hcju/isabelleps huggingface dataset:
Sup_def: 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 A :: "'a set" assumes "class.finite_lattice_complete Inf Sup inf less_eq less sup bot top" shows "Sup A = Finite_Set.fold sup bot A"
is_down: fixes c :: "'a word \<Rightarrow> 'b word" shows "is_down c = (LENGTH('b) \<le> LENGTH('a))"
transaction_checks_def: fixes prot_transaction :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_checks prot_transaction = (case prot_transaction of Transaction x1 x2 x3 x4 x5 x6 \<Rightarrow> x4)"
polynomial_def: fixes R :: "('d, 'e) ring_scheme"   and K :: "'d set"   and p :: "'d list" shows "polynomial\<^bsub>R\<^esub> K p = (p = [] \<or> set p \<subseteq> K \<and> lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>)"
trans_Int: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "trans r"   and "trans s" shows "trans (r \<inter> s)"
assms_5: shows "a \<otimes> g (deg R g) [^] n \<noteq> \<zero>"
LeadsETo_def: fixes A :: "'d set"   and CC :: "'d set set"   and B :: "'d set" shows "A LeadsTo[CC] B = {F. F \<in> (reachable F \<inter> A) leadsTo[(\<inter>) (reachable F) ` CC] B}"
dirichlet_prod_neutral_intro: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> complex"   and n :: "nat" shows "(\<Sum>k | k \<in> {1..n} \<and> coprime k n. f k n) = (\<Sum>k = 1..n. f k n * dirichlet_prod_neutral (gcd k n))"
assms: shows "interaction_bounded_by (consider \<circ> g) gpv n"
PAC_checker_specification_spec_def: fixes spec :: "int mpoly" shows "PAC_checker_specification_spec spec = (\<lambda>(\<V>, A) (b, B). (\<not> is_failed b \<longrightarrow> \<Union> (vars ` set_mset A) \<subseteq> \<V>) \<and> (is_success b \<longrightarrow> PAC_Format\<^sup>*\<^sup>* (\<V>, A) B) \<and> (is_found b \<longrightarrow> PAC_Format\<^sup>*\<^sup>* (\<V>, A) B \<and> spec \<in> pac_ideal (set_mset A)))"
kstar_unfoldl: fixes f :: "'a \<Rightarrow> 'a set" shows "\<eta> \<squnion> f \<circ>\<^sub>K kstar f \<le> kstar f"
rep_eq: fixes x :: "'c set"   and xa :: "'c \<Rightarrow> 'd set" shows "apply_bcontfun ` PiC x xa = Pi x xa \<inter> bcontfun"
top_eq_inf_iff: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "(top = inf x y) = (x = top \<and> y = top)"
ag_inv_unique: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e"   and y :: "'e" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<plusminus>\<^bsub>A\<^esub> y = \<zero>\<^bsub>A\<^esub>" shows "y = -\<^sub>a\<^bsub>A\<^esub> x"
noprishr_parts_knowledge_equiv_NS: fixes X :: "msg"   and A :: "agent"   and evs :: "event list"   and B :: "agent" assumes "X \<notin> {Key (priEK A), Key (priSK A), Key (shrK A)}"   and "X \<in> parts (knows A evs)"   and "evs \<in> ns_public" shows "X \<in> parts (knows B evs)"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> TargetTerm TQ"
hoare_sound_main: fixes A :: "triple set"   and C :: "triple set"   and t :: "etriple" shows "(A |\<turnstile> C \<longrightarrow> A |\<Turnstile> C) \<and> (A |\<turnstile>\<^sub>e t \<longrightarrow> A |\<Turnstile>\<^sub>e t)"
pfsubst_imp_simp: fixes p :: "('a, 'b + 'd, 'c) formula"   and q :: "('a, 'b + 'd, 'c) formula"   and \<sigma> :: "'d \<Rightarrow> ('a, 'b, 'c) formula" shows "PFsubst (p \<rightarrow> q) \<sigma> = (PFsubst p \<sigma> \<rightarrow> PFsubst q \<sigma>)"
pr_conv_2_to_1_def: shows "pr_conv_2_to_1 = (\<lambda>f x. f (c_fst x) (c_snd x))"
ord_le_eq_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
list_of_code: fixes as :: "'a iarray" shows "IArray.list_of as = map ((!!) as) [0..<IArray.length as]"
valid_cube_valid_swap: fixes C :: "real \<times> real \<Rightarrow> real \<times> real" assumes "valid_two_cube C" shows "valid_two_cube (prod.swap \<circ> C \<circ> prod.swap)"
si: fixes A :: "nat set" assumes "finite A" shows "(\<lambda>x. \<Sum>i\<in>I \<inter> A. orthonormal_coeff S w f i * w i x) square_integrable S"
