hcju/isabelleps huggingface dataset:
arcs_elem_2: fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)" shows "b \<in> set (j # xs)"
not_hd_in_tl: fixes x :: "'b"   and xs :: "'b list" assumes "x \<noteq> hd xs"   and "x \<in> set xs" shows "x \<in> set (tl xs)"
semantics_pltl_sugar_6: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p G\<^sub>p \<phi> = (\<forall>i. w [i..] \<Turnstile>\<^sub>p \<phi>)"
rept_ex: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<mu> \<in> \<K> (abss s)" shows "rept s \<mu> \<in> K s \<and> abst (rept s \<mu>) = \<mu>"
xt1_5: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "y \<le>\<^sub>o\<^sub>w x"   and "x \<le>\<^sub>o\<^sub>w y" shows "x = y"
\<mu>': fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length fs" shows "\<mu>' i j = d j * (fs ! i \<bullet> fs ! j) - \<sigma> j i j"
normal_moment_odd: shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k + 1)) 0"
3: fixes y :: "'b"   and s1' :: "'s1"   and y' :: "'b"   and s2' :: "'s2"   and s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a" assumes "((y, s1'), y', s2') \<in> set_spmf (joint_oracle s1 s2 x)"   and "X s1 s2"   and "x \<in> outs_\<I> \<I>" shows "bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else y = y' \<and> X s1' s2')"
eventually_nhds_x_imp_x: fixes "open" :: "'e set \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "class.topological_space open"   and "eventually P (topological_space.nhds open x)" shows "P x"
tendsto_unique: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> a) F"   and "(f \<longlongrightarrow> b) F" shows "a = b"
iter_narrow_termination: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and n :: "'a acom \<Rightarrow> nat"   and C :: "'a acom" assumes "\<And>C. P C \<Longrightarrow> P (f C)"   and "\<And>C1 C2. \<lbrakk>P C1; P C2\<rbrakk> \<Longrightarrow> P (C1 \<triangle> C2)"   and "\<And>C1 C2. \<lbrakk>P C1; P C2; C1 \<le> C2\<rbrakk> \<Longrightarrow> f C1 \<le> f C2"   and "\<And>C1 C2. \<lbrakk>P C1; P C2; C2 \<le> C1; C1 \<triangle> C2 < C1\<rbrakk> \<Longrightarrow> n (C1 \<triangle> C2) < n C1"   and "P C"   and "f C \<le> C" shows "\<exists>C'. iter_narrow f C = Some C'"
derives1_drop_prefixword: fixes w :: "symbol list"   and a :: "symbol list"   and b :: "symbol list" assumes "is_word w"   and "derives1 (w @ a) b" shows "derives1 a (drop (length w) b)"
intros_1: fixes i :: "'a"   and I :: "'a set"   and P :: "'a set set" assumes "i \<in> I" shows "i \<in> gen P I"
intro: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and W :: "'i set" assumes "vectorspace K V"   and "LinearCombinations.submodule K W V" shows "subspace K W V"
simps_2: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" shows "interpret h FalseF xs = False"
run_sdrop: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) stream"   and p :: "nat \<times> 'b"   and k :: "nat" assumes "grun A w V r p" shows "grun A w V (sdrop k r) (gtarget (stake k r) p)"
vec_elements_def: fixes v :: "'b Matrix.vec" shows "vec_elements v = set (map (($v) v) [0..<dim_vec v])"
f: shows "Fvars \<phi> \<subseteq> {inp}"
strict_trans1: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" assumes "b \<subseteq># a"   and "c \<subset># b" shows "c \<subset># a"
reg_tr: shows "trace (S t i) (t ! i # take (j - 1 - (i + 1)) (drop (i + 1) t)) (S t (j - 1))"
isAbr_simps_1: fixes s :: "'e" shows "isAbr (Normal s) = False"
enc_respects_pred_iff_source_target_rel_respects_pred_encRL: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_respects_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_respects_pred Rel Pred)"
principal_dchar_minus1: fixes n :: "nat" assumes "0 < n" shows "principal_dchar n (n - Suc 0) = 1"
