hcju/isabelleps huggingface dataset:
basis_exists: fixes V :: "('c, 'd) vec set" obtains B :: "('c, 'd) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B" and "card B = vec.dim V"
sub2_wit_gram_schmidt_sub_triv'': fixes fs :: "'a vec list"   and mus :: "'a list list"   and gs :: "'a vec list" assumes "sub2_wit [] fs = (mus, gs)"   and "set fs \<subseteq> carrier_vec n" shows "norms_mus' fs [] [] = (map sq_norm (rev gs), rev mus)"
vars_setsum: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "finite S" shows "vars (sum f S) \<subseteq> (\<Union>m\<in>S. vars (f m))"
linear_independent_injective_image: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and s :: "('c, 'd) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.independent s"   and "inj_on f (vec.span s)" shows "vec.independent (f ` s)"
f_def: shows "f = (\<lambda>a. h [^]\<^bsub>H\<^esub> get_exp gen a)"
topspace_map_Proj: shows "topspace (map_topology Proj euclidean) = UNIV"
update_by_fun_pair_eq_update_by_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and k :: "'a"   and f :: "'d \<Rightarrow> 'd" assumes "oalist_inv_raw x xs" shows "update_by_fun_pair x k f xs = update_by_pair x (k, f (lookup_pair x xs k)) xs"
zcount_uminus: fixes A :: "'b zmultiset"   and x :: "'b" shows "zcount (- A) x = - zcount A x"
Rats_cases': fixes x :: "'b" assumes "x \<in> \<rat>" obtains a :: "int" and b :: "int" where "0 < b" and "coprime a b" and "x = of_int a / of_int b"
assms_2: shows "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t"
assms: shows "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)"
P: shows "pre @ [Broadcast (r, Expunge e mo r)] prefix of k"
distinct_1: fixes x2 :: "'b" shows "PUnit \<noteq> PVar x2"
stereographic_inv_stereographic': fixes z :: "complex"   and z1 :: "complex"   and z2 :: "complex"   and X :: "real"   and Y :: "real"   and Z :: "real" assumes "z = z1 / z2"   and "z2 \<noteq> 0"   and "X = Re (2 * z / (1 + z * cnj z))"   and "Y = Im (2 * z / (1 + z * cnj z))"   and "Z = ((cmod z)\<^sup>2 - 1) / (1 + (cmod z)\<^sup>2)" shows "\<exists>k. k \<noteq> 0 \<and> (cor X + \<i> * cor Y, cor (1 - Z)) = k *\<^sub>s\<^sub>v (z1, z2)"
safe_convert_multiway: fixes \<phi> :: "Formula.formula" assumes "safe_formula \<phi>" shows "safe_formula (convert_multiway \<phi>)"
card_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
alw_coinduct: fixes X :: "'d stream \<Rightarrow> bool"   and x :: "'d stream"   and \<phi> :: "'d stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<phi> x"   and "\<And>x. \<lbrakk>X x; \<not> alw \<phi> (stl x)\<rbrakk> \<Longrightarrow> X (stl x)" shows "alw \<phi> x"
ctermsp_cterms_eq: fixes \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" shows "ctermsp \<Gamma> = (\<lambda>x. x \<in> cterms \<Gamma>)"
elims: fixes x :: "('d, 'c) trie"   and xa :: "'d list"   and y :: "'c option" assumes "lookup_trie x xa = y" obtains v :: "'c option" and m :: "('d \<times> ('d, 'c) trie) list" where "x = Trie v__ m__" and "xa = []" and "y = v__"   | v :: "'c option" and m :: "('d \<times> ('d, 'c) trie) list" and k :: "'d" and ks :: "'d list" where "x = Trie v m" and "xa = k # ks" and "y = (case map_of m k of None \<Rightarrow> None | Some st \<Rightarrow> lookup_trie st ks)"
inorder_splay: fixes x :: "'b"   and t :: "'b tree" shows "inorder (splay x t) = inorder t"
mem_permute_set: fixes x :: "'a"   and p :: "perm"   and S :: "'a set" shows "(x \<in> p \<bullet> S) = (- p \<bullet> x \<in> S)"