induct: fixes x1 :: "(nat \<Rightarrow> nat) \<times> 'a memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and x2 :: "(nat \<Rightarrow> nat) \<times> 'a memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and P :: "(nat \<Rightarrow> nat) \<times> 'a memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> (nat \<Rightarrow> nat) \<times> 'a memref list \<times> nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>f x2"   and "\<And>\<O>' volatile \<O> A R \<S>' \<S> W L \<R>' \<R> m a sop v rs. \<lbrakk>\<O>' = (if volatile then \<O> \<union> A - R else \<O>); \<S>' = (if volatile then \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L else \<S>); \<R>' = (if volatile then Map.empty else \<R>)\<rbrakk> \<Longrightarrow> P (m, Write\<^sub>s\<^sub>b volatile a sop v A L R W # rs, \<O>, \<R>, \<S>) (m(a := v), rs, \<O>', \<R>', \<S>')"   and "\<And>m volatile a t v rs \<O> \<R> \<S>. P (m, Read\<^sub>s\<^sub>b volatile a t v # rs, \<O>, \<R>, \<S>) (m, rs, \<O>, \<R>, \<S>)"   and "\<And>m p p' is rs \<O> \<R> \<S>. P (m, Prog\<^sub>s\<^sub>b p p' is # rs, \<O>, \<R>, \<S>) (m, rs, \<O>, \<R>, \<S>)"   and "\<And>m A L R W rs \<O> \<R> \<S>. P (m, Ghost\<^sub>s\<^sub>b A L R W # rs, \<O>, \<R>, \<S>) (m, rs, \<O> \<union> A - R, augment_rels (dom \<S>) R \<R>, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)" shows "P x1 x2"
comm_groupE_4: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
double_dual_iso: shows "double_dual_iso \<in> Group.iso G (Characters (Characters G))"
agg_sum_bot_bot: shows "aggregation.sum_0 (\<lambda>k. zero) {k. True} = zero"
lem2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and a :: "'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. f x * (g x * a)) = a * (\<Sum>x\<in>A. f x * g x)"
ordered_resolvent_upon_is_resolvent: fixes atom_ordering :: "('a \<times> 'a) set"   and P1 :: "'a Literal set"   and A :: "'a"   and P2 :: "'a Literal set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.strictly_maximal_literal atom_ordering P1 (Pos A)"   and "propositional_atoms.strictly_maximal_literal atom_ordering P2 (Neg A)" shows "propositional_atoms.ordered_resolvent atom_ordering P1 P2 (propositional_atoms.resolvent_upon P1 P2 A)"
assms: shows "i \<in>\<^sub>\<circ> vcard xs"
gcd_dvd1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) a"
prems: fixes C :: "'a tensor" assumes "C \<in> set (C__ # CS)" shows "dims C = ds"
step_1_main_dims_main: fixes n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (step_1_main n i j A) = dim_row A \<and> dim_col (step_1_main n i j A) = dim_col A"
simps_16: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = sip"
sup_ge2: fixes y :: "'d"   and x :: "'d" shows "y \<le> x \<sqinter> y"
alpha_abs_lst_stronger1: fixes as :: "atom list"   and x :: "'a"   and p' :: "perm"   and as' :: "atom list"   and x' :: "'a" assumes "(as, x) \<approx>lst (=) supp p' (as', x')" shows "\<exists>p. (as, x) \<approx>lst (=) supp p (as', x') \<and> supp p \<subseteq> set as \<union> set as'"
JHS_Tr1: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and g :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "isom_Gchains (r * s - 1) (rtos r s) (Qw_cmpser G (cmp_rfn G r f s g)) (Qw_cmpser G (cmp_rfn G s g r f))"
Rep_cases: fixes y :: "'a" assumes "y \<in> A" obtains x :: "'b" where "y = Rep x"
ext_inject: fixes orderA :: "('a, 'c) gorder_scheme"   and orderB :: "('b, 'd) gorder_scheme"   and lower :: "'a \<Rightarrow> 'b"   and upper :: "'b \<Rightarrow> 'a"   and more :: "'e"   and orderA' :: "('a, 'c) gorder_scheme"   and orderB' :: "('b, 'd) gorder_scheme"   and lower' :: "'a \<Rightarrow> 'b"   and upper' :: "'b \<Rightarrow> 'a"   and more' :: "'e" shows "(\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr> = \<lparr>orderA = orderA', orderB = orderB', lower = lower', upper = upper', \<dots> = more'\<rparr>) = (orderA = orderA' \<and> orderB = orderB' \<and> lower = lower' \<and> upper = upper' \<and> more = more')"
