hcju/isabelleps huggingface dataset:
measurable_orthogonal_image: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set" assumes "orthogonal_transformation f"   and "S \<in> lmeasurable" shows "f ` S \<in> lmeasurable"
is_red_subset: fixes A :: "('t \<Rightarrow>\<^sub>0 'e) set"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and B :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "is_red A p"   and "A \<subseteq> B" shows "is_red B p"
execute_heap: fixes f :: "heap \<Rightarrow> 'a \<times> heap" shows "execute (Heap_Monad.heap f) = Some \<circ> f"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>B g" shows "g \<cong>\<^sub>B f"
extend_transient: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and A :: "'c set" assumes "Extend h" shows "(extend h F \<in> transient (extend_set h A)) = (F \<in> transient A)"
independent_extend_basis: fixes B :: "'b set" assumes "local.independent B" shows "local.independent (local.extend_basis B)"
Q_wrt_elem: shows "partition_without elem P = P - {Y} \<union> {Y - {elem}} - {{}}"
dom_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "local.dom f = (if arr f then MkIde (Dom f) else null)"
outs'_gpv_cases: fixes x :: "'d"   and gpv :: "('e, 'd, 'f) gpv" assumes "x \<in> outs'_gpv gpv" obtains generat :: "('e, 'd, ('e, 'd, 'f) rpv) generat" where "generat__ \<in> set_spmf (the_gpv gpv)" and "x \<in> generat_outs generat__"   | generat :: "('e, 'd, ('e, 'd, 'f) rpv) generat" and c :: "('e, 'd, 'f) rpv" and input :: "'f" where "generat \<in> set_spmf (the_gpv gpv)" and "c \<in> generat_conts generat" and "x \<in> outs'_gpv (c input)"
take_update_last: fixes n :: "nat"   and list :: "'a list"   and x :: "'a" assumes "n < length list" shows "(take (Suc n) list)[n := x] = take n list @ [x]"
sel_1: fixes x1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x2 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x3 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x4 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x5 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x6 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x7 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x8 :: "((nat, 'b \<times> 'a) mapping \<Rightarrow> (nat, 'b \<times> 'a) mapping) \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and x9 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x10 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.lt (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x1"
cases: fixes x :: "'e \<times> ('f, 'g, 'h, 'i) SLin_state_scheme" obtains s1 :: "'e" and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" where "x = (s1, s2)"
vote_set_def: fixes vs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and Q :: "'c set" shows "vote_set vs Q \<equiv> {uu_. \<exists>a r v. uu_ = (r, v) \<and> ((r, a), v) \<in> map_graph (\<lambda>(x, y). vs x y) \<and> a \<in> Q}"
rbt_insert_def: shows "rbt_insert = rbt_insert_with_key (\<lambda>_ _ nv. nv)"
prems_1: fixes w :: "'a" shows "(\<Sum>i\<le>Suc n. c i * w ^ i) = (0::'a)"
suffixes_append: fixes xs :: "'c list"   and ys :: "'c list" shows "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))"
source_stability: fixes \<sigma> :: "'b" shows "\<exists>v. put \<sigma> v = \<sigma>"
assms_2: shows "\<forall>x\<in>{a..b} - s. f x = g x"
iMODb_iprev_nth: fixes a :: "nat"   and c :: "nat"   and r :: "nat"   and m :: "nat" assumes "a \<le> c" shows "[ r, mod m, c ] \<leftarrow> a = r + m * (c - a)"
l_comb_jointfun_jj: fixes H :: "'a set"   and A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'b"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "t \<in> {j. j \<le> m} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> H" shows "\<Sigma>\<^sub>e M (\<lambda>j. jointfun n s m t j \<cdot>\<^sub>s jointfun n f m g j) n = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n"
stake_smap: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and s :: "'c stream" shows "stake n (smap f s) = map f (stake n s)"
