hcju/isabelleps huggingface dataset:
facet_of_halfspace_le: fixes F :: "'b set"   and a :: "'b"   and b :: "real" shows "(F facet_of {x. a \<bullet> x \<le> b}) = (a \<noteq> (0::'b) \<and> F = {x. a \<bullet> x = b})"
retraction: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" assumes "ChamberComplexRetraction X f"   and "v \<in> \<Union> X" shows "f (f v) = f v"
ts_inv_okI: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option" assumes "\<And>t. (ts t = None) = (I t = None)" shows "ts_inv_ok ts I"
IH: fixes z :: "'a" shows "rGamma z = pochhammer z n * rGamma (z + of_nat n)"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "C.iso (\<epsilon> a)"
proved_terms_closed: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> B" shows "is_closed B"
qSwap_sym: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "X #[[x \<and> y]]_zs = X #[[y \<and> x]]_zs"
id_iso: fixes G :: "('b, 'c) monoid_scheme" shows "id \<in> Group.iso G G"
Atoms_def: fixes istate :: "'a istate" shows "Atoms istate = (case istate of IState x1 x2 x3 x4 x5 \<Rightarrow> x3)"
invertible_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "invertible A = (\<exists>A'. A ** A' = mat (1::'b) \<and> A' ** A = mat (1::'b))"
complete_algebra_summable_geometric: fixes x :: "'b" assumes "norm x < 1" shows "summable ((^) x)"
T1x: fixes x :: "'a" shows "subprob_space (K1.T x)"
lookup_map_scale: fixes c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (c \<cdot> p) = (\<lambda>x. c * lookup p x)"
odiff_ge_iff: fixes z :: "V"   and x :: "V"   and y :: "V" assumes "z \<unlhd> x" shows "(y \<le> odiff x z) = (z + y \<le> x)"
simps: fixes n :: "nat" shows "\<S> n = F \<union> {q. \<exists>j\<ge>the smallest_accepting_rank. state_rank q n = Some j}"
Tree\<^sub>\<alpha>_free_3: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]"   and \<alpha> :: "'g"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>"
eq2: shows "lst_conv \<rho> = lst \<sigma>3'"
perm_vars_distinct: fixes n :: "nat"   and vs :: "nat list" assumes "perm_vars n vs" shows "distinct vs"
con_rews_2: fixes discr :: "bool discr" shows "CB\<cdot>discr \<noteq> \<bottom>"
tab_in_hom_1: shows "\<guillemotleft>\<rho> : src f \<rightarrow> trg r\<guillemotright>"
ideD_1: fixes a :: "'a" assumes "Right.ide a" shows "Right.arr a"
tau_transition_trans: fixes P :: "'state"   and Q :: "'state"   and R :: "'state" assumes "P \<Rightarrow> Q"   and "Q \<Rightarrow> R" shows "P \<Rightarrow> R"
that: fixes S :: "'a set" assumes "a \<in> S"   and "pairwise orthogonal S"   and "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"   and "independent S"   and "card S = DIM('a)"   and "span S = UNIV" shows "thesis"
fmran'I: fixes m :: "'e \<rightharpoonup>\<^sub>f 'd"   and x :: "'e"   and y :: "'d" assumes "fmlookup m x = Some y" shows "y \<in> fmran' m"
prec_s: shows "precondition (next_state s execs) (next_action s execs)"
set_lebesgue_integral_def: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (lcs x xa) = (\<lambda>xb. max (lookup x xb) (lookup xa xb))"
S_def: shows "S \<equiv> count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M)"
R_def: shows "R \<equiv> ((A ===> (=)) ===> A ===> (=)) ===> A ===> (=)"
RedCastNull: fixes C :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>Cast C null,s,b\<rangle> \<rightarrow> \<langle>null,s,b\<rangle>"
poly_induct2: fixes P :: "'a poly \<Rightarrow> 'b poly \<Rightarrow> bool"   and p :: "'a poly"   and q :: "'b poly" assumes "P 0 0"   and "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)" shows "P p q"
sep_disj_zero: fixes x :: "'a" shows "x ## (0::'a)"
commute: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "linordered_monoid.min a b = linordered_monoid.min b a"
list_succ_not_in: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "list_succ xs x = x"
