hcju/isabelleps huggingface dataset:
findn_fresh: fixes x :: "name"   and \<theta>_n :: "(name \<times> coname \<times> trm) list" assumes "x \<sharp> \<theta>_n" shows "\<theta>_n nmaps x to None"
rel_cset_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f cset"   and b :: "'g cset" shows "rel_cset R a b = ((\<forall>t\<in>rcset a. \<exists>u\<in>rcset b. R t u) \<and> (\<forall>t\<in>rcset b. \<exists>u\<in>rcset a. R u t))"
assms: shows "run (f \<bind> (\<lambda>_. g)) \<sigma> \<sigma>'' res"
tauChainInduct2: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "\<And>P. Prop P P"   and "\<And>P P' P''. \<lbrakk>\<Psi> \<rhd> P' \<longmapsto> \<tau> \<prec> P''; \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'; Prop P P'\<rbrakk> \<Longrightarrow> Prop P P''" shows "Prop P P'"
sall_const_TT: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "sall\<cdot>(\<Lambda> x. TT)\<cdot>xs = TT"
lfinite_llist_of_stream: fixes xs :: "'a stream" shows "linfinite (llist_of_stream xs)"
choiceSet_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "choiceSet A r \<equiv> {x \<in> A. \<forall>y\<in>A. x \<^bsub>r\<^esub>\<preceq> y}"
invariant_def: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" shows "(A \<TTurnstile> (I \<rightarrow>) P) = (\<forall>s\<in>reachable A I. P s)"
pc: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "0 \<le> awalk_cost c p"
B: fixes x :: "'n" assumes "x \<in> cbox a b" shows "norm x \<le> B"
satisfies_bounded_fut: fixes idx :: "'n"   and \<psi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and b :: "bool" assumes "local.wf idx \<psi>"   and "#\<^sub>V \<AA> = idx" shows "(\<AA> \<Turnstile>\<^sub>b fut b idx \<psi>) = (if b then \<exists>k. (SNOC (zero idx) ^^ k) \<AA> \<Turnstile>\<^sub>b \<psi> else \<forall>k. (SNOC (zero idx) ^^ k) \<AA> \<Turnstile>\<^sub>b \<psi>)"
linorder_not_less: fixes x :: "'e"   and y :: "'e" shows "(\<not> x < y) = (y \<le> x)"
i: shows "\<forall>k\<in>K. \<exists>X. k = Pi\<^sub>E UNIV X \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
parts_keyfree: shows "parts keyfree \<subseteq> keyfree"
path_edges: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Digraph G"   and "Digraph.walk G xs \<and> distinct xs"   and "(v, w) \<in> Digraph.edges_of_walk xs" shows "\<exists>xs_pre xs_post. xs = xs_pre @ v # w # xs_post \<and> (v, w) \<notin> Digraph.edges_of_walk (xs_pre @ [v]) \<and> (v, w) \<notin> Digraph.edges_of_walk (w # xs_post)"
finfun_rec_const: fixes g :: "('a, 'c) finfun"   and c :: "'b" shows "finfun_rec (\<lambda>b. finfun_comp (Pair b) g) (\<lambda>a b c. finfun_update c a (b, finfun_apply g a)) (finfun_const c) = finfun_comp (Pair c) g"
sqrt_unique: fixes m :: "nat"   and n :: "nat" assumes "m\<^sup>2 \<le> n"   and "n < (Suc m)\<^sup>2" shows "Discrete.sqrt n = m"
body: shows "P,E(this \<mapsto> Class (last Cs'), pns [\<mapsto>] Ts) \<turnstile> \<langle>new_body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"
upd_incoming_def: fixes n :: "('d, 'f) node_scheme"   and ns :: "('d, 'e) node_scheme set" shows "upd_incoming n ns \<equiv> upd_incoming_f n ` ns"
atLeastPlusOneLessThan_greaterThanLessThan_integer: fixes l :: "integer"   and u :: "integer" shows "{l + 1..<u} = {l<..<u}"
fps_deriv_sub: fixes f :: "'b fps"   and g :: "'b fps" shows "fps_deriv (f - g) = fps_deriv f - fps_deriv g"
sep_disj_add: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y ## z"   and "x ## y" shows "x ## y + z = x + y ## z"