finite_Sup_fun: fixes A :: "('a \<Rightarrow> 'b) set" shows "Sup A = Finite_Set.fold sup bot A"
rel_spmf_mono: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b spmf"   and g :: "'d spmf"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
inf_commute: fixes x :: "'b upred"   and y :: "'b upred" shows "(x \<and> y) = (y \<and> x)"
produced_by_scan_step_def: fixes x :: "item"   and k :: "nat"   and u :: "nat" shows "produced_by_scan_step x k u = (\<exists>k' u' y X. indexle k' u' k u \<and> y \<in> \<J> k' u' \<and> item_end y = k' \<and> X \<in> \<T> k' u' \<and> x = inc_item y (k' + length (chars_of_token X)) \<and> next_symbol y = Some (terminal_of_token X))"
divide_self_if: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> '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 a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a a = (if a = zero then zero else one)"
SystemKSubst_def: shows "SystemKSubst = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Geq (Const 0) (Const 0) && Geq ($f fid1 local.empty) (Const 0)) else if C = pid2 then Some ([[DiffAssign vid1 ($f fid1 local.empty)]]Geq (Differential (Var vid1)) (Differential (Const 0))) else None, SPrograms = \<lambda>c. if c = vid1 then Some (EvolveODE (OProd (OSing vid1 ($f fid1 local.empty)) (OSing vid2 (Var vid1))) (Geq (Const 0) (Const 0) && Geq ($f fid1 local.empty) (Const 0))) else None, SODEs = Map.empty\<rparr>"
multpw_mult1_commute: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns" shows "mult1 s O multpw ns \<subseteq> multpw ns O mult1 s"
assms_3: fixes t :: "'a net_state"   and a :: "'b node_action"   and t' :: "'a net_state"   and i :: "nat"   and d :: "nat"   and di :: "nat" assumes "(t, a, t') \<in> T" shows "a \<noteq> i:newpkt(d, di)"
x1ImportantProp1: shows "p x1 \<notin> p ` snd ` f"
size_del_min: fixes t :: "'b tree" assumes "braun t" shows "size (del_min t) = size t - 1"
l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas DocumentClass.type_wf"
static_completeness: fixes p :: "'a pfm" assumes "static p"   and "\<forall>M w. M, w \<Turnstile>\<^sub>! p" shows "\<turnstile>\<^sub>! p"
simps_4: fixes as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" shows "transform_alts as (Arg v # S) = Arg v # transform_alts as S"
inner_prod_cnj: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" assumes "dim_vec u = dim_vec v" shows "\<langle>v|u\<rangle> = cnj \<langle>u|v\<rangle>"
scc_of_other: fixes u :: "'a" assumes "u \<notin> G.scc_of (tail G a)" shows "H.scc_of u = G.scc_of u"
d_IN_add: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_IN (\<lambda>e. f e + g e) x = d_IN f x + d_IN g x"
S3RPC: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCNext crCh rmCh rst p \<and> $S3 rmhist p \<longrightarrow> RPCFwd crCh rmCh rst p \<or> RPCFail crCh rmCh rst p"
case_prodI: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and b :: "'c" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
assms_3: shows "InvariantImpliedLiterals (getF state) (getM state)"
rt_2: shows "flow0 (flow0 x t) (return_time P (flow0 x t)) \<in> P"
\<mu>_stable_subfrmlsn: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltln" assumes "\<mu>_stable \<phi> w"   and "\<psi> \<in> subfrmlsn \<phi>" shows "\<mu>_stable \<psi> w"
partial_magma_axioms: shows "partial_magma B.VV.comp"
cosh_real_eq_iff: fixes x :: "real"   and y :: "real" shows "(cosh x = cosh y) = (\<bar>x\<bar> = \<bar>y\<bar>)"
conj_one_right: fixes x :: "'a" shows "x AND - (1::'a) = x"
