hcju/isabelleps huggingface dataset:
consistency_def: fixes C :: "'b fm set set" shows "consistency C \<equiv> \<forall>S\<in>C. (\<forall>p. \<not> (pi S p \<and> (\<^bold>\<not> Pro p) \<in> S)) \<and> \<^bold>\<bottom> \<notin> S \<and> (\<forall>Z. (\<^bold>\<not> \<^bold>\<not> Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<and> B) \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<or> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<or> B) \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<and> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {\<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<longrightarrow> B) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<longrightarrow> B)) \<in> S \<longrightarrow> S \<union> {A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A. tautology A \<longrightarrow> S \<union> {A} \<in> C) \<and> (\<forall>A i. \<not> (K i A \<in> S \<and> (\<^bold>\<not> K i A) \<in> S))"
intros: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "\<And>s i. i < PL.brn c \<longrightarrow> (s, PL.eff aval tval cval c s i) \<in> indis \<and> PL_Indis.discr aval tval cval indis (PL.cont aval tval cval c s i)" shows "PL_Indis.discr aval tval cval indis c"
distinct_simple_match_to_of_match_portlist_hlp: fixes ifs :: "'b list"   and ps :: "16 word \<times> 16 word" assumes "distinct ifs" shows "distinct (if fst ps = 0 \<and> snd ps = - 1 then [None] else if fst ps \<le> snd ps then map (Some \<circ> (\<lambda>pfx. (pfxm_prefix pfx, ~~ pfxm_mask pfx))) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst ps) (snd ps))) else [])"
assms_1: shows "qGoodBinp qbinp \<or> qGoodBinp qbinp'"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>F.right_map f : F.right_map a \<Rightarrow>\<^sub>C F.right_map b\<guillemotright>"
elims: fixes x :: "('b, 'c, 'a) unboxed"   and y :: "'a option" assumes "cast_Ubx2 x = y" obtains xa :: "'a" where "x = OpUbx2 xa" and "y = Some xa"   | v :: "'b" where "x = OpDyn v__" and "y = None"   | v :: "'c" where "x = OpUbx1 v" and "y = None"
\<theta>_eq: fixes n :: "nat" shows "\<theta> (real n) = (\<Sum>p\<in>P n. ln (real p))"
C_eq_Until_InsertDeniesQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)))) = Cp p"
rel_map_2: fixes S1a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S2a :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e + 'g"   and g1 :: "'i \<Rightarrow> 'f"   and g2 :: "'j \<Rightarrow> 'h"   and y :: "'i + 'j" shows "rel_sum S1a S2a x (map_sum g1 g2 y) = rel_sum (\<lambda>x y. S1a x (g1 y)) (\<lambda>x y. S2a x (g2 y)) x y"
ereal_infty_less_2: fixes x :: "ereal" shows "(- \<infinity> < x) = (x \<noteq> - \<infinity>)"
that: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "m = EnvT f"   and "m' = EnvT g"   and "(R ===> M) f g" shows "thesis"
sappend_strict2: fixes xs :: "[:'a:]" shows "xs :@ \<bottom> = \<bottom>"
\<R>\<^sub>2_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "strong_low_bisim_mm (\<R>\<^sub>2 \<Gamma>')"
var_rename_def: fixes n :: "nat"   and v :: "'f \<times> nat" shows "var_rename n v \<equiv> Var (fst v, snd v + Suc n)"
bset_quot_map: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "Quotient (rel_bset R) (map_bset Abs) (map_bset Rep) (rel_bset T)"
prefixes_are_paths: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and x :: "(symbol \<times> character list) list" assumes "p \<in> \<P> k u"   and "is_prefix x p" shows "x \<in> \<P> k u"
prems_2: shows "(l # ls) ! i = (p, Write volatile a (D, f) A L R W # is, \<theta>, sb, \<D>, \<O>)"
l2norm_lnorm: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" shows "l2norm S f = lnorm (lebesgue_on S) 2 f"
dims2_def: shows "st.dims2 = nths (replicate (Suc (Suc k)) 2) (- {0..<Suc k})"
check_in_heap_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (check_in_heap ptr) h"
assms_5: fixes v :: "nat \<Rightarrow> 'a" shows "v \<Turnstile>\<^sub>t t1 = v \<Turnstile>\<^sub>t t2"