valid_behaviours_def: fixes state :: "'c \<Rightarrow> 'd set \<times> 'd set"   and msg :: "'d \<times> ('d, 'c) operation" shows "valid_behaviours state msg \<equiv> case msg of (i, Create j e) \<Rightarrow> i = j | (i, Delete is e) \<Rightarrow> is = fst (state e) \<union> snd (state e) | (i, Append j e) \<Rightarrow> i = j | (i, Expunge e mo j) \<Rightarrow> i = j \<and> mo \<in> snd (state e) | (i, Store e mo j) \<Rightarrow> i = j \<and> mo \<in> snd (state e)"
finite_filter: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite S" shows "finite (Set.filter P S)"
frontier_eq_empty: fixes S :: "'a set" shows "(frontier S = {}) = (S = {} \<or> S = UNIV)"
distr: fixes t :: "real" assumes "a \<le> t"   and "t \<le> b" shows "prob {x \<in> space M. X x \<le> t} = (t - a) / (b - a)"
fps_X_nth: fixes n :: "nat" shows "fps_nth fps_X n = (if n = 1 then 1::'a else (0::'a))"
\<Psi>_Fun_mapsto: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<Psi>.FUN (y, x) \<in> HomD.set (y, G x) \<rightarrow> HomC.set (F y, x)"
C_6: fixes C :: "c \<Rightarrow> w \<Rightarrow> bool"   and A :: "c \<Rightarrow> w \<Rightarrow> bool"   and B :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. (\<lambda>w. C c \<sqsubseteq> (\<lambda>v. A c v = B c v)) \<sqsubseteq> (\<lambda>w. ob (C c) (A c) = ob (C c) (B c))"
simps_1: fixes P :: "'c" shows "strip (SKIP {P}) = com.SKIP"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [Label n es0 LI] : ts _> ts'"
Busemann_function_eq_additive_strength: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion"   and n :: "nat"   and x :: "'b" assumes "isometry f"   and "Gromov_extension f xi = xi" shows "\<bar>Busemann_function_at xi ((f ^^ n) x) x - real n * additive_strength f xi\<bar> \<le> 2 * deltaG TYPE('b)"
enab_gamma2: shows "\<turnstile> $pc2 = #g \<longrightarrow> Enabled gamma2"
hmem_induct: fixes P :: "hf \<Rightarrow> bool"   and x :: "hf" assumes "\<And>x. (\<And>y. y \<^bold>\<in> x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P x"
simps: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com \<times> 'b"   and a2 :: "'b" shows "PL.MtransT tval aval a1 a2 = (\<exists>cf cf' s''. a1 = cf \<and> a2 = s'' \<and> PL.MtransC tval aval cf cf' \<and> PL.transT tval aval cf' s'')"
inf_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<cdot> z = x \<cdot> (y \<cdot> z)"
type_update_select_length1_2: fixes cts :: "ct list"   and tm :: "checker_type" assumes "type_update_select (TopType cts) = tm"   and "length cts = 1"   and "tm \<noteq> Bot" shows "tm = TopType [TAny]"
measurable_bernoulli: shows "bernoulli \<in> stock_measure REAL \<rightarrow>\<^sub>M subprob_algebra (stock_measure BOOL)"
case_eq_if: fixes f :: "'e \<Rightarrow> 'e stream \<Rightarrow> 'd"   and stream :: "'e stream" shows "(case stream of x ## xa \<Rightarrow> f x xa) = f (shd stream) (stl stream)"
eq: shows "eq_for_lvar_code t (lhs eq) = eq"
bdd_above_top: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and A :: "'c set" assumes "class.order_top less_eq less top" shows "preorder.bdd_above less_eq A"
i_less_greatest: shows "i < (GREATEST m. \<not> is_zero_row_upt_k m k A)"
assms: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x =tag= y"   and "x @ z \<in> A" shows "y @ z \<in> A"
iTILL_set_UN_set_eq: shows "iTILL_set = iTILL_UN_set"
strict_trans1: fixes a :: "'h multiset"   and b :: "'h multiset"   and c :: "'h multiset" assumes "a \<subseteq># b"   and "b \<subset># c" shows "a \<subset># c"