reads_restrict1: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'e"   and n :: "'a"   and \<sigma>' :: "'b \<Rightarrow> 'e" assumes "IFC prob"   and "\<sigma> \<restriction> IFC_def.reads prob n = \<sigma>' \<restriction> IFC_def.reads prob n" shows "\<forall>x\<in>IFC_def.reads prob n. \<sigma> x = \<sigma>' x"
ucast_ucast_eq_mask_shift: fixes x :: "'b word"   and low_bits :: "nat" assumes "x \<le> mask (low_bits + LENGTH('c))" shows "UCAST('c \<rightarrow> 'b) (UCAST('b \<rightarrow> 'c) (x >> low_bits)) = x >> low_bits"
assms_5: shows "val_Zp b < val_Zp c"
cancel_separate: fixes x :: "'a"   and y :: "'a" assumes "x ; y \<le> 1'" shows "x\<^sup>\<star> ; y\<^sup>\<star> \<le> x\<^sup>\<star> + y\<^sup>\<star>"
icard_insert_disjoint: fixes x :: "'a"   and A :: "'a set" assumes "x \<notin> A" shows "icard (insert x A) = eSuc (icard A)"
sup_commute: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
sexp_case_Numb: fixes c :: "'d \<Rightarrow> 'c"   and d :: "nat \<Rightarrow> 'c"   and e :: "('d, unit) Old_Datatype.node set \<Rightarrow> ('d, unit) Old_Datatype.node set \<Rightarrow> 'c"   and k :: "nat" shows "sexp_case c d e (Numb k) = d k"
wf_envs_progress_regex_le_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and db :: "(String.literal, event_data list set list) mapping"   and r :: "Formula.formula Regex.regex" assumes "wf_envs \<sigma> j P P' V db" shows "progress_regex \<sigma> P r j \<le> j"
two: shows "Missing_List.min_list ys \<le> Missing_List.min_list xs"
nonzero_mult_div_cancel_right: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.semidom_divide divide plus minus zero times one"   and "b \<noteq> zero" shows "divide (times a b) b = a"
extreme_point_of_face: fixes F :: "'b set"   and S :: "'b set"   and v :: "'b" assumes "F face_of S" shows "(v extreme_point_of F) = (v extreme_point_of S \<and> v \<in> F)"
weight_le_less_iff: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" assumes "weight_le s t" shows "weight_less s t = (weight s < weight t)"
replication_numbers_non_empty: assumes "\<V> \<noteq> {}" shows "replication_numbers \<noteq> {}"
subdg_arrD: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
cr_finfun_def: shows "cr_finfun \<equiv> \<lambda>x y. x = finfun_apply y"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "ide u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> s \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((s \<star> \<theta>) \<cdot> \<a>[s, s\<^sub>0, w] \<cdot> (\<sigma> \<star> w)) \<cdot> \<nu> = \<omega>"
is_real_interval: fixes S :: "real set" assumes "is_interval S" shows "\<exists>a b. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
SMP_SMP_subset: fixes N :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "N \<subseteq> SMP M" shows "SMP N \<subseteq> SMP M"
h4: shows "satisfy_equation p qs (get_subsets (reduce_system p (qs, matrix, subsets, signs))) (get_signs (reduce_system p (qs, matrix, subsets, signs)))"
cases: fixes r :: "BlockProc" obtains block :: "DiskBlock" and proc :: "Proc" where "r = \<lparr>block = block, proc = proc\<rparr>"
INum_def: shows "INum = (\<lambda>(a, b). of_int a / of_int b)"
NextSendUpd'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and tt :: "('loc \<times> 't) set" shows "NextSendUpd' c0 c1 p tt = (cri.next_sendupd' (exchange_config c0) (exchange_config c1) p tt \<and> prop_config c1 = prop_config c0 \<and> init c1 = init c0)"
simps_2: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" shows "repeat_stabilize (Suc n) f v = (let v_new = f v in if v = v_new then v else repeat_stabilize n f v_new)"
