hcju/isabelleps huggingface dataset:
reflexion_def: fixes P :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" shows "reflexion P C \<sigma> k C' = (\<exists>L1 t s Cl_P Cl_C trms_C. eligible_literal L1 P \<sigma> \<and> L1 \<in> cl_ecl P \<and> Cl_C = cl_ecl C \<and> Cl_P = cl_ecl P \<and> orient_lit_inst L1 t s neg \<sigma> \<and> ck_unifier t s \<sigma> k \<and> C = Ecl Cl_C trms_C \<and> trms_C = get_trms Cl_C (dom_trms Cl_C (subst_set (trms_ecl P \<union> {t}) \<sigma>)) k \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P - {L1})"
strong_reduction_simulation_impl_SOSou: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"   and "strong_reduction_simulation (Rel\<inverse>) (STCal Source Target)" shows "\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)"
intruder_synth_code: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" shows "(M \<turnstile>\<^sub>c t) = intruder_synth' public arity M t"
wf_mdeclI: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "methd" assumes "wf_mhead G (pid C) sig (mhead m)"   and "unique (lcls (mbody m))"   and "\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None"   and "\<forall>(vn, T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T"   and "jumpNestingOkS {Ret} (stmt (mbody m))"   and "is_class G C"   and "\<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile>stmt (mbody m)\<Colon>\<surd>"   and "\<exists>A. \<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A \<and> Result \<in> nrm A" shows "wf_mdecl G C (sig, m)"
paths_mono_iff: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "(paths t \<sqsubseteq> paths t') = (t \<sqsubseteq> t')"
decr_is_closed: fixes t :: "Core.term"   and lev :: "nat" assumes "is_closed t" shows "decr lev t = t"
make\<Pi>\<^sub>1_inject: fixes x :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and y :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(make\<Pi>\<^sub>1 x = make\<Pi>\<^sub>1 y) = (x = y)"
lhss_union_distrib: fixes A :: "('b \<times> 'c) set"   and B :: "('b \<times> 'c) set" shows "lhss (A \<union> B) = lhss A \<union> lhss B"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. \<lbrakk>x < y \<Longrightarrow> P (y # xs); \<not> x < y \<Longrightarrow> P (y # xs)\<rbrakk> \<Longrightarrow> P (x # y # xs)" shows "P a0"
bisimulation_into_delay_flip: shows "bisimulation_into_delay trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
inj_on_last_index: fixes xs :: "'a list" shows "inj_on (last_index xs) (set xs)"
cong_add_lcancel_nat: fixes a :: "nat"   and x :: "nat"   and y :: "nat"   and n :: "nat" shows "[a + x = a + y] (mod n) = [x = y] (mod n)"
assms_2: fixes x :: "'oid" assumes "x \<in> set xs" shows "x < oid"
step_eq_fetch': fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and ins :: "abc_inst"   and s :: "nat"   and b :: "cell" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "abc_fetch as ap = Some ins"   and "start_of ly as \<le> s"   and "s < start_of ly (Suc as)" shows "fetch tp s b = fetch (ci ly (start_of ly as) ins) (Suc s - start_of ly as) b"
df: fixes x :: "real" shows "(f has_real_derivative k) (at x)"
alphaAbs_ex_equal_or_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_ex_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
assms_6: shows "snd (rep_nat_term a) = snd (rep_nat_term b)"
step_push_ubx1: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and n :: "'ubx1"   and H :: "'henv"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IPushUbx1 n" shows "Subx.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (OpUbx1 n # \<Sigma>) # st))"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>a as b bs. \<lbrakk>key b < key a \<Longrightarrow> P (a # as) bs; \<not> key b < key a \<Longrightarrow> P as (b # bs)\<rbrakk> \<Longrightarrow> P (a # as) (b # bs)"   and "\<And>bs. P [] bs"   and "\<And>v va. P (v # va) []" shows "P a0 a1"