Gp_mult_induced: fixes G :: "('e, 'f) Group_scheme"   and L :: "'e set"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "G \<guillemotright> L "   and "a \<in> L"   and "b \<in> L" shows "a \<cdot>\<^bsub>\<natural>\<^bsub>G\<^esub>L\<^esub> b = a \<cdot>\<^bsub>G\<^esub> b"
ProcSpec: fixes P :: "'a set"   and init :: "'a \<Rightarrow> 'a"   and P' :: "'b \<Rightarrow> 'a set"   and Q' :: "'b \<Rightarrow> 'a set"   and return :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> 'a set"   and A' :: "'b \<Rightarrow> 'a set"   and A :: "'a set"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'c, 'd) com option"   and \<Theta> :: "('a set \<times> 'c \<times> 'a set \<times> 'a set) set"   and F :: "'d set"   and c :: "'a \<Rightarrow> 'a \<Rightarrow> ('a, 'c, 'd) com"   and Q :: "'a set"   and p :: "'c" assumes "P \<subseteq> _Collect s (\<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A))"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) Call p (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return c Q,A"
DERIV_ln: fixes x :: "real" assumes "0 < x" shows "(ln has_real_derivative inverse x) (at x)"
compact_max_time_flow_in_closed: fixes M :: "'a set"   and t :: "real"   and x :: "'a" assumes "closed M"   and "t \<in> existence_ivl0 x" shows "compact {s \<in> {0..t}. flow0 x ` {0..s} \<subseteq> M}"
StuckProp: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> \<Rightarrow> Stuck"
has_derivative_def: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "D f \<mapsto> f' F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit F) - f' (y - netlimit F)) /\<^sub>R (\<parallel>y - netlimit F\<parallel>)) \<longlongrightarrow> (0::'c)) F)"
F: shows "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> v"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
exec_gpv_parametric: fixes S :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and CALL :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod (=) S)))) (rel_fun (rel_gpv A CALL) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
hext_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "h \<unlhd> h' \<equiv> \<forall>a C S. h a = \<lfloor>(C, S)\<rfloor> \<longrightarrow> (\<exists>S'. h' a = \<lfloor>(C, S')\<rfloor>)"
ntrs_stack_decomp_s: fixes u :: "'a"   and s :: "'a list"   and r :: "'a list"   and c :: "'a list multiset"   and ee :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # s @ r, c), ee, s', c') \<in> ntrs fg" obtains v :: "'a" and u' :: "'a" and p :: "'b" where "s' = v # u' # s @ r" and "((u # s, c), ee, v # u' # s, c') \<in> ntrs fg" and "(u, Call p, u') \<in> edges fg"
less_1: fixes y :: "('s, 'v) tm" assumes "hsize y < hsize s"   and "y >\<^sub>t y" shows "False"
id_act: fixes b :: "'b" shows "b <o \<one> = b"
progress: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> M : \<tau>" shows "NF M \<or> (\<exists>M'. M \<rightarrow>\<beta> M')"
strict_domE: fixes n :: "'node"   and m :: "'node"   and g :: "'g" assumes "n \<noteq> m \<and> dominates g n m" obtains "m \<in> set (\<alpha>n g)" and "\<And>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<Longrightarrow> n \<in> set (butlast ns)"
funionM_comp_fun_idem: fixes return :: "'d fset \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d fset \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad_commute return bind"   and "monad_duplicate return bind" shows "comp_fun_idem (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A |\<union>| B))))"
eq_Nil_null: fixes xs :: "'a list" shows "(xs = []) = List.null xs"
abs_eq: fixes x :: "moebius_mat" shows "unit_circle_fix (abs_moebius x) = unit_circle_fix_mmat x"
is_singletonE: fixes A :: "'b set" assumes "is_singleton (A::'b set)" obtains x :: "'b" where "(A::'b set) = {x::'b}"
remove_term_sum: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'a mpoly" shows "remove_term m p + MPoly_Type.monom m (MPoly_Type.coeff p m) = p"