entries_rbtreeify_f: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "entries (fst (rbtreeify_f n kvs)) = take n kvs"
semiring_0_cancel_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semiring_0_cancel plus minus zero times \<equiv> class.cancel_comm_monoid_add plus minus zero \<and> class.semiring plus times"
\<Z>_srngD_1: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_vfsequence \<alpha> \<SS>"
ih_3: shows "{$$} \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>"
function_one_eval: fixes a :: "'c" assumes "a \<in> S" shows "\<one>\<^bsub>F\<^esub> a = \<one>"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> nfa.states M" shows "nfa.nxt M q x \<subseteq> nfa.states M"
antisym: fixes b :: "'k"   and a :: "'k" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
zero_hom_0_axioms_def: fixes hom :: "'b \<Rightarrow> 'c" shows "zero_hom_0_axioms hom \<equiv> \<forall>x. hom x = (0::'c) \<longrightarrow> x = (0::'b)"
wf_HasFoo: shows "wf_idecl tprg (HasFoo, HasFooInt)"
list2listNMT: fixes x :: "'b list"   and sem :: "'b \<Rightarrow> 'c" assumes "x \<noteq> []" shows "map sem x \<noteq> []"
\<tau>move2Sync3: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) (Suc (length (compE2 o'))) None"
subgraph': fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and V :: "'c \<Rightarrow> bool" assumes "E x y"   and "V x"   and "V y" shows "Subgraph_Node_Defs.E' E V x y"
embed_o: fixes P :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "\<guillemotleft> P \<guillemotright> \<circ> f = \<guillemotleft> P \<circ> f \<guillemotright>"
fMax_eqI: fixes A :: "'a fset"   and x :: "'a" assumes "\<And>y. y |\<in>| A \<Longrightarrow> y \<le> x"   and "x |\<in>| A" shows "fMax A = x"
timpls_transformable_to_pred_inv_2: fixes A :: "'e set set"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to_pred A (Fun f T) t" shows "length T = length (args t)"
op_dg_components_1: fixes \<CC> :: "V" shows "op_dg \<CC>\<lparr>Obj\<rparr> = \<CC>\<lparr>Obj\<rparr>"
hyps: fixes k :: "nat"   and cs :: "('b, 'a, 'c) com list" assumes "f k = (cs, css, Stuck)" shows "False"
c: shows "list_all (check_poly_weak_mono_discrete p) (poly_vars_list p)"
vertex_in_separator_1: fixes s :: "'c"   and t :: "'c"   and v :: "'a" assumes "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<in> left_part s t"   and "v \<in> left_part t s" shows "v \<in> bag s"
m_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.m p q = card (GAUSS.E p (int q))"
hom_in_carr: fixes f :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme"   and x :: "'e" assumes "f \<in> hom G H"   and "x \<in> carrier G" shows "f x \<in> carrier H"
dghm_const_components_3: fixes \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" shows "dghm_const \<CC> \<DD> a f\<lparr>HomDom\<rparr> = \<CC>"
set_of_embed_is: fixes xs :: "'a list" shows "set (List_Supplement.embed xs) = {[x] |x. x \<in> set xs}"
shift_map_keys_def: fixes t :: "'a"   and f :: "'e \<Rightarrow> 'd"   and m :: "('t, 'e) fmap" shows "shift_map_keys t f m = fmmap f (shift_keys t m)"
simps_1: fixes A :: "'b mat" shows "A ^\<^sub>m 0 = 1\<^sub>m (dim_row A)"
that_3: fixes CTa :: "nat \<Rightarrow> classDef option"   and D :: "nat"   and Ca :: "nat"   and esa :: "exp list" assumes "x = CTa"   and "xa = Cast D (New Ca esa)"   and "xb = New Ca esa"   and "CTa \<turnstile> Ca <: D" shows "thesis"
