hcju/isabelleps huggingface dataset:
PiE_iff: fixes f :: "'g \<Rightarrow> 'h"   and I :: "'g set"   and X :: "'g \<Rightarrow> 'h set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
dbm_abstr_zone_eq: fixes v :: "'e \<Rightarrow> nat"   and n :: "nat"   and cc :: "('e, 'f) cconstraint" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "[abstr cc (\<lambda>i j. \<infinity>) v]\<^bsub>v,n\<^esub> = {u. u \<turnstile> cc}"
simps_6: fixes lformula0 :: "'a \<Rightarrow> bool"   and k :: "'b"   and \<phi> :: "('a, 'b) aformula" shows "Formula_Operations.lformula lformula0 (FEx k \<phi>) = Formula_Operations.lformula lformula0 \<phi>"
lprob_comp_last: fixes p :: "real"   and xs :: "bool list"   and x :: "bool" shows "lprob_comp p (xs @ [x]) = lprob_comp p xs * (if x then p else 1 - p)"
monad_alt_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "monad_alt return bind alt \<equiv> monad return bind \<and> monad_alt_axioms bind alt"
return_time_plane_has_derivative_blinfun: fixes S :: "'a set"   and i :: "'a"   and c :: "real"   and x :: "'a" assumes "returns_to {x \<in> S. x \<bullet> i = c} x"   and "closed S"   and "f (poincare_map {x \<in> S. x \<bullet> i = c} x) \<bullet> i \<noteq> 0"   and "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. x \<bullet> i = c} x). x \<bullet> i = c \<longrightarrow> x \<in> S"   and "x \<notin> S \<or> x \<bullet> i \<noteq> c" shows "(return_time {x \<in> S. x \<bullet> i = c} has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (blinfun_inner_left i) (f (poincare_map {x \<in> S. x \<bullet> i = c} x)))) o\<^sub>L (blinfun_inner_left i o\<^sub>L Dflow x (return_time {x \<in> S. x \<bullet> i = c} x)))) (at x)"
invariantsHoldInState_def: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "invariantsHoldInState state F0 decisionVars \<equiv> InvariantImpliedLiterals (getF state) (getM state) \<and> InvariantVarsM (getM state) F0 decisionVars \<and> InvariantVarsF (getF state) F0 decisionVars \<and> InvariantConsistent (getM state) \<and> InvariantUniq (getM state) \<and> InvariantEquivalent F0 (getF state)"
sets_measure_of: fixes A :: "'d set set"   and \<Omega> :: "'d set"   and \<mu> :: "'d set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
sorted_list_of_set_inject: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "linorder.sorted_list_of_set less_eq A = linorder.sorted_list_of_set less_eq B"   and "finite A"   and "finite B" shows "A = B"
balanced_alt_def_all: shows "\<forall>ps\<subseteq>\<V>. int (card ps) = \<t> \<longrightarrow> int (\<B> index ps) = \<Lambda>\<^sub>t"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "comm_monoid_cancel G"   and "gcd_condition_monoid_axioms G" shows "gcd_condition_monoid G"
eq_imp_below: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below"   and "x = y" shows "below x y"
perfect_map_id: fixes X :: "'b topology" shows "perfect_map X X id"
rotate2_scaleR: fixes k :: "real"   and x :: "(real, 2) vec" shows "rotate2 (k *\<^sub>R x) = k *\<^sub>R rotate2 x"
poly_0: fixes x :: "'b" shows "poly 0 x = (0::'b)"
ope: fixes T :: "'a set" assumes "T \<in> \<G>" shows "openin (top_of_set S) T \<and> S \<subseteq> closure T"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>trg\<^sub>D f : trg\<^sub>D a \<Rightarrow>\<^sub>D trg\<^sub>D b\<guillemotright>"
assms: shows "\<sharp> (upd \<circ> map_hd)"
assms_2: shows "eq_\<I>_gpv A \<I> gpv2 gpv2'"