Top_prop: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "CLF cl f"   and "x \<in> pset cl" shows "(x, Top cl) \<in> order cl"
oracle_decrypt0'_simps: fixes key :: "'prf_key \<times> 'upf_key"   and bad :: "bool"   and L :: "(bool list \<times> bool list \<times> 'hash) set"   and x' :: "bool list"   and c' :: "bool list"   and t' :: "'hash" shows "oracle_decrypt0' key (bad, L) (x', c', t') = return_spmf (if (x', c', t') \<in> L \<or> length x' \<noteq> prf_dlen then (None, bad, L) else (decrypt key (x', c', t'), bad \<or> upf_fun (snd key) (x' @ c') = t', L))"
holo2: shows "(deriv ^^ n) f holomorphic_on (*) u ` S"
mult_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "pl.eq_m (f * pl.Mp g) (f * g)"
28487: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - y \<sqinter> - x"
induced_automorphism_half_chamber_system_pgallery_map_g: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "order.greater_eq (g \<turnstile> ChamberComplex.chamber_system X) (set Cs)"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "SimplicialComplex.pmaxsimpchain X (OpposedThinChamberComplexFoldings.induced_automorphism X f g \<Turnstile> Cs)"
contract_vacuity: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "AGM_Contraction Cn contraction"   and "K = Cn A"   and "\<phi> \<notin> K" shows "contraction K \<phi> = K"
vars_of_cnfi_def: fixes cnfi :: "(lit \<times> lit) list" shows "vars_of_cnfi cnfi \<equiv> remdups (concat (map (\<lambda>(l1, l2). [var l1, var l2]) cnfi))"
vfinite_vsubset: fixes B :: "V"   and A :: "V" assumes "vfinite B"   and "A \<subseteq>\<^sub>\<circ> B" shows "vfinite A"
card_funcset: fixes A :: "'a set"   and B :: "'b set"   and d :: "'b" assumes "finite A"   and "finite B" shows "card (extensionalD d A \<inter> (A \<rightarrow> B)) = card B ^ card A"
inverse_arrows_can_2: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "B.inverse_arrows (B.can u t) (B.can t u)"
assms_1: fixes m :: "nat" assumes "0 < m"   and "m \<le> n" shows "f m = f' m"
inv_on_left_moving_nonempty: fixes as :: "nat"   and am :: "nat list"   and s :: "nat"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving (as, am) (s, [], r) ires = False"
chain_prefix_limit: fixes w :: "nat \<Rightarrow> 'a list"   and k :: "nat" assumes "chain w" shows "w k \<le>\<^sub>F\<^sub>I Word_Prefixes.limit w"
set_verticesFrom: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "set (verticesFrom f v) = \<V> f"
cos_eq_zero_imp_norm_ge: fixes z :: "complex" assumes "cos z = 0" shows "pi / 2 \<le> cmod z"
simps_93: fixes x41 :: "'b formula"   and x42 :: "'b formula" shows "atoms (x41 \<^bold>\<and> x42) = atoms x41 \<union> atoms x42"
CONS_CONS_eq: fixes K :: "('b, unit) Old_Datatype.node set"   and M :: "('b, unit) Old_Datatype.node set"   and L :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" shows "(CONS K M = CONS L N) = (K = L \<and> M = N)"
f_not_a: fixes a :: "'a"   and c :: "'a"   and b :: "'b" assumes "a \<in> basisA"   and "c \<in> basisA"   and "a \<noteq> c" shows "f a b c = (0::'b)"
is_none_bind: fixes f :: "'c option"   and g :: "'c \<Rightarrow> 'b option" shows "Option.is_none (Option.bind f g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
simps_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list"   and y21 :: "'a"   and y22 :: "('a, 'b) Term.term list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