assms_2: shows "Domainp A = (\<lambda>x. x \<in> \<UU>)"
bheight_paint_Black: fixes t :: "('b \<times> color) tree" shows "bheight (paint Black t) \<le> bheight t + 1"
valid_ownership_and_sharing_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (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" assumes "valid_ownership_and_sharing \<S> (t # ts)" shows "valid_ownership_and_sharing \<S> ts"
simps_9: fixes bbs :: "bool list"   and bs :: "int list"   and i :: "int"   and b :: "Cooper.num" shows "Ifm bbs bs (Dvd i b) = (i dvd Inum bs b)"
IH: fixes x2a :: "(('b, 'c, 'a) prot_fun, 'd) Term.term"   and t :: "(('b, 'c, 'a) prot_fun, 'd) Term.term"   and u :: "(('b, 'c, 'a) prot_fun, 'd) Term.term" assumes "x2a \<in> set T"   and "timpls_transformable_to TI x2a t"   and "timpls_transformable_to TI t u" shows "timpls_transformable_to TI x2a u"
1: shows "map (vec \<circ> (\<lambda>i. tensor_from_vec (tl (dims A)) (fixed_length_sublist (vec A) (prod_list (tl (dims A))) i))) [0..<hd (dims A)] = map (fixed_length_sublist (vec A) (prod_list (tl (dims A)))) [0..<hd (dims A)]"
eq: shows "spoly g1 g2 = fst (crit_pair g1 g2) + - snd (crit_pair g1 g2)"
vreal_of_real_def: fixes x :: "real" shows "x\<^sub>\<real> = (if x \<in> \<nat> \<Rightarrow> nat_of_real x\<^sub>\<nat> | otherwise \<Rightarrow> vreal_of_real_impl'' x)"
vimage_Collect_eq: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" shows "f -` Collect P = {y. P (f y)}"
array_length_new_array: fixes a :: "'a"   and n :: "nat" shows "array_length (new_array a n) = n"
transI: fixes r :: "('b \<times> 'b) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
Cons: fixes "is" :: "nat list"   and ds :: "nat list"   and i :: "nat"   and d :: "nat" assumes "is \<lhd> ds"   and "i < d" shows "i # is \<lhd> d # ds"
irreducibleI': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a one"   and "\<And>b. dvd.dvd times b a \<Longrightarrow> dvd.dvd times a b \<or> dvd.dvd times b one" shows "comm_semiring_1.irreducible times one zero a"
induct: fixes P1 :: "('c, 'd) Term.term \<Rightarrow> bool"   and P2 :: "('c, 'd) Term.term list \<Rightarrow> bool"   and "term" :: "('c, 'd) Term.term" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 term"
outSetOfComponentsLimit: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "outSetOfComponents i chSet \<subseteq> AbstrLevel i"
Int_mono2: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<inter> B \<subseteq> C"
exp_sum: fixes I :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite I" shows "exp (sum f I) = (\<Prod>x\<in>I. exp (f x))"
succs_is_node: fixes ks :: "int list"   and l :: "int"   and x :: "int"   and n :: "nat" assumes "dioph_is_node ks l x" shows "list_all (dioph_is_node ks l) (dioph_succs n ks x)"
derangements_def: fixes S :: "nat set" shows "derangements S = {p. p permutes S \<and> (\<forall>x\<in>S. p x \<noteq> x)}"
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> assn) \<Rightarrow> ('e \<Rightarrow> 'f \<Rightarrow> assn) \<Rightarrow> ('c, 'e, 'a) pac_step \<Rightarrow> ('d, 'f, 'b) pac_step \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a1 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a2 :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and a3 :: "('c, 'e, 'a) pac_step"   and a4 :: "('d, 'f, 'b) pac_step" assumes "\<And>R1 R2 R3 p1 p2 i r p1' p2' i' r'. P R1 R2 R3 (Add p1 p2 i r) (Add p1' p2' i' r')"   and "\<And>R1 R2 R3 p1 p2 i r p1' p2' i' r'. P R1 R2 R3 (Mult p1 p2 i r) (Mult p1' p2' i' r')"   and "\<And>R1 R2 R3 p1 p1'. P R1 R2 R3 (Del