fag_single_additionTr1: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<forall>s. s \<in> add_set f {a} (Suc n) \<longrightarrow> s \<^sub>f+ \<^sub>i- a \<in> add_set f {a} n"
lincomb_cong: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "A = B"   and "A \<subseteq> carrier M"   and "\<And>v. v \<in> A \<Longrightarrow> a v = b v"   and "b \<in> B \<rightarrow> carrier R" shows "lincomb a A = lincomb b B"
interior_of_closure_of: fixes X :: "'a topology"   and S :: "'a set" shows "X interior_of S = topspace X - X closure_of (topspace X - S)"
xor_def: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl x y = disj (conj x (compl y)) (conj (compl x) y)"
app1: fixes env :: "v sem_env"   and s1 :: "'a state"   and es :: "exp0 list"   and s2 :: "'a state"   and vs :: "v list"   and env' :: "v sem_env"   and e :: "exp0"   and bv :: "'a state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 (rev es) (s2, Rval vs)"   and "do_opapp (rev vs) = Some (env', e)"   and "Big_Step_Unclocked_Single.evaluate env' s2 e bv" shows "Big_Step_Unclocked_Single.evaluate env s1 (App Opapp es) bv"
eventually_nhds_ball: fixes d :: "real"   and z :: "'a" assumes "0 < d" shows "\<forall>\<^sub>F x in nhds z. x \<in> ball z d"
integrable_completion: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "integrable (completion M) f = integrable M f"
cleanChans_length: fixes dchans :: "integer list" shows "length (cleanChans dchans (gState.channels g)) = length (gState.channels g)"
alphaBinp_def: shows "alphaBinp \<equiv> {(qbinp, qbinp'). sameDom qbinp qbinp' \<and> liftAll2 ($=) qbinp qbinp'}"
actHn_def: fixes k :: "cnf" shows "actHn k \<equiv> {nid. \<parallel>nid\<parallel>\<^bsub>k\<^esub> \<and> honest nid}"
eq: fixes v :: "'a vec" shows "fs ! i - (fs ! i - v) = v"
nh_case: fixes M :: "(action \<times> nat) list"   and ns :: "nat list" assumes "\<not> halts M ns" shows "{\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)}"
bl_img: shows "blocks_image B \<pi> \<subseteq># \<B>'"
abs_idempotent: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (abs a) = abs a"
cBang: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "P \<parallel> !P \<longmapsto> a<x> \<prec> P'"   and "\<And>C. F C (P \<parallel> !P) a x P'" shows "F C (!P) a x P'"
dests_inv: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:21, PRrep-:10, PRerr-:1} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>))) \<and> (l \<in> {PAodv-:16..PAodv-:19} \<union> {PPkt-:8..PPkt-:11} \<union> {PRreq-:10..PRreq-:13} \<union> {PRreq-:22..PRreq-:25} \<union> {PRrep-:11..PRrep-:14} \<union> {PRerr-:2..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> iD (rt \<xi>) \<and> the (dests \<xi> ip) = sqn (rt \<xi>) ip)) \<and> (l = PPkt-:14 \<longrightarrow> dip \<xi> \<in> iD (rt \<xi>)))"
0: shows "rawpsubst (rawpsubst (imp \<phi>1 \<phi>2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = imp (rawpsubst (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubst (rawpsubst \<phi>2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
invariantsHoldInState_def: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "invariantsHoldInState state F0 decisionVars \<equiv> InvariantImpliedLiterals F0 (getM state) \<and> InvariantVarsM (getM state) F0 decisionVars \<and> InvariantConsistent (getM state) \<and> InvariantUniq (getM state)"
induct: fixes P :: "'b tree \<Rightarrow> nat \<times> nat \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "nat \<times> nat" assumes "\<And>t w. \<lbrakk>\<And>x xa. \<lbrakk>\<not> r t < fst w; fst w \<noteq> l t; x = lchild t; xa = rchild t; l xa \<le> fst w\<rbrakk> \<Longrightarrow> P xa w; \<And>x xa. \<lbrakk>\<not> r t < fst w; fst w \<noteq> l t; x = lchild t; xa = rchild t; \<not> l xa \<le> fst w\<rbrakk> \<Longrightarrow> P x w\<rbrakk> \<Longrightarrow> P t w" shows "P a0 a1"