smallest_accepting_rank\<^sub>\<R>_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F" shows "mojmir_to_rabin.smallest_accepting_rank\<^sub>\<R> \<Sigma> \<delta> q\<^sub>0 w F \<equiv> if accept\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0, semi_mojmir_def.initial q\<^sub>0, {(mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta> q\<^sub>0 F \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta> q\<^sub>0 F j, mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F j) |j. j < semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0}) w then Some (LEAST i. accepting_pair\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0) (semi_mojmir_def.initial q\<^sub>0) (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta> q\<^sub>0 F \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta> q\<^sub>0 F i, mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F i) w) else None"
cos_theta_div_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "cos (grover_state.\<theta> n f / 2) = sqrt (real (grover_state.N n - grover_state.M n f) / real (grover_state.N n))"
symmetric_P: fixes I :: "'c"   and J :: "'c" assumes "\<exists>a b. Q I a b"   and "\<exists>a b. Q I a b"   and "P I J" shows "P J I"
valid_path_imp_reverse: fixes g :: "real \<Rightarrow> 'a" assumes "valid_path g" shows "valid_path (reversepath g)"
le_add_same_cancel1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
BackwardSlice_axioms: shows "BackwardSlice sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val backward_slice"
Trans: shows "[a\<frown>b]P \<longmapsto> Rs"
o_def: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "f \<circ> g = (\<lambda>x. f (g x))"
X_def: shows "X = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. {h \<in> A \<rightarrow>\<^sub>E UNIV. h x = h y})"
assms_5: shows "\<forall>x\<in>set eq. case x of (d, e, f) \<Rightarrow> d * ((sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a))\<^sup>2 + e * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) + f = 0"
comm_ring_hom_axioms: shows "comm_ring_hom rebase"
ndepth_K0: fixes x :: "'a + nat" shows "ndepth (Abs_Node (\<lambda>k. Inr 0, x)) = 0"
lessThan_iff: fixes i :: "'c"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c" shows "(i \<in> ord.lessThan less k) = less i k"
A: shows "((u # s @ r, c), ee, s', c') \<in> ntrp fg"
BseqD: fixes X :: "nat \<Rightarrow> 'b" assumes "Bseq X" shows "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
excess'_other: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat"   and x :: "nat" assumes "push_effect_locale c s t f l u v"   and "x \<noteq> u"   and "x \<noteq> v" shows "Network.excess c (Network.push_effect c f (u, v)) x = Network.excess c f x"
refl: fixes ko :: "'a key_order"   and a :: "'a" shows "ko.le ko a a"
append_vec_add: fixes v :: "'b vec"   and n :: "nat"   and v' :: "'b vec"   and w :: "'b vec"   and m :: "nat"   and w' :: "'b vec" assumes "v \<in> carrier_vec n"   and "v' \<in> carrier_vec n"   and "w \<in> carrier_vec m"   and "w' \<in> carrier_vec m" shows "v @\<^sub>v w + (v' @\<^sub>v w') = (v + v') @\<^sub>v w + w'"
cdfi: shows "d O f\<inverse> \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
acc_sim1: fixes r :: "nat \<Rightarrow> 'u" assumes "is_acc r" shows "G'.is_acc (f \<circ> r)"
sig_red_single_tail_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg (\<prec>) p q f t" shows "punit.lt (rep_list q) = punit.lt (rep_list p)"
cong_dvd_modulus_poly: fixes x :: "'c poly"   and y :: "'c poly"   and m :: "'c poly"   and n :: "'c poly" assumes "[x = y] (mod m)"   and "n dvd m" shows "[x = y] (mod n)"
inf_dense: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> bot"   and "y \<noteq> bot" shows "x \<sqinter> y \<noteq> bot"
closed\<^sub>T2: shows "T\<^sub>2 closed_in \<Gamma>"
Obj_pbeta: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option" assumes "Obj f T => Obj f' T" shows "dom f' = dom f \<and> (\<exists>L. finite L \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (f' l) = \<sigma>[s,p] t))) \<and> (\<forall>l\<in>dom f. body (the (f l)))"
bool_to_ternary_simp2: fixes Y :: "bool" shows "(bool_to_ternary Y = TernaryFalse) = (\<not> Y)"
plus_commute: fixes x :: "nat discr Box"   and y :: "nat discr Box" shows "x + y = y + x"