p1) (Del p1')"   and "\<And>R1 R2 R3 i x p1 i' x' p1'. P R1 R2 R3 (Extension i x p1) (Extension i' x' p1')"   and "\<And>R1 R2 uu_ v va vb vc vd ve vf vg. P R1 R2 uu_ (Mult v va vb vc) (Add vd ve vf vg)"   and "\<And>R1 R2 uu_ v va vb vc vd ve vf. P R1 R2 uu_ (Mult v va vb vc) (Extension vd ve vf)"   and "\<And>R1 R2 uu_ v va vb vc vd. P R1 R2 uu_ (Mult v va vb vc) (Del vd)"   and "\<And>R1 R2 uu_ v va vb vc vd ve vf. P R1 R2 uu_ (Extension v va vb) (Add vc vd ve vf)"   and "\<And>R1 R2 uu_ v va vb vc vd ve vf. P R1 R2 uu_ (Extension v va vb) (Mult vc vd ve vf)"   and "\<And>R1 R2 uu_ v va vb vc. P R1 R2 uu_ (Extension v va vb) (Del vc)"   and "\<And>R1 R2 uu_ v va vb vc vd. P R1 R2 uu_ (Del v) (Add va vb vc vd)"   and "\<And>R1 R2 uu_ v va vb vc vd. P R1 R2 uu_ (Del v) (Mult va vb vc vd)"   and "\<And>R1 R2 uu_ v va vb vc. P R1 R2 uu_ (Del v) (Extension va vb vc)"   and "\<And>R1 R2 uu_ vd ve vf vg v va vb vc. P R1 R2 uu_ (Add vd ve vf vg) (Mult v va vb vc)"   and "\<And>R1 R2 uu_ vd ve vf v va vb vc. P R1 R2 uu_ (Extension vd ve vf) (Mult v va vb vc)"   and "\<And>R1 R2 uu_ vd v va vb vc. P R1 R2 uu_ (Del vd) (Mult v va vb vc)"   and "\<And>R1 R2 uu_ vc vd ve vf v va vb. P R1 R2 uu_ (Add vc vd ve vf) (Extension v va vb)"   and "\<And>R1 R2 uu_ vc vd ve vf v va vb. P R1 R2 uu_ (Mult vc vd ve vf) (Extension v va vb)"   and "\<And>R1 R2 uu_ vc v va vb. P R1 R2 uu_ (Del vc) (Extension v va vb)"   and "\<And>R1 R2 uu_ va vb vc vd v. P R1 R2 uu_ (Add va vb vc vd) (Del v)"   and "\<And>R1 R2 uu_ va vb vc vd v. P R1 R2 uu_ (Mult va vb vc vd) (Del v)"   and "\<And>R1 R2 uu_ va vb vc v. P R1 R2 uu_ (Extension va vb vc) (Del v)" shows "P a0 a1 a2 a3 a4"
pref_comp_sym: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie> v" shows "v \<bowtie> u"
one_is_inc_zero: shows "1 = inc\<cdot>0"
transition_idx_in: fixes i :: "nat"   and j :: "nat"   and s :: "nat \<Rightarrow> 'b"   and T :: "nat \<Rightarrow> ('b \<times> 'b) set"   and k :: "nat" assumes "i < j"   and "(s j, s i) \<in> T k" shows "(s j, s i) \<in> T (transition_idx s T {i, j})"
sets_T: fixes s :: "'s" shows "K.T.events s = sets (stream_space (count_space UNIV))"
is_monomial_setI: fixes A :: "('d \<Rightarrow>\<^sub>0 'e) set" assumes "\<And>p. p \<in> A \<Longrightarrow> is_monomial p" shows "is_monomial_set A"
fps_X_mult_nth: fixes f :: "'a fps"   and n :: "nat" shows "fps_nth (fps_X * f) n = (if n = 0 then 0::'a else fps_nth f (n - 1))"
axioms_1: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e" assumes "diff k charts1 charts2 f" shows "c_manifolds k charts1 charts2"
cf_pos_cf_pos_poly: fixes p :: "int poly" assumes "p \<noteq> 0" shows "cf_pos (cf_pos_poly p)"
g_def: shows "g \<equiv> \<lambda>x. e * (\<Sum>i\<le>n. xf i x)"
dom_closed: fixes f :: "'a" assumes "\<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>" shows "\<guillemotleft>local.dom f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
character_data_ptr_kinds_M_def: shows "character_data_ptr_kinds_M \<equiv> l_ptr_kinds_M.a_ptr_kinds_M character_data_ptr_kinds"
bij_betw_imageI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "inj_on f A"   and "f ` A = B" shows "bij_betw f A B"
assms_1: shows "u *\<^sub>R b + v *\<^sub>R a = u *\<^sub>R a + v *\<^sub>R b"
bangInduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and Prop :: "'d \<Rightarrow> 'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) residual \<Rightarrow> bool"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> Rs"   and "\<And>\<alpha> P' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>)\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"   and "\<And>\<alpha> P' C. \<lbrakk>\<Psi> \<rhd> !P \<longmapsto> \<alpha> \<prec> P'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>); \<And>C. Prop C \<Psi> (!P) (\<alpha> \<prec> P')\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"   and "\<And>M N P' K xvec P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''; \<And>C. Prop C \<Psi> (!P) (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''); \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C; distinct xvec\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>M xvec N P' K P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''; \<And>C. Prop C \<Psi> (!P) (K\<lparr>N\<rparr> \<prec> P''); \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C; distinct xvec\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>Rs C. \<lbrakk>\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs; \<And>C. Prop C \<Psi> (P \<parallel> !P) Rs; guarded P\<rbrakk> \<Longrightarrow> Prop C \<Psi> (!P) Rs" shows "Prop C \<Psi> (!P) Rs"
gso_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "gs'.gso i \<in> Rn"
W1_def: shows "W1 = local.span (set (cols A))"
rtrancl_eta_App: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* s'"   and "t \<rightarrow>\<^sub>\<eta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<eta>\<^sup>* s' \<degree> t'"
p\<^sub>1_simps: shows "trg r\<^sub>0s\<^sub>1.p\<^sub>1 = src r\<^sub>0"
NSDERIV_DERIV_iff: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" shows "NSDERIV f x :> D = (f has_field_derivative D) (at x)"
nsZeroI: fixes x :: "'a"   and n :: "nat" assumes "x = \<zero>" shows " n \<times>\<^bsub>R\<^esub> x = \<zero>"
hyp_edge_to_valid_in_port: fixes "is" :: "nat list"   and c :: "'form"   and n :: "nat"   and s :: "'subst"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" assumes "is \<in> it_paths (it' c)"   and "c \<in> set conclusions"   and "tree_at (it' c) is = HNode n s ants" shows "valid_in_port (hyp_edge_to c is)"
A: fixes e :: "'b" shows "(case Inl e of Inl e \<Rightarrow> e) = e"
h1: shows "\<not> is_final (steps0 (1, tp) A stp)"
pop_code_def: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat" shows "pop_code node_eq_impl node_hash_impl \<equiv> \<lambda>(x, xa, xb, xc). let y' = as_length xa - 1 in WHILE\<^sub>T (\<lambda>(xe, xf). xe < (if y' + 1 = as_length xa then as_length x else as_get xa (y' + 1))) (\<lambda>(xe, xf). RETURN (Suc xe, Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl (as_get x xe) (- 1) xf)) (as_get xa y', xb) \<bind> (\<lambda>(xd, y). RETURN y) \<bind> (\<lambda>xd. let y'a = as_take (as_top xa) x; y'b = as_pop xa in RETURN (y'a, y'b, xd, xc))"
meet_le2: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<sqsubseteq> y"
suminf_0_le: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>n. 0 \<le> f n" shows "0 \<le> (\<Sum>n. f n)"
dimension_def: shows "finite_dimensional_vector_space.dimension {1::'b} = card {1::'b}"
cases: fixes x :: "(nat set \<times> 'a option list set) \<times> nat set \<times> 'a option list" obtains s :: "nat set" and tab :: "'a option list set" and st :: "nat set" and tup :: "'a option list" where "x = ((s, tab), st, tup)"
klavu: shows "a = k\<^sup>2 - l\<^sup>2 \<and> v = 2 * k * l \<and> \<bar>u\<bar> = k\<^sup>2 + l\<^sup>2"
deriv_in_subterms_4: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "p1 \<oplus> p2 \<in> subterms p" shows "p2 \<in> subterms p"
smap_applicative: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b stream" shows "smap f x = pure f \<diamondop> x"
NormalizeD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P Language.normalize c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
covered: fixes K :: "'a set" assumes "K \<in> \<D>" shows "\<exists>x\<in>S \<inter> K. K \<subseteq> \<gamma> x"