WT_calleeI: fixes \<I> :: "('b, 'c) \<I>"   and callee :: "'b \<Rightarrow> ('c \<times> 'd) spmf" assumes "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call" shows "\<I> \<turnstile>c callee \<surd>"
H_def: shows "local.H = Q\<^sub>1 AND Neg (PfP \<guillemotleft>HVar X0 HIMP HFm Q\<^sub>1\<guillemotright>) XOR Q\<^sub>2 AND Neg (PfP \<guillemotleft>HVar X0 HAND HNeg (HFm Q\<^sub>1) HIMP HFm Q\<^sub>2\<guillemotright>)"
Exit_source: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and a :: "'a" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "valid_edge a"   and "sourcenode a = Exit" shows "False"
pickInp_psubstInp_qPsubstInp: fixes inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodInp inp"   and "goodEnv rho" shows "(pickInp inp %[rho], pickInp inp %[[pickE rho]]) \<in> alphaInp"
order_bot_ow_axioms: shows "order_bot_ow U \<bottom>\<^sub>o\<^sub>w (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
carrier_FactGroup: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" shows "carrier (G Mod N) = (#>\<^bsub>G\<^esub>) N ` carrier G"
derivable_clauses_are_well_constrained: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> k C'" shows "well_constrained C"
extensional0_add: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "extensional0 S f"   and "extensional0 S g" shows "extensional0 S (f + g)"
diff: fixes x :: "'a"   and y :: "'a" shows "T (x - y) = T x - T y"
enc_atom_welldef: fixes I :: "('b + 'b set) list"   and I' :: "('b + 'b set) list"   and m :: "nat" assumes "\<forall>x a. enc_atom I x a = enc_atom I' x a"   and "m < length I" shows "case (I ! m, I' ! m) of (Inl p, Inl q) \<Rightarrow> p = q | (Inl p, Inr ba) \<Rightarrow> True | (Inr P, Inl a) \<Rightarrow> True | (Inr P, Inr Q) \<Rightarrow> P = Q"
simps: fixes i :: "int"   and j :: "int" shows "local.opt_bst i j = (if j < i then \<langle>\<rangle> else argmin (local.wpl i j) (map (\<lambda>k. \<langle>local.opt_bst i (k - 1), k, local.opt_bst (k + 1) j\<rangle>) [i..j]))"
ImpInit_def: fixes p :: "PrIds" shows "ImpInit p = (\<not> Calling memCh p \<and> MClkInit crCh cst p \<and> RPCInit rmCh rst p \<and> PInit ires p)"
ereal_less_minus: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "(x < z - y) = (x + y < z)"
case_eq_if: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
carde: shows "1 + (m - card X - card Y - 1) = m - card X - card Y"
coretract_axioms_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "coretract_axioms G \<equiv> \<forall>y. y \<in> carrier \<Y>\<^bsub>G\<^esub> \<longrightarrow> y \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y)"
wf_rexp_of: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "local.wf n (rexp_of n \<phi>)"
jordan_matrix_concat_diag_block_mat: fixes jbs :: "(nat \<times> 'b) list list" shows "jordan_matrix (concat jbs) = diag_block_mat (map jordan_matrix jbs)"
in_domains_char: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(a \<in> domains f) = ((f \<noteq> Null \<and> Dom f \<in> {A. |A| <o \<AA>} \<and> Cod f \<in> {A. |A| <o \<AA>} \<and> Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)) \<and> a = MkIde (Dom f))"
distinct_21: fixes x3 :: "'b"   and x51 :: "'b rexp"   and x52 :: "'b rexp" shows "Atom x3 \<noteq> Times x51 x52"
disjoint_add_mset_1: fixes B :: "'b multiset"   and a :: "'b"   and A :: "'b multiset" shows "(B \<inter># (a, A) = {#}) = (a \<notin># B \<and> B \<inter># A = {#})"
pos_prod_le: fixes c :: "'b"   and x :: "'b" assumes "(0::'b) < c" shows "c * x \<le> (0::'b) \<equiv> x \<le> (0::'b)"