single_nz_val_access1: fixes i :: "nat"   and n :: "nat"   and v :: "'a" assumes "i < n" shows "single_nz_val n i v $ i = v"
invertible_mult: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and B :: "(('b, 'e) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "invertible A"   and "invertible B" shows "invertible (A ** B)"
simps_2: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e a (Suc r) ts) = (\<exists>e1 a1 ts1 e2 a2 ts2. BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e1 a1 r ts1) \<and> BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e2 a2 r ts2) \<and> BinomialHeapStruc.Node e a (Suc r) ts = BinomialHeapStruc.link (BinomialHeapStruc.Node e1 a1 r ts1) (BinomialHeapStruc.Node e2 a2 r ts2))"
strict_mono_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "Fun.strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
residualGraph_zero_flow: shows "cf_of (\<lambda>_. 0::'capacity) = c"
purgeIdle_map_Idle: fixes s :: "'St list" shows "purgeIdle (map Idle s) = []"
X_subseteq_XH_largest: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "X \<subseteq> XH_largest ds X"
linear: fixes f :: "'c \<Rightarrow> real" assumes "bounded_linear f" shows "real_polynomial_function f"
distinct_disc_17: fixes event :: "('d, 'e, 'f) event" assumes "isRecv event" shows "\<not> isRecvMarker event"
fail_cont_def: fixes fail :: "'e" shows "fail_cont fail = ContT (\<lambda>_. fail)"
vwalk_length_def: fixes p :: "'a list" shows "vwalk_length p \<equiv> length (vwalk_arcs p)"
N_def: shows "N \<equiv> stock_measure (PRODUCT t1 t2)"
simps_5: fixes v :: "'b"   and va :: "'b"   and vb :: "'b"   and vc :: "'b"   and l :: "'b set list" shows "smaller (DenyAllFromTo v va) (DenyAllFromTo vb vc) l = (DenyAllFromTo v va = DenyAllFromTo vb vc \<or> (if bothNet (DenyAllFromTo v va) = bothNet (DenyAllFromTo vb vc) then case DenyAllFromTo vb vc of DenyAllFromTo a b \<Rightarrow> DenyAllFromTo v va = DenyAllFromTo b a | _ \<Rightarrow> True else position (bothNet (DenyAllFromTo v va)) l \<le> position (bothNet (DenyAllFromTo vb vc)) l))"
rel_resource_conversep: fixes A :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_resource A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_resource A B)\<inverse>\<inverse>"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_elem_length_eq: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)" shows "length G = length F"
bit_b'': fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) \<and> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0]"
satC_tC: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and c :: "('a, 'b) lit list"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP"   and "Signature.wtC wtFsym wtPsym arOf resOf parOf c"   and "Struct.wtE (ProblemIkTpart_TEModel.intT prot protFw eintT eintF) \<xi>" shows "Struct.satC eintF eintP (ProblemIkTpart_TEModel.eenv prot protFw eintT eintF \<xi>) (ProblemIkTpart.tC resOf prot protFw c) = Struct.satC (ProblemIkTpart_TEModel.intF arOf resOf prot protFw eintT eintF) (ProblemIkTpart_TEModel.intP parOf prot protFw eintT eintF eintP) \<xi> c"
prems_9: shows "fst (hd xs) < fst (hd ys)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G G G"
funpow_shift1: fixes bs :: "'b list"   and n :: "nat"   and p :: "'b poly" shows "Ipoly bs ((shift1 ^^ n) p) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
Diff_UNIV: fixes A :: "'c set" shows "A - UNIV = {}"
case_option_rule: fixes v :: "'a option"   and P :: "assn"   and fn :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and fs :: "'a \<Rightarrow> 'b Heap" assumes "v = None \<Longrightarrow> <P> fn <Q>"   and "\<And>x. v = Some x \<Longrightarrow> <P> fs x <Q>" shows "<P> case v of None \<Rightarrow> fn | Some x \<Rightarrow> fs x <Q>"