monomial_0D: fixes t :: "'a"   and c :: "'b" assumes "monomial c t = 0" shows "c = (0::'b)"
SoundWhileProp_def: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "SoundWhileProp n G c B b A = ((\<forall>m. G \<Turnstile>\<^sub>m c : B) \<longrightarrow> (\<forall>s. \<not> evalB b s \<longrightarrow> A s s) \<longrightarrow> (\<forall>s. evalB b s \<longrightarrow> (\<forall>r. B s r \<longrightarrow> (\<forall>t. A r t \<longrightarrow> A s t))) \<longrightarrow> G \<Turnstile>\<^sub>n While b c : (\<lambda>s t. A s t \<and> \<not> evalB b t))"
neq_Max_imp_inext_neq_iMin: fixes t :: "nat"   and I :: "nat set" assumes "t \<in> I"   and "t \<noteq> Max I \<or> infinite I" shows "inext t I \<noteq> iMin I"
assms_1: fixes \<epsilon> :: "real" assumes "l < \<epsilon>"   and "\<epsilon> < inverse C" shows "(\<lambda>n. norm (f n)) \<in> \<omega>((^) \<epsilon>)"
all_deducible_sound: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" assumes "Sound R" shows "entails_formula S (all_deducible_clauses R S)"
bin_to_bl_aux_alt: fixes n :: "nat"   and w :: "int"   and bs :: "bool list" shows "bin_to_bl_aux n w bs = bin_to_bl n w @ bs"
map_comp_simps_2: fixes m2 :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and k' :: "'c"   and m1 :: "'c \<Rightarrow> 'e option" assumes "m2 k = Some k'" shows "(m1 \<circ>\<^sub>m m2) k = m1 k'"
assms_2: shows "exec_gpv2 \<equiv> exec_gpv"
inv3: shows "user\<^sub>1 \<noteq> owner att"
indexed_eval_aux_is_inj: fixes Ps :: "('d multiset \<Rightarrow> 'a) list"   and i :: "'d"   and Qs :: "('d multiset \<Rightarrow> 'a) list" assumes "list_all carrier_coeff Ps"   and "list_all (\<lambda>P. index_free P i) Ps"   and "list_all carrier_coeff Qs"   and "list_all (\<lambda>Q. index_free Q i) Qs"   and "indexed_eval_aux Ps i = indexed_eval_aux Qs i"   and "length Ps = length Qs" shows "Ps = Qs"
simps_12: fixes x1 :: "'c" shows "Notified \<noteq> Suspend x1"
IH: assumes "finite F"   and "\<And>a. a \<in> F \<Longrightarrow> infsetsum'_converges f (B a)"   and "\<And>a a'. \<lbrakk>a \<in> F; a' \<in> F; a \<noteq> a'\<rbrakk> \<Longrightarrow> B a \<inter> B a' = {}"   and "\<And>a. a \<in> F \<Longrightarrow> infsetsum'_converges f (B a)"   and "\<And>a a'. \<lbrakk>a \<in> F; a' \<in> F; a \<noteq> a'\<rbrakk> \<Longrightarrow> B a \<inter> B a' = {}" shows "(\<Sum>a\<in>F. infsetsum' f (B a)) = infsetsum' f (\<Union> (B ` F))"
Diff_Int_distrib: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
column_def: fixes j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" shows "column j A = (\<chi>i. A $ i $ j)"
ord_residue_mult_group: fixes a :: "nat" assumes "a \<in> totatives p" shows "units.ord a = ord p a"
Real_Alg_Quotient_def: shows "Real_Alg_Quotient \<equiv> quot_type.abs (\<lambda>x y. real_of_3 x = real_of_3 y) Abs_real_alg"
cont_apply: fixes t :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "cont t"   and "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. f x (t x))"
is_weakly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "is_weakly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_dom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<union> added_pointers \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
subset_eq_atLeast0_lessThan_finite: fixes N :: "nat set"   and n :: "nat" assumes "N \<subseteq> {0..<n}" shows "finite N"
uminus_apply: fixes A :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(- A) x = - A x"
assms_6: fixes z :: "complex" assumes "z \<in> eball 0 r"   and "z \<noteq> 0" shows "eval_fps g z \<noteq> 0"
sum_atLeastAtMost_code: fixes f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" shows "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a. (+) (f a)) a b (0::'a)"