src_port_ipv4_TCPUDP_def: fixes x :: "int \<times> ((int \<times> int \<times> int \<times> int) \<times> int \<times> protocol) \<times> ((int \<times> int \<times> int \<times> int) \<times> int \<times> protocol) \<times> 'a" shows "src_port x \<equiv> (fst \<circ> snd \<circ> fst \<circ> snd) x"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(ev x1 = ev y1) = (x1 = y1)"
mylens_composition: shows "x +\<^sub>L y +\<^sub>L more\<^sub>L \<approx>\<^sub>L 1\<^sub>L"
hyps: shows "abc_steps_l (0, args) (A [+] B) stp = abc_steps_l (0, args) A stp"
axioms_2: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "vmc2_path G P v0 p \<sigma> \<sigma>'" shows "vmc_path G P v0 p \<sigma>"
downclosed_downset: fixes x :: "'a" shows "downclosed_set (\<down> x)"
bad_sticky1: fixes s2 :: "'a" assumes "bad2 s2" shows "callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"
diffeomorphism_scaleR: fixes c :: "real"   and k :: "enat" assumes "c \<noteq> 0" shows "Smooth.diffeomorphism k UNIV UNIV ((*\<^sub>R) c) (\<lambda>x. x /\<^sub>R c)"
aux4: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x (a # p) = \<lfloor>a\<rfloor>"   and "a \<notin> set p" shows "applied_rule_rev C x p = \<bottom>"
close_neg_def: fixes M :: "'a formula set" shows "close_neg M \<equiv> M \<union> {\<^bold>\<not> (Atom a) |a. Atom a \<notin> M}"
simps_1: fixes n :: "nat"   and \<omega> :: "nat \<Rightarrow> 'a" shows "C n 0 \<omega> = return (Pi\<^sub>M {0..<n} M) \<omega>"
R42_4: shows "pmf (sds R42) d = 1 / 2"
dl_iso_left_invertible: fixes pf :: "('c, 'd) lifted_pair dense" shows "dl_iso_inv (dl_iso pf) = pf"
Let_is_CSP: fixes A :: "('c, 'd) action"   and v :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)" shows "is_CSP_process (action.relation_of A ;; R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = decrease v (alpha_rp.more A)))"
inject_2: fixes x21 :: "'a"   and x22 :: "'a convnet"   and y21 :: "'a"   and y22 :: "'a convnet" shows "(Conv x21 x22 = Conv y21 y22) = (x21 = y21 \<and> x22 = y22)"
simple_path_assoc: fixes p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and r :: "real \<Rightarrow> 'a" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart r" shows "simple_path (p +++ q +++ r) = simple_path ((p +++ q) +++ r)"
MVT_ivl': fixes D :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and J :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and J0 :: "'b"   and a :: "'a"   and b :: "'a" assumes "\<And>x. x \<in> D \<Longrightarrow> (f has_derivative J x) (at x within D)"   and "\<And>x. x \<in> D \<Longrightarrow> J0 \<le> J x (a - b)"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> b + x *\<^sub>R (a - b) \<in> D" shows "f b + J0 \<le> f a"
strict_mono_eq: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x = f y) = (x = y)"
IdInj: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d"   and Y :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>"   and "id\<^bsub>C\<^esub> X = id\<^bsub>C\<^esub> Y" shows "X = Y"
enforce_return_spmf: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "enforce_spmf P (return_spmf x) = (if P x then return_spmf x else return_pmf None)"
distinct_12: fixes list :: "char list"   and var :: "NominalSequents.var"   and form :: "form" shows "ff \<noteq> list \<nabla> [var].form"
vrrestriction_atD_2: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A" shows "r\<lparr>a\<rparr> = b"