show8: fixes A :: "real"   and d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and s :: "nat" assumes "1 < A"   and "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "0 < s"   and "convergent_prod d"   and "\<forall>n\<ge>s. (\<Prod>j. d (n + j)) < A / real_of_int (a n) powr (1 / real_of_int (2 ^ n))" shows "\<forall>n\<ge>s. (\<Prod>j. d (j + n)) < A / (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))"
lift_is_hom: fixes f :: "'b \<Rightarrow> bool"   and gens :: "'b set" assumes "f \<in> gens \<rightarrow> carrier C2" shows "C2.lift f \<in> hom \<F>\<^bsub>gens\<^esub> C2"
FSwSbImorph_iff: fixes h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'h, 'j, 'm) model_scheme" shows "FSwSbImorph h hA MOD MOD' = (FSbImorph h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD')"
that: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "thesis"
NatTransFtor2: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTransP \<eta>" shows "Functor (NTCod \<eta>)"
insert_spec_none: fixes xs :: "'c list"   and oid :: "'c" shows "set (insert_spec xs (oid, None)) = set xs \<union> {oid}"
assms_5: shows "f (deg R f) \<otimes> g (deg R g) [^] n \<noteq> \<zero>"
r2_trans: fixes t :: "trm"   and t' :: "trm"   and s :: "trm" assumes " t \<mapsto>\<^sup>* t' " shows " App s t \<mapsto>\<^sup>* App s t' "
ev_signal_respects_policy: fixes tid :: "thread_id_t"   and u :: "partition_id_t"   and s :: "state_t"   and partner :: "thread_id_t"   and ipt :: "int_point_t" assumes "\<not> Policy.ifp (Step.partition tid) u"   and "atomic_step_invariant s"   and "atomic_step_precondition s tid (SK_EV_SIGNAL EV_SIGNAL_FINISH partner)"   and "ipt = SK_EV_SIGNAL EV_SIGNAL_FINISH partner" shows "vpeq u s (atomic_step_ev_signal tid partner s)"
supp_of_finite_union: fixes S :: "'b set"   and T :: "'b set" assumes "finite S"   and "finite T" shows "supp (S \<union> T) = supp S \<union> supp T"
def_scheme: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "f X Y = (\<lambda>\<tau>. if (\<upsilon> X) \<tau> = true \<tau> \<and> (\<upsilon> Y) \<tau> = true \<tau> then (X \<triangleq> Y) \<tau> else invalid \<tau>)"
F_def: shows "F = (\<lambda>x. \<Sum>n. arctan_coeffs n * x ^ n)"
ennreal_top_minus: fixes x :: "ennreal" shows "\<top> - x = \<top>"
open_octo_def: fixes U :: "octo set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
ofilter_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. under r a \<subseteq> A)"
plus_enat_def: fixes m :: "enat"   and n :: "enat" shows "m + n = (case m of enat m \<Rightarrow> case n of enat n \<Rightarrow> enat (m + n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
ideal_generated_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.ring plus zero minus uminus times" shows "ring.ideal_generated plus zero uminus times S = \<Inter> {I. ring.ideal plus zero uminus times I \<and> S \<subseteq> I}"
the_inverseD_3: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and g :: "'a" assumes "BinOpSetGroup G binop e"   and "g \<in> G" shows "binop (BinOpSetGroup.the_inverse G binop e g) g = e"
list_collect_set_def: fixes f :: "'b \<Rightarrow> 'a set"   and l :: "'b list" shows "list_collect_set f l \<equiv> \<Union> {f a |a. a \<in> set l}"
cond_3': assumes "s\<^sub>o \<le> ego.s_stop"   and "ego.s_stop < other.s_stop" shows "collision {0..} = (a\<^sub>e < a\<^sub>o \<and> v\<^sub>o < v\<^sub>e \<and> s\<^sub>o - s\<^sub>e \<le> snd_safe_distance \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o < 0)"
a: shows "supp x - as \<inter> supp x = supp (p \<bullet> x) - as' \<inter> supp (p \<bullet> x)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (gcd_integer x xa) = gcd (int_of_integer x) (int_of_integer xa)"
sdrop_until: fixes q :: "'a stream \<Rightarrow> bool"   and j :: "nat"   and \<omega> :: "'a stream"   and p :: "'a stream \<Rightarrow> bool" assumes "q (sdrop j \<omega>)"   and "\<forall>k<j. p (sdrop k \<omega>)" shows "(p until q) \<omega>"