map_poly_idI': fixes p :: "'a poly"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" shows "p = map_poly f p"
n_mult_glb: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(n z \<sqsubseteq> n x \<and> n z \<sqsubseteq> n y) = (n z \<sqsubseteq> n x \<cdot> n y)"
cases: fixes x :: "V list" obtains "x = []"   | \<alpha> :: "V" and \<alpha>s :: "V list" where "x = \<alpha> # \<alpha>s"
returns_result_select_result: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
finite_if_finite_subsets_card_bdd: fixes F :: "'d set"   and C :: "nat" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C"
a_all_ptrs_in_heap_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_child_nodes get_disconnected_nodes h = ((\<forall>ptr\<in>fset (object_ptr_kinds h). set |h \<turnstile> get_child_nodes ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)) \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)))"
simps_17: fixes vb :: "'b"   and vc :: "value list"   and v :: "'b"   and va :: "value list" shows "GExp_Lexorder.less_gexp_aux (In vb vc) (In v va) = (vb < v \<or> vb = v \<and> vc < va)"
huntington1: fixes x :: "'a"   and y :: "'a" shows "n (n (t x \<cdot> n y) \<cdot> n (t x \<cdot> t y)) = t x"
proj6_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and npre :: "nat set" shows "\<pi>\<^sub>6 (addpre v npre) = \<pi>\<^sub>6 v"
intros: fixes F :: "'fenv_inca"   and main :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and H :: "'c" assumes "Finca_get F main = Some fd"   and "arity fd = 0" shows "Sinca.load (Prog F H main) (State F H [Frame main 0 []])"
terminate: fixes s :: "'a" assumes "P s"   and "\<not> b s" shows "Q s"
Red: shows "((e, s), e'', s'') \<in> Red P E"
i_expand_i_take_Suc: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" assumes "0 < k" shows "f \<odot> k \<Down> Suc n = f \<Down> (n div k) \<odot> k @ f (n div k) # NoMsg\<^bsup>n mod k\<^esup>"
assms_1: shows "poincare_collinear {0\<^sub>h, of_complex x, of_complex y}"
refl: fixes x :: "('a, 'b) rose_tree\<^sub>m" assumes "x \<in> UNIV" shows "blinding_of_tree h bo x x"
loop_back': fixes A :: "abc_inst list"   and gap :: "'a list"   and xs :: "nat list"   and n :: "nat"   and x :: "nat"   and y :: "nat"   and m :: "nat"   and anything :: "nat list" assumes "length A = length gap + 4"   and "length xs = n"   and "x \<le> y" shows "\<exists>stp. abc_steps_l (length A, xs @ m # (y - x) # x # anything) (A @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]) stp = (length A, xs @ m # y # 0 # anything)"
inv_unique: fixes y :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int"   and y' :: "nat \<Rightarrow> int" assumes "y \<otimes> x = \<one>"   and "x \<otimes> y' = \<one>"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "y' \<in> carrier Zp" shows "y = y'"
cases: fixes x :: "'b hp list" obtains "x = []"   | xa :: "'b" and hsl :: "'b hp list" and hsr :: "'b hp list" where "x = Hp xa hsl # hsr"
IH: fixes gpv :: "('c, 'a, 'b) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "exec_gpv' (mk_lossless_gpv (responses_\<I> \<I>) x gpv) s = exec_gpv' gpv s"
expand: fixes optionT :: "('e, 'd) optionT"   and optionT' :: "('e, 'd) optionT" assumes "run_option optionT = run_option optionT'" shows "optionT = optionT'"
linear_construct: fixes B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "cindependent B" shows "clinear (cconstruct B f)"
is_glb_maximal: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_lb below S x"   and "x \<in> S" shows "po.is_glb below S x"
assms_1: shows "P \<turnstile> \<sigma> -jvm\<rightarrow> (x, h, [], sh)"
alphaAbs_qAbs_iff_ex_equal_or_qAFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
d_OUT_h_minus: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and x :: "'a" assumes "flow_network \<Delta> g"   and "cycles \<Delta> \<noteq> {}" shows "KIR (flow_network.h_minus \<Delta> g i) x"