subset_box_1: fixes c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" shows "(cbox c d \<subseteq> cbox a b) = ((\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i) \<longrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i))"
strategyproofness_2: shows "(pmf (sds R12) d + pmf (sds R12) b < pmf (sds R10) d + pmf (sds R10) b \<or> pmf (sds R12) d + (pmf (sds R12) b + pmf (sds R12) a) < pmf (sds R10) d + (pmf (sds R10) b + pmf (sds R10) a)) \<or> pmf (sds R12) d + pmf (sds R12) b = pmf (sds R10) d + pmf (sds R10) b \<and> pmf (sds R12) d + (pmf (sds R12) b + pmf (sds R12) a) = pmf (sds R10) d + (pmf (sds R10) b + pmf (sds R10) a)"
pred_intros_logic_3: fixes Q :: "'a \<Rightarrow> bool"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "Measurable.pred M Q"   and "Measurable.pred M P" shows "Measurable.pred M (\<lambda>x. Q x \<and> P x)"
iso_def: fixes "from" :: "'a \<Rightarrow> 'b"   and to :: "'b \<Rightarrow> 'a" shows "Derive.iso from to = ((\<forall>a. to (from a) = a) \<and> (\<forall>b. from (to b) = b))"
simps_1: fixes x :: "'c"   and ys :: "'c list" shows "replace x ys [] = []"
rel_set_neg_distr_cond_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_set_neg_distr_cond A A' = (rel_set (A OO A') \<le> rel_set A OO rel_set A')"
group_strategy_proofI: fixes ds :: "'b set"   and \<phi> :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('c \<Rightarrow> 'a set \<Rightarrow> 'a set) \<Rightarrow> 'b set \<Rightarrow> 'a set" assumes "\<And>Pd Pd' Ch ds'. \<lbrakk>ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch; ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (override_on Pd Pd' ds') Ch; ds' \<subseteq> ds; ds' \<noteq> {}; \<forall>d\<in>ds'. \<exists>y\<in>\<phi> (override_on Pd Pd' ds') Ch ds. y \<in> AboveS (Pd d) (ManiPref'.dX (\<phi> Pd Ch ds) d)\<rbrakk> \<Longrightarrow> False" shows "ManiPref'.group_strategy_proof ds \<phi>"
f_inj: shows "inj_on f (Sup_llist (lmap g xs))"
ex_final1_conv_ex_final2: shows "(\<exists>x1. final1 x1) = (\<exists>x2. final2 x2)"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright> \<equiv> a \<in> A.domains f \<and> b \<in> A.codomains f"
markedElementsTrailMemPrefixAreMarkedElementsPrefix: fixes M :: "('a \<times> bool) list"   and p :: "('a \<times> bool) list"   and m :: "'a" assumes "uniq (elements M)"   and "isPrefix p M"   and "m \<in> set (elements p)"   and "m \<in> set (markedElements M)" shows "m \<in> set (markedElements p)"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "a *\<^sub>\<real> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_mult"
surjective: fixes r :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" shows "r = \<lparr>CatDom = CatDom r, CatCod = CatCod r, MapM = (##) r, \<dots> = Functor.more r\<rparr>"
length_mtf2: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "length (mtf2 n x xs) = length xs"
T_nonTickFree_imp_decomp: fixes t :: "'c event list"   and P :: "'c process" assumes "t \<in> T P"   and "\<not> tickFree t" shows "\<exists>s. t = s @ [tick]"
distinct_mtf2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "distinct (mtf2 n x xs) = distinct xs"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "KA.basis A" shows "KA.dim = card A"
of_int_sint: fixes a :: "'b word" shows "word_of_int (sint a) = a"
less_self: fixes t :: "'b list"   and a :: "'b" shows "t < t @ [a]"
SR_B_inj: shows "inj_on (\<lambda>x. x mod int p) B"
t_star_def: fixes X :: "('a \<times> ('b \<times> 'a) list) set" shows "t_star X \<equiv> \<Union> (range (trace_dioid.power X))"
assms: shows "r \<in> p_e modRules2 S T"
min_less_iff_disj: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" shows "(signed.min x y <s z) = (x <s z \<or> y <s z)"
