "'a set" assumes "class.finite TYPE('a)" shows "finite A"
adv: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>" shows "advantage \<A> (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "F.F g (insert x A) = (if x \<in> A then F.F g A else g x \<^bold>* F.F g A)"
cong_solve_coprime_poly: fixes a :: "'c poly"   and n :: "'c poly" assumes "coprime a n" shows "\<exists>x. [a * x = 1] (mod n)"
\<Z>_wellformed: fixes t :: "symbol \<times> character list"   and k :: "nat"   and u :: "nat" assumes "t \<in> \<Z> k u" shows "wellformed_token t"
assms: shows "knows P [sE m] \<or> knows Q [sE m]"
bisim_simple_\<R>\<^sub>3: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and lc :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set"   and lc' :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.\<R>\<^sub>3_aux ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type lc \<Gamma> \<S> P lc'" shows "fst (fst lc) = fst (fst lc')"
of_nat_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
ord_elems: fixes a :: "'a" assumes "finite (carrier (mult_of R))"   and "a \<in> carrier (mult_of R)" shows "{a [^] x |x. x \<in> UNIV} = {a [^] x |x. x \<in> {0..G.ord a - 1}}"
and_and_not: fixes a :: "'a word"   and b :: "'a word" shows "(a AND b) AND NOT b = 0"
vector_derivative_at: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real" assumes "(f has_vector_derivative f') (at x)" shows "vector_derivative f (at x) = f'"
widen_refT: fixes T :: "ty"   and P :: "'b prog"   and U :: "ty" assumes "is_refT T"   and "P \<turnstile> U \<le> T" shows "is_refT U"
tail_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "tail G (rev_G a) = head G a"
id_o: fixes g :: "'d \<Rightarrow> 'e" shows "id \<circ> g = g"
two_orders: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "x \<le> y"
f1: shows "fst (memory_read 10 address s1) \<noteq> None \<and> fst (memory_read 10 address s2) \<noteq> None"
completeness_def: shows "\<Sigma>_AND.completeness \<equiv> \<forall>h w e. (h, w) \<in> Rel_AND \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (\<Sigma>_AND.completeness_game h w e) True = 1"
quotient_of_split_asm: fixes P :: "int \<times> int \<Rightarrow> bool"   and q :: "rat" shows "P (quotient_of q) = (\<nexists>a b. 0 < b \<and> coprime a b \<and> q = Rat.Fract a b \<and> \<not> P (a, b))"
vectorize_monomial: fixes v :: "'t"   and c :: "'c" shows "vectorize_poly (monomial c v) = monomial (monomial c (pp_of_term v)) (component_of_term v)"
image_mset_subseteq_mono: fixes A :: "'b multiset"   and B :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "A \<subseteq># B" shows "image_mset f A \<subseteq># image_mset f B"
l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs \<and> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
avg_correct: fixes s :: "(nat, unit) HashMap.hashmap" assumes "my_invar s" shows "hs_ctx.avg s = average (my_\<alpha> s)"
A: shows "A \<in> carrier_mat (dim_col A) (dim_col A)"
cases: fixes a1 :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and a2 :: "'a"   and a3 :: "String.literal"   and a4 :: "'a list"   and a5 :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" assumes "unique_static_operation a1 a2 a3 a4 a5" obtains operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list" and \<tau> :: "'a" and name :: "String.literal" and \<pi> :: "'a list" and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" where "a1 = operations" and "a2 = \<tau>" and "a3 = name" and "a4 = \<pi>" and "a5 = oper" and "static_operation' operations \<tau> name \<pi> oper" and "\<not> static_operation_not_unique operations \<tau> name \<pi> oper"
simp_from_to: fixes i :: "int"   and j :: "int" shows "{i..j} = (if j < i then {} else insert i {i + 1..j})"
merge0_correct: shows "\<forall>I I'. oneone I \<longrightarrow> oneone I' \<longrightarrow> PlaneGraphIso.compat I' I \<longrightarrow> map_of (merge0 I' I) = map_of I ++ map_of I'"
arity_def: fixes T :: "Unary_PCF.type" shows "arity T = ar T"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
gcd_poly_code_def: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd_poly_code p q = (if p = 0 then normalize q else if q = 0 then normalize p else smult (gcd (content p) (content q)) (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
r_dovedelay_def: shows "r_dovedelay \<equiv> Cn 3 r_and [Cn 3 r_dovetail [recf.Id 3 1, recf.Id 3 0], Cn 3 r_ifle [recf.Id 3 2, recf.Id 3 0, r_constn 2 0, r_constn 2 1]]"
eval_det: fixes lc :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)"   and lc' :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)"   and lc'' :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)" assumes "eval_abv\<^sub>A lc lc'"   and "eval_abv\<^sub>A lc lc''" shows "lc' = lc''"
fields_def2: fixes P :: "'b prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "fields P C = FDTs"
test_prop: fixes P :: "'a \<Rightarrow> bool" shows "(\<forall>x. test x \<longrightarrow> P x) = (\<forall>x. P (t x))"
bijI: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f"   and "surj f" shows "bij f"
observable3_def: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set" shows "observable3 ob P \<equiv> ob -` ob ` P \<subseteq> P"
Max_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and N :: "'a set" assumes "class.linorder less_eq less"   and "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "less_eq (linorder.Max less_eq M) (linorder.Max less_eq N)"
agents_ge_4: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "sds_impossibility agents alts sds" shows "4 \<le> card agents"
plus_stream_shd: fixes x :: "'a stream"   and y :: "'a stream" shows "shd (x + y) = shd x + shd y"
R_def: shows "R \<equiv> SOME R. new_last \<leadsto>R\<leadsto>\<^bsub>H\<^esub> v1 \<and> (\<forall>R'. new_last \<leadsto>R'\<leadsto>\<^bsub>H\<^esub> v1 \<longrightarrow> length R \<le> length R')"
enum_ex_H2_def: fixes P :: "H2 \<Rightarrow> bool" shows "enum_class.enum_ex P = (P H1 \<or> P H2)"
assms_2: shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M = ennreal x"
unique_connecting_path_decomp_1: fixes v :: "'a"   and w :: "'a"   and ps :: "'a list"   and u :: "'a"   and ps' :: "'a list" assumes "v \<in> V"   and "w \<in> V"   and "v \<leadsto> w = ps @ u # ps'" shows "ps @ [u] = v \<leadsto> u"
is_non_volatile_Read\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_non_volatile_Read\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
in_Joints_intervalD: fixes x :: "'a"   and xs :: "'a list"   and X :: "'a \<times> 'a pdevs"   and XS :: "('a \<times> 'a pdevs) list"   and p :: "nat" assumes "x # xs \<in> Joints (X # XS)" shows "x \<in> {Inf_aform' p X..Sup_aform' p X} \<and> xs \<in> Joints XS"