H_1: shows "x \<in> extended_distance_set \<and> y \<in> extended_distance_set \<and> extended_distance x y = 0"
monom_poly_assemble: fixes p :: "linear_poly" assumes "is_monom p" shows "monom_coeff p *R lp_monom 1 (monom_var p) = p"
simps: fixes valid_edge :: "'g \<Rightarrow> bool"   and src :: "'g \<Rightarrow> 'h"   and trg :: "'g \<Rightarrow> 'h"   and knd :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and E :: "'h"   and X :: "'h"   and a :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" shows "lift_valid_edge valid_edge src trg knd E X a = ((\<exists>aa e. a = e \<and> valid_edge aa \<and> (src aa \<noteq> E \<or> trg aa \<noteq> X) \<and> e = (Node (src aa), knd aa, Node (trg aa))) \<or> (\<exists>e. a = e \<and> e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E)) \<or> (\<exists>e. a = e \<and> e = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)) \<or> (\<exists>e. a = e \<and> e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit)))"
cbox_translation: fixes c :: "'b"   and a :: "'b"   and b :: "'b" shows "cbox (c + a) (c + b) = (+) c ` cbox a b"
float_distinct_1: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (IEEE.is_nan a \<and> is_infinity a)"
Rep_ttree: fixes x :: "'b ttree" shows "Rep_ttree x \<in> {xss. [] \<in> xss \<and> downset xss}"
unreach_equiv: fixes Q :: "'a set"   and R :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "a \<in> Q"   and "b \<in> R"   and "a \<in> \<emptyset> Q b" shows "b \<in> \<emptyset> R a"
EM_remainder'_integral_conv_Suc: shows "integral {real_of_int a..real_of_int b} (\<lambda>t. pbernpoly n t *\<^sub>R f t) = (bernoulli (Suc n) / real (Suc n)) *\<^sub>R (f (real_of_int b) - f (real_of_int a)) - integral {real_of_int a..real_of_int b} (\<lambda>t. pbernpoly (Suc n) t *\<^sub>R f' t) /\<^sub>R real (Suc n)"
s0_is_state_def: fixes SES :: "('a, 'b) SES_rec" shows "s0_is_state SES \<equiv> s0\<^bsub>SES\<^esub> \<in> S\<^bsub>SES\<^esub>"
assms: shows "\<forall>x\<in>support_set p. \<forall>w i. p x i w = p x (Suc i) w"
hyps_5: fixes \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "(\<N>\<lbrace>as @ \<Gamma>\<rbrace>\<rho>) f|` domA (as @ \<Gamma>) \<sqsubseteq> (\<N>\<lbrace>\<Delta>\<rbrace>\<rho>) f|` domA (as @ \<Gamma>)"
compact_Un: fixes s :: "'a set"   and t :: "'a set" assumes "compact s"   and "compact t" shows "compact (s \<union> t)"
STL2_gen: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> Init F"
IH: fixes x2a :: "('fun, 'var) Term.term" assumes "x2a \<in> set T"   and "\<forall>x\<in>fv x2a. \<exists>a. \<Gamma> (Var x) = Var a"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<theta> ` fv x2a)" shows "subterms (x2a \<cdot> \<theta>) = subterms x2a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
par_Inf_distrib: fixes D :: "'a::refinement_lattice set"   and c :: "'a::refinement_lattice" assumes "(D::'a::refinement_lattice set) \<noteq> {}" shows "(c::'a::refinement_lattice) \<parallel> \<Sqinter> (D::'a::refinement_lattice set) = \<Sqinter> ((\<parallel>) c ` D)"
eq: shows "Polynomial.smult (\<alpha> i) (F (i - 2) - Polynomial.smult (inverse (\<alpha> i)) (Q i) * F (i - 1)) = Polynomial.smult (\<alpha> i) (Polynomial.smult (inverse (\<alpha> i)) (Q i) * F (i - 1) + Polynomial.smult (inverse (\<alpha> i)) (Polynomial.smult (\<beta> i) (F i)) - Polynomial.smult (inverse (\<alpha> i)) (Q i) * F (i - 1))"
assms_4: shows "v \<in> set (delete_effects_of op)"
awalk_hd_in_verts: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" shows "u \<in> verts G"
simps_2: fixes "is" :: "'b set"   and ia :: "'b interrupt_action"   and ias :: "'b interrupt_action list" shows "interrupt_actions_ok is (ia # ias) = (interrupt_action_ok is ia \<and> interrupt_actions_ok (redT_updI is ia) ias)"