god_is_greatest: shows "\<not> (\<^bold>\<exists>x. G x (\<^bold>\<tau>x. \<Phi> x) \<and> C x)"
segment_a_a'_neD: assumes "segment G.face_cycle_succ a a' \<noteq> {}" shows "segment G.face_cycle_succ a a' \<in> H.face_cycle_sets"
strict_implies_order: fixes a :: "'b"   and b :: "'b" assumes "order.greater b a" shows "order.greater_eq b a"
comp_converter_id_left: fixes conv :: "('h, 'i, 'j, 'k) converter" shows "1\<^sub>C \<odot> conv = conv"
complete_def: fixes V :: "nat set" shows "Ugraph_Lemmas.complete V = (V, all_edges V)"
unhindered_criterion: fixes X :: "'v set" assumes "\<not> hindered \<Gamma>"   and "X \<subseteq> A \<Gamma>"   and "finite X" shows "integral\<^sup>N (count_space X) (weight \<Gamma>) \<le> integral\<^sup>N (count_space (\<^bold>E `` X)) (weight \<Gamma>)"
d_1: shows "D \<in> N_of_state (lnth Sts i) \<union> P_of_state (lnth Sts i) \<union> Q_of_state (lnth Sts i)"
hyps_8: shows "A\<^sub>P \<sharp>* \<Psi>"
axioms_1: fixes G :: "('d, 'e) monoid_scheme"   and act :: "'f \<Rightarrow> 'd \<Rightarrow> 'f" assumes "action G act" shows "Group.group G"
wffp: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" shows "GM_sec_model.wffp do sl = (sl \<noteq> [] \<and> (\<forall>i. Suc i < length sl \<longrightarrow> sl ! Suc i \<in> GM_sec_model.\<delta> do (sl ! i)))"
subset_dfs_is_node: fixes q0 :: "bool list" assumes "nfa_is_node A q0" shows "list_all (nfa_is_node A) (snd (subset_dfs A q0))"
rv1_Nil: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and x :: "'d" shows "rv1 s1 s2 [] x = Inl x"
ineq: fixes p :: "nat" assumes "prime p"   and "\<forall>q\<in>P. \<bar>\<beta> q\<bar> < int p"   and "cmod (\<Prod>\<alpha>\<in>Roots'. of_int (l ^ n) * prod ((-) \<alpha>) (Roots' - {\<alpha>})) < real p" shows "fact (p - 1) ^ n \<le> C' * C p ^ n"
IH_3: assumes "\<not> (n = 0 \<or> n = 1)"   and "n mod 2 \<noteq> 0"   and "n div 2 \<le> length kvs" shows "length (snd (rbtreeify_f (n div 2) kvs)) = length kvs - n div 2"
idempotent_def: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" shows "Idem\<^bsub>L\<^esub> f \<equiv> \<forall>x\<in>carrier L. f (f x) .=\<^bsub>L\<^esub> f x"
not_isOp_Var: fixes s :: "'sort"   and xs :: "'varSort"   and X :: "'var" shows "\<not> isOp s (Var xs X)"
NR\<^sub>N_bounds: fixes n :: "nat"   and u :: "nat"   and l :: "nat" shows "NR\<^sub>N n u l \<le> of_nat (l * n) + of_nat u * NH\<^sub>N n"
assms_2: shows "Deliver (i1, Store e mo i2) \<in> set xs"
multi_nonempty_split: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "\<exists>A a. M = add_mset a A"
SUP_le_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and u :: "'b" shows "(Sup (f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
scaleR_add_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and x :: "'c" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR (a + b) x = plus (scaleR a x) (scaleR b x)"
eq_1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s) (Abs_fv x \<tau>' t)"
compP\<^sub>1_def: shows "compP\<^sub>1 \<equiv> compP (\<lambda>b (pns, body). compE\<^sub>1 (case b of Static \<Rightarrow> pns | NonStatic \<Rightarrow> this # pns) body)"
PFun_inj: fixes F :: "ZF"   and U :: "ZF"   and V :: "ZF"   and x :: "ZF"   and y :: "ZF" assumes "Elem F (PFun U V)"   and "Elem x F"   and "Elem y F"   and "Fst x = Fst y" shows "Snd x = Snd y"
I2c: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s \<and> HInv2 s \<and> HInv3 s" shows "HInv3 s'"
neutr_eq_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "({#} = a \<union># b) = (a = {#} \<and> b = {#})"
rep_nat_term_of_exps: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "rep_nat_term (Term_Order.of_exps a b i) = (pp_of_fun (\<lambda>x. if x = 0 then a else if x = 1 then b else 0), if \<exists>y. snd (rep_nat_term y) = i then i else 0)"
seqseq_holds: fixes n :: "nat" shows "convergent (\<lambda>k. f (nat.seqseq (Suc n) k) (r n))"