clocked_evaluate: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r :: "(v, v) result" shows "(\<exists>k. BigStep.evaluate True env (update_clock (\<lambda>_. k) s) e (s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error)) = (\<exists>k. BigStep.evaluate True env (update_clock (\<lambda>_. k) s) e (update_clock (\<lambda>_. 0) s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error))"
step: shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css' @ css, t)"
ide_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "ide (mkIde A)"
comm_monoid_axioms: shows "comm_monoid (+) (0::'c)"
algebra_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "algebra \<Omega> M \<equiv> ring_of_sets \<Omega> M \<and> algebra_axioms \<Omega> M"
e1: fixes e :: "nat \<Rightarrow> real" assumes "e \<in> funcset UNIV {- 1..1}" shows "\<exists>e'\<in>funcset UNIV {- 1..1}. pdevs_val e (snd X) = pdevs_val e' (snd Y)"
eq: fixes k :: "nat" shows "- 1 \<cdot>\<^sub>m g k = - g k"
decode_0: fixes e :: "nat" shows "local.decode e 0 = 0"
assms_2: fixes a :: "'a" assumes "ide a" shows "\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
span_listI: fixes qs :: "'d list"   and bs :: "('c \<Rightarrow>\<^sub>0 'd) list" shows "sum_list (map2 (\<cdot>) qs bs) \<in> phull (set bs)"
RTranCl_inv: fixes P :: "'a \<Rightarrow> bool"   and succs :: "'a \<Rightarrow> 'a list"   and g :: "'a"   and g' :: "'a" assumes "invariant P succs"   and "g [succs]\<rightarrow>* g'"   and "P g" shows "P g'"
simps_1: fixes uu :: "'a" shows "isSome (Some uu) = True"
bigo_useful_add: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "g \<in> O(h)" shows "f + g \<in> O(h)"
asm\<^sub>X: shows "x\<^sub>C \<in> (conc.restrict_modes mdss (- range var\<^sub>C_of) ! i) AsmNoReadOrWrite"
filterlim_real_of_int_at_top: shows "filterlim real_of_int at_top at_top"
FixVars_def: shows "FixVars TYPE('a) TYPE('b) \<equiv> var_infinite undefined \<and> var_regular undefined \<and> varSort_lt_var undefined undefined"
adjoint_transpose_left_1: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src k = trg v"   and "src k\<^sup>* = trg u" shows "k.trnl\<^sub>\<eta> v \<in> hom (k \<star> v) u \<rightarrow> hom v (k\<^sup>* \<star> u)"
decisive5: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} y x"
P: shows "\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow> appendL\<cdot>(unfold\<cdot>ha\<cdot>sa)\<cdot>(unfold\<cdot>hb\<cdot>sb0) \<sqsubseteq> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Left\<cdot>sa)"
g: shows "continuous_on (f ` {x - d..x + d}) g"
Start_nsys_xcpts1_2: shows "Start \<noteq> ClassCast"
hom_carrier: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "h ` carrier G \<subseteq> carrier H"
A_def: shows "A = {0<..(int p - 1) div 2}"
starfun_ln_not_eq_zero: fixes x :: "real star" assumes "0 < x"   and "x \<noteq> 1" shows "(*f* real_ln) x \<noteq> 0"
map_append_append_maps: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'a list"   and zs :: "'a list" assumes "map f xs = ys @ zs" obtains xs' :: "'b list" and xs'' :: "'b list" where "map f xs' = ys" and "map f xs'' = zs" and "xs = xs' @ xs''"
nsum_zeroA: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j = \<zero>" shows "\<Sigma>\<^sub>e A f n = \<zero>"
Suc_diff_le: fixes n :: "nat"   and m :: "nat" assumes "order.greater_eq m n" shows "Suc m - n = Suc (m - n)"
eq_\<I>_gpv_Done2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and x :: "'b" shows "eq_\<I>_gpv A \<I> gpv (Done x) = (lossless_spmf (the_gpv gpv) \<and> (\<forall>a\<in>set_spmf (the_gpv gpv). eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>) a (Pure x)))"
distinct_27: fixes x41 :: "'b formula"   and x42 :: "'b formula"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "(x41::'b formula) \<^bold>\<and> (x42::'b formula) \<noteq> (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
inj_semiring_hom_axioms: shows "inj_semiring_hom (inv hom)"
steps_c\<^sub>1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