igWlsAbsIsInBar_def: fixes MOD :: "('u, 'v, 'varSort, 'sort, 'w, 'x, 'y, 'z, 'aa) model_scheme" shows "igWlsAbsIsInBar MOD \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
b_def: shows "b = (\<lambda>p. ind prime p * ln (real p))"
case_2: fixes f1 :: "'d"   and f2 :: "dag \<Rightarrow> ref \<Rightarrow> dag \<Rightarrow> 'd"   and x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "(case Node x21 x22 x23 of Tip \<Rightarrow> f1 | Node x xa xb \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
ext: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.comp g f \<noteq> CC.null" shows "CC.seq g f"
parallel_converter_def: shows "parallel_converter \<equiv> \<lambda>uu uua. corec_converter (\<lambda>(conv1, conv2) ac. case ac of Inl a \<Rightarrow> map_gpv (map_prod Inl (\<lambda>uu. Inr (uu, conv2))) id (run_converter conv1 a) | Inr b \<Rightarrow> map_gpv (map_prod Inr (\<lambda>uu. Inr (conv1, uu))) id (run_converter conv2 b)) (uu, uua)"
prefix_of_node_history_def: fixes xs :: "'evt list"   and i :: "nat" shows "xs prefix of i \<equiv> \<exists>ys. xs @ ys = history i"
pq'_def: shows "pq' = endpoint_in_S (Rep_hyp2 b') (Rep_hyp2 a')"
LTL_costsMoney: fixes i :: "(String.literal \<times> value list) stream" shows "alw (\<lambda>xs. nxt (state_eq (Some 2)) xs \<longrightarrow> check_exp (Ge (V (Rg 2)) (L (Num 100))) xs) (watch drinks i)"
id2: shows "map snd (map (\<lambda>(A, jbs). (A, jordan_matrix jbs)) Ms) = map jordan_matrix (map snd Ms)"
exhaust: fixes y :: "ty" obtains "y = Void"   | "y = Boolean"   | "y = Integer"   | "y = NT"   | x5 :: "String.literal" where "y = Class x5"   | x6 :: "ty" where "y = x6\<lfloor>\<rceil>"
restrictP_subsume1: fixes A :: "'b set"   and B :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B" shows "r |` A |` B = r |` A"
has_derivative_continuous_on: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)" shows "continuous_on s f"
infinite_subset: fixes A :: "'c set"   and B :: "'c set" assumes "infinite A"   and "A \<subseteq> B" shows "infinite B"
has_derivative_unique: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and x :: "'b"   and F' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative F) (at x)"   and "(f has_derivative F') (at x)" shows "F = F'"
all_b: shows "\<forall>b\<in>{rotate n w |n. 0 \<le> n \<and> n < \<^bold>|w\<^bold>|}. b \<le>lex w' \<longrightarrow> w' = b"
dist_commute_lessI: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist y x < e" shows "dist x y < e"
vcard_xs: shows "vcard xs = ZFC_in_HOL.succ (vcard xs')"
"4_4": fixes t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,t'\<rangle>) = fv t \<union> fv t'"
rel_cong: fixes x :: "'c stream"   and ya :: "'c stream"   and y :: "'d stream"   and xa :: "'d stream"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> sset ya; yb \<in> sset xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "stream_all2 R x y = stream_all2 Ra ya xa"
assms_1: shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> k"
pi_hm: shows "proper_it' hm_iteratei hm_iteratei"
hyps_2: fixes b :: "(name \<times> ty) list"   and ba :: "ty" assumes "b \<turnstile> u : ba" shows "b \<turnstile> u' : ba"
Times_mset_empty: fixes A :: "'a multiset"   and B :: "'b multiset" shows "(A \<times># B = {#}) = (A = {#} \<or> B = {#})"
scp_slice: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "(\<pi>, i) \<in> IFC_def.scp prob" shows "\<pi> i \<in> IFC.slice prob"