ltake_lfilter: fixes x :: "nat"   and P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" shows "\<exists>n. ltake (enat x) (lfilter P lst) = lfilter P (ltake (enat n) lst) \<and> ldrop (enat x) (lfilter P lst) = lfilter P (ldrop (enat n) lst)"
remove_child_removes_child: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "child \<notin> set children"
atom_name_sort: fixes a :: "name" shows "sort_of (atom a) = Sort ''SyntaxN.name'' []"
vector_sadd_rdistrib: fixes a :: "'d"   and b :: "'d"   and x :: "('d, 'e) vec" shows "(a + b) *s x = a *s x + b *s x"
S0: shows "\<T> = (\<lambda>x. ball x (r x)) ` S0"
ordstNS_refl: fixes X :: "char list" shows "ordstNS X X"
eq_states_def: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" shows "s \<approx> s' \<equiv> \<exists>\<alpha> \<beta>. bij \<alpha> \<and> bij \<beta> \<and> \<R>\<^sub>G \<alpha> \<beta> s = s'"
enum_finite_5_def: shows "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
pointfree_idE: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f \<circ> g = id" shows "f (g x) = x"
function_case_inner: fixes I :: "('e, 'f, 'd) interp"   and args :: "'d \<Rightarrow> ('e, 'd) trm"   and \<nu> :: "(real, 'd) vec"   and f :: "'e" assumes "\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)"   and "((\<lambda>v. \<chi>i. sterm_sem I (args i) v) has_derivative (\<lambda>v. \<chi>i. frechet I (args i) \<nu> v)) (at \<nu>)" shows "((\<lambda>v. Functions I f (\<chi>i. sterm_sem I (args i) v)) has_derivative frechet I ($f f args) \<nu>) (at \<nu>)"
lim_imp_Liminf: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
space\<^sub>N_def: fixes N :: "'c quasinorm" shows "space\<^sub>N N = {f. eNorm N f < \<infinity>}"
t2: assumes "(Q\<^sup>T - 1\<^sub>m (dim_row Q)) *\<^sub>v v = 0\<^sub>v (dim_vec v)" shows "Q\<^sup>T *\<^sub>v v - v = 0\<^sub>v (dim_vec v)"
take_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'a" shows "take i (replicate k x) = replicate (min i k) x"
iso_iff: shows "(hom_boundary (int n) (nsphere n) (equator n) \<in> Group.iso (relative_homology_group (int n) (nsphere n) (equator n)) (homology_group (int n - 1) (subtopology (nsphere n) (equator n)))) = (carrier (homology_group (int n - 1) (subtopology (nsphere n) (equator n))) \<subseteq> hom_boundary (int n) (nsphere n) (equator n) ` carrier (relative_homology_group (int n) (nsphere n) (equator n)) \<and> (\<forall>x\<in>carrier (relative_homology_group (int n) (nsphere n) (equator n)). hom_boundary (int n) (nsphere n) (equator n) x = \<one>\<^bsub>homology_group (int n - 1) (subtopology (nsphere n) (equator n))\<^esub> \<longrightarrow> x = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>))"
op_ASSERT_bind_def: fixes I :: "bool"   and m :: "'a nres" shows "op_ASSERT_bind I m \<equiv> ASSERT I \<bind> (\<lambda>_. m)"
f7: shows "\<forall>s sa sb p w wa sc. \<not> low_equal s sa \<or> sb \<noteq> snd (fst (div_write_new_val p w wa s)) \<or> sc \<noteq> snd (fst (div_write_new_val p w wa sa)) \<or> low_equal sb sc"
max_stepsize_nonneg: fixes n :: "nat" shows "0 \<le> max_stepsize n"
idom_hom_axioms: shows "idom_hom (\<lambda>a. [:a:])"
summable_NSsummable_iff: fixes f :: "nat \<Rightarrow> real" shows "summable f = NSsummable f"
assms: shows "k icd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
simps_1: fixes f :: "'fsym" shows "GE_arOf (Oldf f) = arOf f"
f_group: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (local.group xs)" shows "\<exists>x'\<in>set xs. f x = f x'"
vlrestriction_vempty: fixes A :: "V" shows "0 \<restriction>\<^sup>l\<^sub>\<circ> A = 0"
iteration_unfoldl: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<infinity> = x\<^sup>\<infinity>"
valid_line_drop_B: fixes xs :: "color list" assumes "valid_line (B # xs)" shows "valid_line xs"
simps_8: fixes e\<^sub>1 :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "fv (e\<^sub>1\<bullet>F{D} := e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
N_pos_simps_1: fixes i :: "nat" shows "i \<noteq> Suc i mod N"