fresh_atom_bullet: fixes x :: "name"   and t :: "Lambda_Z.term" assumes "atom x \<sharp> t" shows "atom x \<sharp> t\<^sup>\<bullet>"
rbt_insertwk_rbt_sorted: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and x :: "'c" assumes "rbt_sorted t" shows "rbt_sorted (rbt_insert_with_key f k x t)"
varInFormulaVars: fixes variable :: "nat"   and formula :: "Literal list list" shows "(variable \<in> vars formula) = (\<exists>literal. literal el formula \<and> var literal = variable)"
fact2: fixes i :: "'c" assumes "i \<in> ODE_vars I ODE" shows "(\<lambda>t. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol t) $ i else 0) = (\<lambda>t. ODE_sem I ODE (sol t) $ i)"
is_oc_projection_def: fixes w :: "rat vec"   and S :: "rat vec set"   and v :: "rat vec" shows "gs.is_oc_projection w S v = (w \<in> Rn \<and> v - w \<in> gs.span S \<and> (\<forall>u. u \<in> S \<longrightarrow> w \<bullet> u = 0))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'b"   and f3 :: "'c tree \<Rightarrow> 'b"   and up :: "'c up" shows "P (case up of Same \<Rightarrow> f1 | Bal x \<Rightarrow> f2 x | Unbal x \<Rightarrow> f3 x) = ((up = Same \<longrightarrow> P f1) \<and> (\<forall>x2. up = Bal x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. up = Unbal x3 \<longrightarrow> P (f3 x3)))"
simps_2: fixes f :: "float interval \<Rightarrow> float interval option" shows "lift_un None f = None"
Bolzano_Weierstrass_imp_seq_compact: fixes s :: "'b set" assumes "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)" shows "seq_compact s"
neq_top_trans: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> \<top>"   and "x \<le> y" shows "x \<noteq> \<top>"
Disamis: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set" assumes "A I C"   and "B Q C" shows "A I B"
A: shows "dfa_accepting M (dfa_steps M (dfa_trans M p b) bs) \<noteq> dfa_accepting M (dfa_steps M (dfa_trans M q b) bs)"
ide_exp: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "ide (exp b c)"
lcmof_leastUpper: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme"   and b :: "'c"   and x :: "'c" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "(x \<in> carrier G \<and> x lcmof\<^bsub>G\<^esub> a b) = is_lub (division_rel G) x {a, b}"
3: shows "s \<odot>\<^bsub>P\<^esub> f = s \<odot>\<^bsub>P\<^esub> monom P (f 0) 0 \<oplus>\<^bsub>P\<^esub> X_poly R \<otimes>\<^bsub>P\<^esub> (s \<odot>\<^bsub>P\<^esub> poly_shift f)"
inj: shows "inj_on (\<lambda>m. {i. (\<exists>k. i = 2 ^ k * m) \<and> i \<le> n}) {m. m \<le> n \<and> odd m}"
m_def: shows "m = (GREATEST k. k < length vs \<and> vs ! k \<noteq> 0)"
subset_insertI: fixes B :: "'f set"   and a :: "'f" shows "B \<subseteq> insert a B"
R: fixes P :: "assn"   and s :: "nat"   and bi :: "nat \<Rightarrow> bool Heap"   and f :: "nat \<Rightarrow> nat Heap" assumes "wf (measure ((-) n))"   and "P \<Longrightarrow>\<^sub>A is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p))"   and "\<And>s. <is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p)) * true> bi s <\<lambda>r. is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p)) * \<up> (r = (s < n \<and> snd (xs ! s) < p))>\<^sub>t"   and "\<And>s. s < n \<and> snd (xs ! s) < p \<Longrightarrow> <is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p)) * true> f s <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p)) * \<up> ((r, s) \<in> measure ((-) n))>\<^sub>t"   and "\<And>s. \<not> (s < n \<and> snd (xs ! s) < p) \<Longrightarrow> is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p)) \<Longrightarrow>\<^sub>A is_pfa c xs (a, n) * \<up> (s \<le> n \<and> (\<forall>j<s. snd (xs ! j) < p) \<and> (s < n \<longrightarrow> p \<le> snd (xs ! s)))" shows "<P> heap_WHILET bi f s <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p) \<and> (r < n \<longrightarrow> p \<le> snd (xs ! r)))>\<^sub>t"
resolve_rename_eq_Bin_ord_resolve_rename: fixes C :: "'a literal list"   and D :: "'a literal list" shows "mset ` set (resolve_rename C D) = Bin_ord_resolve_rename (mset C) (mset D)"