set_integrable_UN: fixes I :: "'c set"   and M :: "'d measure"   and A :: "'c \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'e" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "set_integrable M (\<Union> (A ` I)) f"
gen_cong_minimal: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "R \<le> R'"   and "cong.cong rel eval R'" shows "cong.gen_cong rel eval R \<le> R'"
rel_interior_subset: fixes S :: "'a set" shows "rel_interior S \<subseteq> S"
strict_mono_def: fixes f :: "'b \<Rightarrow> 'c" shows "strict_mono f = (\<forall>x y. x < y \<longrightarrow> f x < f y)"
in_existence_between_zeroI: fixes t :: "real"   and x0 :: "'a"   and s :: "real" assumes "t \<in> existence_ivl0 x0"   and "s \<in> {0--t}" shows "s \<in> existence_ivl0 x0"
strict_dominate_def: fixes n1 :: "nat"   and n2 :: "nat" shows "strict_dominate n1 n2 \<equiv> \<forall>pa. path_entry (g_E G) pa n2 \<longrightarrow> n1 \<in> set pa \<and> n1 \<noteq> n2"
cycle_decomp_imp_swapidseq_ext: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'a" assumes "cycle_decomp S p" shows "\<exists>n. swapidseq_ext S n p"
has_disc_negated_positiv_has_disc: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr" shows "(has_disc_negated disc neg m \<or> has_disc_negated disc (\<not> neg) m) = has_disc disc m"
alpha_sym_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "perm"   and x :: "'a"   and y :: "'a"   and bs :: "atom set"   and f :: "'a \<Rightarrow> atom set"   and cs :: "atom set" assumes "R (p \<bullet> x) y \<Longrightarrow> R (- p \<bullet> y) x"   and "(bs, x) \<approx>set R f p (cs, y)" shows "(cs, y) \<approx>set R f (- p) (bs, x)"
disc_eq_case_1: fixes option :: "'a option" shows "(option = None) = (case option of None \<Rightarrow> True | Some uu_ \<Rightarrow> False)"
g1_def: fixes x :: "nat"   and u :: "nat" shows "g1 x u = (if c_in x u = 1 then g x else 0)"
subgame_ParityGame: fixes V' :: "'a set" shows "ParityGame (subgame V')"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (x \<le>lex y \<and> y \<le>lex x)"
card_B: shows "card B = local.dim {x. f x = (0::'c)}"
sfilter_smap': fixes p :: "'a \<rightarrow> tr"   and f :: "'b \<rightarrow> 'a"   and xs :: "[:'b:]" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p\<cdot>(smap\<cdot>f\<cdot>xs) = smap\<cdot>f\<cdot>(sfilter\<cdot>(p oo f)\<cdot>xs)"
negation_type_forall_split: fixes Ms :: "'a negation_type list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<forall>is\<in>set Ms. case is of Pos i \<Rightarrow> P i | Neg i \<Rightarrow> Q i) = ((\<forall>i\<in>set (getPos Ms). P i) \<and> (\<forall>i\<in>set (getNeg Ms). Q i))"
meet_compl2_bot: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqinter> \<^bold>- x = \<bottom>"
scaleC_sum_right: fixes a :: "complex"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "a *\<^sub>C sum f A = (\<Sum>x\<in>A. a *\<^sub>C f x)"
cont_absc_1: fixes k :: "'a \<Rightarrow> nat"   and v :: "'a \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'a"   and A :: "('b \<times> ('a, real) cconstraint \<times> ('a set \<times> 'b) pmf) set \<times> ('b \<Rightarrow> ('a, real) cconstraint)"   and cfg :: "('b \<times> ('a \<Rightarrow> real)) cfg"   and s' :: "'b \<times> ('a \<Rightarrow> real)" assumes "Probabilistic_Timed_Automaton_Regions k v n not_in_X A"   and "cfg \<in> Markov_Decision_Process_Invariant.valid_cfg (Probabilistic_Timed_Automaton.K A) (Probabilistic_Timed_Automaton.S A)"   and "s' \<in> set_pmf (action cfg)" shows "cont (Probabilistic_Timed_Automaton_Regions.absc k A cfg) (Probabilistic_Timed_Automaton_Regions.abss k A s') = Probabilistic_Timed_Automaton_Regions.absc k A (cont cfg s')"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "P 0 0"   and "\<And>k. P 0 (Suc k)"   and "\<And>m. P (Suc m) 0"   and "\<And>m k. \<lbrakk>P m k; P (m - k) (Suc k)\<rbrakk> \<Longrightarrow> P (Suc m) (Suc k)" shows "P a0 a1"