in_S: fixes x :: "'d stream" shows "x \<in> space (stream_space (count_space UNIV))"
run_step_basic_sound: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and b_e :: "b_e"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')" shows "\<lparr>s;vs;vs_to_es ves @ [$b_e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
index_nth_id: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "index xs (xs ! n) = n"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "hom.in_hom f a b = (hom.arr a \<and> hom.arr b \<and> hom.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
anonymous_profile_agent_permutation: fixes R1 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and agents :: "'b set"   and alts :: "'a set" assumes "anonymous_profile R1 = anonymous_profile R2"   and "pref_profile_wf agents alts R1"   and "pref_profile_wf agents alts R2"   and "finite agents" obtains \<pi> :: "'b \<Rightarrow> 'b" where "\<pi> permutes agents" and "R2 \<circ> \<pi> = R1"
invalid_assn_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "invalid_assn R x y \<equiv> \<up> (\<exists>h. h \<Turnstile> R x y) * true"
ID_ceq: fixes eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ID CEQ('c) = Some eq" shows "equal eq"
emb_prj_emb: fixes x :: "'d" assumes "DEFL('d) \<sqsubseteq> DEFL('e)" shows "EMB('e)\<cdot>(PRJ('e)\<cdot>(EMB('d)\<cdot>x)) = EMB('d)\<cdot>x"
SFAccInit: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and h :: "heap"   and l :: "sfields"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and v :: "val" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v',(h', l', sh')\<rangle>"   and "sh' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v,(h', l', sh')\<rangle>"
ordLeq_ordLess_Un_ordIso: shows "ordLeq = ordLess \<union> ordIso"
nth_equalityI: fixes xs :: "'c list"   and ys :: "'c list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) local.map \<Omega>.map trm"
not_in_pow: fixes var :: "nat"   and p :: "real mpoly"   and i :: "nat" assumes "var \<notin> vars p" shows "var \<notin> vars (p ^ i)"
Lxy_distinct: fixes xs :: "'b list"   and S :: "'b set" assumes "distinct xs" shows "distinct (Lxy xs S)"
cases: fixes G :: "prog"   and a1 :: "state"   and a2 :: "obj_tag"   and a3 :: "loc"   and a4 :: "state" assumes "G\<turnstile>a1 \<midarrow>halloc a2\<succ>a3\<rightarrow> a4" obtains x :: "abrupt" and s :: "st" and oi :: "obj_tag" where "a1 = (Some x__, s__)" and "a2 = oi__" and "a3 = undefined" and "a4 = (Some x__, s__)"   | s :: "st" and a :: "loc" and x :: "abopt" and oi' :: "obj_tag" and oi :: "obj_tag" where "a1 = Norm s" and "a2 = oi" and "a3 = a" and "a4 = (x, init_obj G oi' (Inl a) s)" and "new_Addr (heap s) = Some a" and "(x, oi') = (if atleast_free (heap s) (Suc (Suc 0)) then (None, oi) else (Some (Xcpt (Loc a)), CInst (SXcpt OutOfMemory)))"
12: shows "w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T)\<^sup>T = w \<squnion> w\<^sup>T"
permutes_inv_o_1: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "p \<circ> inv p = id"
join_self_below_3: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y = x) = (y \<sqsubseteq> x)"
ibind_strict_2: fixes f :: "unit \<Rightarrow> 'b nres" shows "FAIL \<bind> f = FAIL"
viewRn_list: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node"   and s :: "('c, 'd) FingerTreeStruc" assumes "viewRn t = Some (nd, s)" shows "toList t = toList s @ nodeToList nd"
sterms_reflD: fixes q :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> sterms \<Gamma> p"   and "not_choice p"   and "not_call p" shows "q = p"
assms_2: fixes x :: "complex" assumes "x \<in> s" shows "(f has_field_derivative f' x) (at x within s)"
per_col_eq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C"   and "Col A B C"   and "B \<noteq> C" shows "A = B"
Send_ev': shows "ev' = Send i'' (occurs_on ev') s u'' u''' m'"