divide_eq1: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "(f \<in> o[F](\<lambda>x. g x / h x)) = ((\<lambda>x. f x * h x) \<in> o[F](g))"
InitError: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and C' :: "char list" assumes "sh C = \<lfloor>(sfs, Error)\<rfloor>"   and "P \<turnstile> \<langle>RI (C,THROW NoClassDefFoundError) ; Cs \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
dis_rews_1: shows "is_CFn\<cdot>\<bottom> = \<bottom>"
cSup_singleton: fixes x :: "'c multiset" shows "\<Squnion> {x} = x"
set_bits_aux_0: fixes w :: "'b word" shows "local.set_bits_aux w 0 = w"
fps_divide_times2: fixes h :: "'a fps"   and f :: "'a fps"   and g :: "'a fps" assumes "subdegree h \<le> subdegree f" shows "f * g / h = f / h * g"
induct: fixes x1 :: "'a \<Rightarrow> 'a ttree"   and x2 :: "'a set"   and x3 :: "'a list"   and x4 :: "'a list"   and P :: "('a \<Rightarrow> 'a ttree) \<Rightarrow> 'a set \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "substitute'' x1 x2 x3 x4"   and "\<And>f T. P f T [] []"   and "\<And>zs f x xs' xs T ys. \<lbrakk>zs \<in> paths (f x); xs' \<in> xs \<otimes> zs; substitute'' (f_nxt f T x) T xs' ys; P (f_nxt f T x) T xs' ys\<rbrakk> \<Longrightarrow> P f T (x # xs) (x # ys)" shows "P x1 x2 x3 x4"
domain_Epigraph: fixes f :: "'b \<Rightarrow> ereal" shows "domain f = {x. \<exists>y. (x, y) \<in> Epigraph UNIV f}"
execs_Stuck_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>cs,css,s\<rangle> \<Rightarrow> t"   and "s = Stuck" shows "t = Stuck"
intros_18: fixes t :: "t"   and v :: "v"   and s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and tp :: "tp"   and mem' :: "mem"   and vs :: "v list"   and a :: "nat" assumes "types_agree t v"   and "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "store_packed m (nat_of_int k) off (bits v) (tp_length tp) = Some mem'" shows "\<lparr>s;vs;[$C ConstInt32 k, $C v, $Store t (Some tp) a off]\<rparr> \<leadsto>_ i \<lparr>s\<lparr>s.mem := (s.mem s)[j := mem']\<rparr>;vs;[]\<rparr>"
enat_add_left_cancel_less: fixes a :: "enat"   and b :: "enat"   and c :: "enat" shows "(a + b < a + c) = (a \<noteq> \<infinity> \<and> b < c)"
infinite_typed_consts': fixes a :: "'atom" shows "infinite {c. \<Gamma> (Fun c []) = Var a \<and> public c \<and> arity c = 0}"
R: shows "R = {p \<in> KP. fst m < fst p}"
set_complete: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "\<exists>a. ide a \<and> local.set a = A"
IF1rel_F1rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "('d, 'd IF1, 'd IF2) F1"   and b :: "('e, 'e IF1, 'e IF2) F1" shows "IF1rel R (ctor1 a) (ctor1 b) = F1rel R (IF1rel R) (IF2rel R) a b"
IFimpl_rule: fixes s :: "bdd"   and ni1 :: "nat"   and n1 :: "nat ifex"   and ni2 :: "nat"   and n2 :: "nat ifex"   and v :: "nat" assumes "bdd_sane s"   and "(ni1, n1) \<in> Rmi s"   and "(ni2, n2) \<in> Rmi s" shows "ospec (ifmi' v ni1 ni2 s) (\<lambda>(ni, s'). (ni, IFC v n1 n2) \<in> Rmi s' \<and> bdd_sane s' \<and> mi.les s s')"
in_imp_mirror_elem_in: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<in> I" shows "mirror_elem n I \<in> imirror I"
of_nat_1: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
fst_swap: fixes x :: "'c \<times> 'b" shows "fst (prod.swap x) = snd x"
rep_eq: fixes x :: "'c set_dlist"   and xa :: "'d set_dlist" shows "list_of_dlist (product x xa) = rev (concat (map (\<lambda>x. map (Pair x) (list_of_dlist xa)) (list_of_dlist x)))"
ring_hom: shows "ring_hom_ring R (carrier R [X]) poly_of_const"