all: shows "\<forall>us. us \<in> lists G\<^sub>+ \<and> concat us = b \<longrightarrow> \<^bold>|us\<^bold>| = 1"
equality: fixes r :: "'b transition_scheme"   and r' :: "'b transition_scheme" assumes "Label r = Label r'"   and "Arity r = Arity r'"   and "Guards r = Guards r'"   and "Outputs r = Outputs r'"   and "Updates r = Updates r'"   and "more r = more r'" shows "r = r'"
sum: fixes X :: "'a set"   and a :: "real \<times> real"   and f :: "'a \<Rightarrow> real \<times> real" assumes "finite X"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> ccw' 0 a (f x)" shows "ccw' 0 a (sum f X)"
S2: fixes y :: "i" assumes "AxiomsSet6" shows "E y \<^bold>\<leftarrow> E (cod y)"
empty_state_def: fixes e :: "'l" shows "empty_state e \<equiv> \<lparr>counter = 0, discovered = Map.empty, finished = Map.empty, pending = {}, stack = [], tree_edges = {}, back_edges = {}, cross_edges = {}, \<dots> = e\<rparr>"
comp_null_1: fixes f :: "'a" shows "Right.comp Right.null f = Right.null"
cases: fixes x :: "('a \<times> 'b) list" obtains "xa__ = []"   | uu_ :: "'a" and x :: "'b" and xs :: "('a \<times> 'b) list" where "xa__ = (uu_, x) # xs"
has_derivative_locally_injective_blinfun: fixes a :: "'b"   and s :: "'b set"   and g' :: "'c \<Rightarrow>\<^sub>L 'b"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c"   and f :: "'b \<Rightarrow> 'c" assumes "a \<in> s"   and "open s"   and "g' o\<^sub>L f' a = 1\<^sub>L"   and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"   and "isCont f' a" obtains r :: "real" where "0 < r" and "ball a r \<subseteq> s" and "inj_on f (ball a r)"
NE_I_intT: fixes \<sigma> :: "'tp" shows "\<exists>a. I_intT \<sigma> a"
R_def: shows "R = (\<lambda>x. {i. term_pp_rel (=) (lt (seq i), punit.lt (rep_list (seq i))) x})"
parallelI: fixes xs :: "'b list"   and ys :: "'b list" assumes "\<not> xs \<le>p ys"   and "\<not> ys \<le>p xs" shows "xs \<parallel> ys"
negligible_countable_Union: fixes \<F> :: "'c set set" assumes "countable \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> negligible S" shows "negligible (\<Union> \<F>)"
mult_minus1: fixes z :: "'a" shows "- (1::'a) * z = - z"
preserves_arr: fixes f :: "'j" assumes "J.arr f" shows "C.arr (A'.F.map f)"
Ioi_le_Ico: fixes a :: "'a" shows "{a<..} \<subseteq> {a..}"
monom_sum: fixes a :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and n :: "nat" shows "monom (sum a A) n = (\<Sum>x\<in>A. monom (a x) n)"
that: fixes s :: "nat \<Rightarrow> real" assumes "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"   and "(flow0 x \<circ> s) \<longlonglongrightarrow> p"   and "\<forall>n. b \<le> s n" shows "thesis"
twice_lemma: fixes rs :: "nat" shows "rec_exec rec_twice [rs] = 2 * rs"
stuttering_invariant_def: fixes A :: "(nat \<Rightarrow> 'b) set" shows "stuttering_invariant A \<equiv> \<forall>u v. u \<approx> v \<longrightarrow> (u \<in> A) = (v \<in> A)"
elims_3: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v" assumes "\<not> is_trail x xa xb" obtains v :: "'v" and v' :: "'v" where "x = va__" and "xa = []" and "xb = v'a__" and "\<not> (va__ = v'a__ \<and> v'a__ \<in> V)"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and ps :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "\<not> (v = v1 \<and> (v1, w, v2) \<in> E \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> is_trail v2 ps v')"
nsgTr2: fixes a :: "'a"   and b :: "'a"   and a1 :: "'a"   and b1 :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "a1 \<in> carrier G"   and "b1 \<in> carrier G" shows "a \<cdot> b \<cdot> \<rho> (a1 \<cdot> b1) = a \<cdot> (b \<cdot> \<rho> b1 \<cdot> (\<rho> a1 \<cdot> a) \<cdot> \<rho> a)"