vinsertI2: fixes x :: "V"   and A :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "x \<in>\<^sub>\<circ> vinsert y A"
Sup_upper: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
exhaust: fixes y :: "'a discr" obtains x :: "'a" where "y = Discr x"
dns4: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d y \<cdot> d x"
Rats_cases': fixes x :: "'b" assumes "x \<in> \<rat>" obtains a :: "int" and b :: "int" where "0 < b" and "algebraic_semidom_class.coprime a b" and "x = of_int a / of_int b"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "Rep_pos (x - xa) = (if Rep_pos xa \<le> Rep_pos x then Rep_pos x - Rep_pos xa else (0::'a))"
homeomorphic_empty_2: fixes S :: "'d set" shows "{} homeomorphic S = (S = {})"
invariantsHoldInValidRunsFromInitialState: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars" shows "invariantsHoldInState state F0 decisionVars"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs" shows "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs type_wf"
bis_imp_6: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>wT c2" shows "c1 \<approx>T c2"
uminus_uint_def: shows "uminus \<equiv> map_fun Rep_uint Abs_uint uminus"
left_adjoint_functor_def: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" shows "left_adjoint_functor D C F \<equiv> (category C \<and> category D) \<and> functor D C F \<and> left_adjoint_functor_axioms D C F"
distinct_2: fixes x21 :: "tree"   and x22 :: "tree" shows "Branching x21 x22 \<noteq> Leaf"
BRParCases': fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q \<parallel> !Q) \<in> bangRel Rel" obtains R :: "pi" and T :: "pi" and P :: "pi" and Qa :: "pi" where "Pa__ = R \<parallel> P" and "Q \<parallel> !Q = T \<parallel> Qa" and "(R, T) \<in> Rel" and "(P, Qa) \<in> bangRel Rel"
strategyproofness: shows "(pmf (sds R12) c < pmf (sds R9) d \<or> pmf (sds R12) c + pmf (sds R12) d < pmf (sds R9) d + pmf (sds R9) c) \<or> pmf (sds R12) c = pmf (sds R9) d \<and> pmf (sds R12) c + pmf (sds R12) d = pmf (sds R9) d + pmf (sds R9) c"
assms_1: shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t1"
ladder_stepdown_\<alpha>_0_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_stepdown_\<alpha>_0 a D L = Derive a (take (ladder_stepdown_diff L) D)"
induces_meta_adjunction: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "left_adjoint_functor D C F" shows "meta_adjunction C D F (left_adjoint_functor.G D C F) (left_adjoint_functor.\<phi> D C F) (left_adjoint_functor.\<psi> D C F)"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "arr f"   and "cod f = b" shows "b \<cdot> f = f"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>Q' a x. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>Q' a x. \<lbrakk>Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"   and "\<And>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel> Q"
S_inductive_Sup_min_bounds: fixes X :: "'a set"   and b :: "'a" assumes "S_inductive_set D"   and "Chain D X"   and "ub X b" shows "Algebra2.Sup D X \<preceq> b"
prems: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
2: fixes u :: "real"   and A :: "real \<Rightarrow> bool" assumes "\<forall>x. u \<le> \<bar>x\<bar> \<longrightarrow> A x" shows "\<exists>N. \<forall>n\<ge>N. A n"
H: shows "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> S <: T"
"2_2": shows "\<forall>u\<in>set l1. P1 u"
unfold: shows "lall = (\<Lambda> p v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>TT) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(p\<cdot>x andalso lall\<cdot>p\<cdot>xs))))"
carrier_subgroup_generated_subgroup: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "subgroup H G" shows "carrier (subgroup_generated G H) = H"