simps_52: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x4 :: "int" shows "(case byteV x4 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f4 x4"
collapse: fixes list :: "'g list" assumes "list \<noteq> []" shows "hd list # tl list = list"
alloc_def: fixes xs :: "'b list"   and h :: "heap" shows "Array.alloc xs h = (let l = lim h; r = Array l in Let (Array.set r xs (h\<lparr>lim := l + 1\<rparr>)) (Pair r))"
univalent_def: fixes R :: "('c \<times> 'd) set" shows "univalent R = (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> z = y)"
eval_fds_of_real: fixes f :: "real fds"   and s :: "real" assumes "fds_converges f s" shows "eval_fds (fds_of_real f) (of_real s) = of_real (eval_fds f s)"
attach_converter_of_resource_conv_parallel_resource: fixes res :: "('i, 'k) resource"   and res' :: "('j, 'l) resource" shows "converter_of_resource res |\<^sub>\<propto> 1\<^sub>C \<rhd> res' = res \<parallel> res'"
ltl_step_singleton: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and n :: "nat"   and r :: "nat \<Rightarrow>f value option"   and v :: "String.literal \<times> value list"   and aa :: "nat"   and b :: "value option list"   and c :: "nat \<Rightarrow>f value option" assumes "\<exists>t. possible_steps e n r (fst v) (snd v) = {|(aa, t)|} \<and> evaluate_outputs t (snd v) r = b \<and> evaluate_updates t (snd v) r = c" shows "ltl_step e (Some n) r v = (Some aa, b, c)"
P_def: fixes x :: "nat" shows "P x = (let F = from_nat x in (\<exists>A. \<not> A \<Turnstile> F) \<and> (\<exists>A. A \<Turnstile> F))"
relpowp_1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "P ^^ 1 = P"
FAssRedsNone: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
fmranI: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "m $$ x = Some y" shows "y |\<in>| fmran m"
prems_4: shows "reads_consistent False \<O> m (r # sb)"
indexed_padd_zero_2: fixes P :: "'d multiset \<Rightarrow> 'a" assumes "carrier_coeff P" shows "indexed_const \<zero> \<Oplus> P = P"
lm034: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G"   and "finite G" shows "card (pseudoAllocation a) = card G"
add_ac_2: fixes a :: "'e"   and b :: "'e" shows "a + b = b + a"
factorize_polys_irreducible_monic_1: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list" assumes "factorize_polys ps = (fs, data)" shows "distinct fs"
extremum_strict: fixes a :: "'a" shows "\<not> top_class.top < a"
Spy_not_see_PMS: fixes A :: "agent"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list" assumes "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "evs \<in> tls"   and "A \<notin> bad"   and "B \<notin> bad" shows "Nonce PMS \<notin> analz (knows Spy evs)"
events_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P: fixes A :: "'a set" shows "events_of (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = A"
mod_ring_minus_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (minus_p_integer pp) (-)"
tensor_mat_hermitian: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "hermitian m1"   and "hermitian m2" shows "hermitian (tensor_mat dims vars m1 m2)"
preserves_inv: fixes f :: "'a \<times> 'a" assumes "C.VV.iso f" shows "fst (C.VV.inv f) \<star>\<^sub>C snd (C.VV.inv f) = C.inv (fst f \<star>\<^sub>C snd f)"
has_limits_of_shape_if_target_does: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "category J"   and "B.has_limits_of_shape J" shows "has_limits_of_shape J"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_erase m rs (l, r) = (\<exists>ml mr ln rn. l = Bk \<up> ln @ Bk # Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> tl r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs \<and> 0 < mr)"
simps_18: fixes vb :: "'b"   and vc :: "value list"   and v :: "bool" shows "GExp_Lexorder.less_gexp_aux (In vb vc) (Bc v) = False"
positively_invariant_y: shows "g.positively_invariant {p. - snd p \<le> 0}"
d: shows "\<forall>x y. T (i x y) \<and> T x \<longrightarrow> T y"