pair_fin_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_fin_digraph G \<equiv> pair_wf_digraph G \<and> pair_fin_digraph_axioms G"
filtermap_sup: fixes f :: "'c \<Rightarrow> 'b"   and F1 :: "'c filter"   and F2 :: "'c filter" shows "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
call_def: fixes init :: "'d \<Rightarrow> 'd"   and p :: "'e"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "call init p return c = block init (Call p) return c"
congruent_def: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd" shows "f respects r = (\<forall>(y, z)\<in>r. f y = f z)"
le_less: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" shows "(x \<sqsubseteq> y) = (x \<sqsubset> y \<or> x = y)"
the_inv_into_the_inv_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and a :: "'b" assumes "inj_on f A"   and "a \<in> A" shows "the_inv_into (f ` A) (the_inv_into A f) a = f a"
Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile> {P} SKIP {P}"
neg_mask_test_bit: fixes n :: "nat"   and m :: "nat" shows "NOT (mask n) !! m = (n \<le> m \<and> m < LENGTH('b))"
yEqFresh: shows "(y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X')"
semiattractive_axioms: shows "semiattractive UNIV ((\<sqsubseteq>) \<restriction> A)"
irc_def: fixes f :: "'d set \<Rightarrow> 'd set" shows "irc f = (\<forall>B a. a \<notin> f (insert a B) \<longrightarrow> f (insert a B) = f B)"
asymmetric_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "asymmetric x" shows "asymmetric (x \<sqinter> y)"
Inl_in_sum_iff: fixes a :: "V"   and A :: "V"   and B :: "V" shows "(ZFC_Cardinals.Inl a \<in> elts (A \<Uplus> B)) = (a \<in> elts A)"
app_last: fixes t :: "lam"   and ts :: "lam list"   and u :: "lam" shows "(t \<degree>\<degree> ts) \<degree> u = t \<degree>\<degree> (ts @ [u])"
existence_safe: fixes P :: "'prog"   and P' :: "'prog"   and t :: "'test"   and o1 :: "'prog_out" assumes "P \<in> progs"   and "P' \<in> progs"   and "t \<in> tests"   and "o1 \<in> out P t"   and "deselect P o1 P'" shows "\<exists>o2\<in>out P' t. equiv_out o1 o2"
even_succ_div_2: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus one a) (numeral.numeral one plus (num.Bit0 num.One)) = divide a (numeral.numeral one plus (num.Bit0 num.One))"
subst_apply_dom_ident: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<cdot> \<theta> = t"   and "subst_domain \<delta> \<subseteq> subst_domain \<theta>" shows "t \<cdot> \<delta> = t"
det_diagonal: fixes A :: "(('d, 'c) vec, 'c) vec" assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = (0::'d)" shows "Determinants.det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
Pow_fold: fixes A :: "'c set" assumes "finite A" shows "Pow A = Finite_Set.fold (\<lambda>x A. A \<union> insert x ` A) {{}} A"
IH: shows "(\<Sum>i<a * b. f (i div b) * g (i mod b)) = sum f {..<a} * sum g {..<b}"
snd_sylow_axioms_def: fixes p :: "nat"   and m :: "nat" shows "snd_sylow_axioms p m \<equiv> \<not> p dvd m"
Arg_eqvt: fixes \<pi> :: "perm"   and v :: "Vars.var" shows "\<pi> \<bullet> Arg v = Arg (\<pi> \<bullet> v)"
exp_conj_mono_right: fixes Q :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "Q \<tturnstile> R" shows "P && Q \<tturnstile> P && R"
boundOutputChainOpenIH: fixes xvec :: "name list"   and x :: "name"   and B :: "('d, 'e, 'f) boundOutput"   and yvec :: "name list"   and y :: "name"   and B' :: "('d, 'e, 'f) boundOutput" assumes "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>*yvec\<rparr>\<lparr>\<nu>y\<rparr>B'"   and "length xvec = length yvec"   and "x \<sharp> B'"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*yvec\<rparr>[(x, y)] \<bullet> B'"
inj_axis: fixes c :: "'a" assumes "c \<noteq> (0::'a)" shows "inj (\<lambda>k. axis k c)"
exhaust: fixes y :: "'a literal" obtains x1 :: "'a" where "y = Pos x1"   | x2 :: "'a" where "y = Neg x2"