linorder_proper_interval_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and proper_interval :: "'c option \<Rightarrow> 'c option \<Rightarrow> bool" shows "class.linorder_proper_interval less_eq less proper_interval \<equiv> class.linorder less_eq less \<and> class.proper_interval less proper_interval"
dims1_def: shows "st.dims1 = nths dims {0..<Suc k}"
k_th_has_derivative_at_left: shows "((\<lambda>x. nth_derivative k f x 1) has_derivative (\<lambda>x. 0)) (at 0 within {..0})"
perp_not_eq_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "A \<noteq> B"
in_mset_fold_plus_iff: fixes x :: "'k"   and M :: "'k multiset"   and NN :: "'k multiset multiset" shows "(x \<in># fold_mset (+) M NN) = (x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N))"
assms_1: shows "\<Omega> \<inter> space M \<in> sets M"
sup_id_star1: fixes x :: "'a list set" assumes "p_one \<subseteq> x" shows "p_prod x (p_star x) = p_star x"
swaps_inv: fixes sws :: "nat list"   and xs :: "'c list" shows "set (swaps sws xs) = set xs \<and> length (swaps sws xs) = length xs \<and> distinct (swaps sws xs) = distinct xs"
concat_eq_Nil_conv: fixes xss :: "'b list list" shows "(concat xss = []) = (\<forall>xs\<in>set xss. xs = [])"
embed_path_def: fixes embed_elem :: "'a \<Rightarrow> 'b" shows "embed_path embed_elem \<equiv> map (embed_path_elem embed_elem)"
closed_proj_rel: shows "closed {(x, y). proj_rel x y}"
real_lebesgue_integral_def: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) - enn2real (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M)"
of_int_poly_Mp_0: fixes m :: "int"   and a :: "int poly" assumes "poly_mod_prime_type TYPE('a) m" shows "(of_int_poly (poly_mod.Mp m a) = 0) = (poly_mod.Mp m a = 0)"
filterlim_at_top_gt: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" shows "filterlim f at_top F = (\<forall>Z>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
rtranclp_induct2: fixes r :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool"   and ax :: "'c"   and ay :: "'d"   and bx :: "'c"   and "by" :: "'d"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay) (bx, by)"   and "P ax ay"   and "\<And>a b aa ba. \<lbrakk>r\<^sup>*\<^sup>* (ax, ay) (a, b); r (a, b) (aa, ba); P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
powrfinitesum: fixes s :: "nat"   and n :: "nat"   and a :: "real" assumes "s \<le> n" shows "(\<Prod>j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}"
actions_\<E>E_aux: fixes \<sigma> :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and E' :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist"   and a :: "nat" assumes "h.mthr.if.mthr.Runs \<sigma> E'"   and "enat a < llength (lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) E'))" obtains m :: "nat" and n :: "nat" and t :: "'thread_id" and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action" where "lnth (lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) E')) a = (t, \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! n)" and "n < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>" and "enat m < llength E'" and "a = (\<Sum>i<m. length \<lbrace>snd (lnth E' i)\<rbrace>\<^bsub>o\<^esub>) + n" and "lnth E' m = (t, ta)"
1: shows "map ((!) (as @ bs)) (fst xs) = map ((!) as) (fst xs)"
Q_def: shows "Q = {q. q permutes {0..<l}}"
cscc_invar_loc_def: fixes G :: "('c, 'd) graph_rec_scheme"   and v0 :: "'c"   and D0 :: "'c set"   and l :: "'c set list"   and p :: "'c set list"   and D :: "'c set"   and pE :: "('c \<times> 'c) set" shows "cscc_invar_loc G v0 D0 l p D pE \<equiv> invar_loc G v0 D0 p D pE \<and> cscc_invar_ext G l D"
left_right_inverse_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = one" shows "times (power.power one times x n) (power.power one times y n) = one"
lookupEvalHeap_other: fixes v :: "'d"   and \<Gamma> :: "('d \<times> 'e) list"   and f :: "'e \<Rightarrow> 'f" assumes "v \<notin> domA \<Gamma>" shows "evalHeap \<Gamma> f v = \<bottom>"