with_proj_simps_3: fixes G :: "'b pair_pre_digraph" shows "arcs_ends (with_proj G) = parcs G"
wf_list_graph_iff_wf_graph: fixes G :: "'b list_graph" shows "wf_graph (list_graph_to_graph G) = wf_list_graph_axioms G"
assms_1: shows "A O1 P LtA A O1 B"
nbrhd: fixes U :: "'a set" assumes "U \<in> \<C>" shows "open U \<and> (\<exists>T. T \<in> \<B> \<and> U \<subseteq> T)"
is_closed_subst_bv: fixes u :: "Core.term"   and t :: "Core.term" assumes "is_closed (subst_bv u t)" shows "\<not> loose_bvar t 1"
ChurchRosser_unique_final: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b1 :: "'b"   and b2 :: "'b" assumes "relation.is_ChurchRosser r"   and "r\<^sup>*\<^sup>* a b1"   and "r\<^sup>*\<^sup>* a b2"   and "relation.is_final r b1"   and "relation.is_final r b2" shows "b1 = b2"
Sup_in_diff: shows "Red_F (Sup_llist Ns) \<subseteq> Red_F (Sup_llist Ns - (Sup_llist Ns - Liminf_llist Ns))"
x1_mem: shows "flow0 x t1 \<in> {flow0 x t2<--<b}"
usk: fixes k :: "nat" assumes "k < length us" shows "us ! k \<in> carrier_vec n"
mem_diff: fixes c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<forall>x\<in>mds' AsmNoWrite. mem' x = mem'' x" shows "\<langle>c', mds', mem''\<rangle> \<in> loc_reach lc"
simps_1: fixes t :: "'a tree23" shows "inorder2 (T t) = inorder t"
gen_cong_rho: fixes \<rho> :: "'d \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'c"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "\<rho> = eval' \<circ> f"   and "rel' (step.gen_cong R) (f x) (f y)" shows "step.gen_cong R (\<rho> x) (\<rho> y)"
mapping_eqI': fixes m :: "('c, 'd) mapping"   and d :: "'d"   and m' :: "('c, 'd) mapping" assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default d m x = Mapping.lookup_default d m' x"   and "Mapping.keys m = Mapping.keys m'" shows "m = m'"
T_TApp: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmark.type"   and T\<^sub>1\<^sub>2 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall><:T\<^sub>1\<^sub>1. T\<^sub>1\<^sub>2)"   and "\<Gamma> \<turnstile> T\<^sub>2 <: T\<^sub>1\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<bullet>\<^sub>\<tau> T\<^sub>2 : T\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]\<^sub>\<tau>"
induct: fixes x :: "'d"   and \<delta> :: "('d, 'e) ta_rule set"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> b_accessible \<delta>"   and "\<And>q l qs. \<lbrakk>q \<rightarrow> l qs \<in> \<delta>; \<And>x. x \<in> set qs \<Longrightarrow> x \<in> b_accessible \<delta>; \<And>x. x \<in> set qs \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P q" shows "P x"
xor_cancel_right: fixes x :: "'a" shows "x \<oplus> \<sim> x = \<one>"
cBexCI: fixes A :: "'d cset"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "cBall A (\<lambda>x. \<not> P x) \<Longrightarrow> P a"   and "cin a A" shows "cBex A P"
simps_4: fixes b :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "Formula.fvi b (formula.Less t1 t2) = Formula.fvi_trm b t1 \<union> Formula.fvi_trm b t2"
finer_def: fixes P :: "'a set set"   and Q :: "'a set set" shows "finer P Q \<equiv> (\<forall>J\<in>Q. J = \<Union> {I \<in> P. I \<subseteq> J}) \<and> (P \<noteq> {} \<longrightarrow> Q \<noteq> {})"
simps_5: fixes vc :: "'b match_expr"   and va :: "action"   and rs :: "'b rule list" shows "rmMatchFalse (Rule (MatchNot (MatchNot vc)) va # rs) = Rule (MatchNot (MatchNot vc)) va # rmMatchFalse rs"