assms_2: shows "chop (subst \<rho> s) \<noteq> subst \<rho> (chop s)"
ab: fixes k' :: "nat"   and k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k' \<le> k" shows "a k' \<lless> b k (j, i)"
zcf_monom_sub: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "zcf (Cring_Poly.compose R (monom P \<one> (Suc n)) p) = zcf p [^] Suc n"
parallel_reduction_subst: fixes X :: "'a trm"   and X' :: "'a trm"   and M :: "'a trm"   and M' :: "'a trm"   and z :: "'a" assumes "X >> X'"   and "M >> M'" shows "X[z ::= M] >> X'[z ::= M']"
child_eq: fixes p :: "(nat \<times> int) list"   and d :: "nat"   and l :: "nat"   and i :: "int"   and dir :: "dir" assumes "p ! d = (l, i)" shows "\<exists>j. child p dir d = p[d := (l + 1, j)]"
functor_axioms: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) GF.map"
induced_function_integral_nonergodic_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (induced_function A f)"
LCons1: fixes xs :: "'a llist"   and x :: "'a" assumes "lfinite xs"   and "P xs LNil" shows "P (LCons x xs) LNil"
resp: fixes x :: "'a" shows "(\<lambda>u. R `` {u @ [x]}) respects R"
elims: fixes x :: "ty"   and y :: "ty" assumes "ground_type x = y" obtains T :: "ty" where "x = T\<lfloor>\<rceil>" and "y = ground_type T"   | "x = Void" and "y = Void"   | "x = Boolean" and "y = Boolean"   | "x = Integer" and "y = Integer"   | "x = NT" and "y = NT"   | v :: "String.literal" where "x = Class v" and "y = Class v"
infs_any: fixes P :: "'c \<Rightarrow> bool"   and w :: "'c stream" assumes "infs P w" shows "Bex (sset w) P"
mC_C_wt_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_C_wt aval tval cval c d s t P F \<equiv> \<forall>I\<in>P. sum (PL.wt aval tval cval c s) I = sum (PL.wt aval tval cval d t) (F I)"
weakenTransitiveCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<(X \<union> weakenBisim)> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>\<^sub>w<{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<union> weakenBisim \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<union> weakenBisim"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X \<union> weakenBisim" shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
graph_axioms: shows "graph {False, True} Arr (\<lambda>_. False) (\<lambda>_. True)"
continuous_on_ereal: fixes s :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. ereal (f x))"
tObsC_abs_def: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "tObsC_abs t \<equiv> {(tFirst t', tLast t') |t'. t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t}"
LIM_const_not_eq: fixes k :: "'b"   and L :: "'b"   and a :: "'c" assumes "k \<noteq> L" shows "\<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
partition_main_vars_def: shows "partition_main_vars = partition_main (\<le>) id"
length_concat_runs: fixes xs :: "'b list" shows "length (concat (runs xs)) = length xs"
mismatchCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<noteq>b]P \<longmapsto> Rs" obtains P :: "pi" and aa :: "name" and ba :: "name" where "a = aa \<and> b = ba \<and> Pa__ = P" and "P \<longmapsto> Rs" and "aa \<noteq> ba"
sl_cons_right_invert: fixes ba' :: "'e list"   and t :: "'f"   and bc :: "'f list"   and R :: "('e \<times> 'f) set" assumes "(ba', t # bc) \<in> seq_lift R" obtains xs :: "'e list" and x :: "'e" where "ba' = x # xs" and "(xs, bc) \<in> seq_lift R" and "(x, t) \<in> R"
keys_minus: fixes a :: "'c \<Rightarrow>\<^sub>0 'd" shows "Poly_Mapping.keys (- a) = Poly_Mapping.keys a"
list_all3_map_3: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and h :: "'d \<Rightarrow> 'c"   and zs :: "'d list" shows "list_all3 P xs ys (map h zs) = list_all3 (\<lambda>x y z. P x y (h z)) xs ys zs"
r_eq_prim: shows "prim_recfn 2 r_eq"
partitioningCom: fixes a :: "'b set"   and ss :: "'b set list"   and ts :: "'b set set" shows "addSubsetSet a (partitioning ss ts) = partitioning ss (addSubsetSet a ts)"
Call_\<tau>red1r_param: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and es' :: "'addr expr1 list"   and xs' :: "'addr val list"   and v :: "'addr val"   and M :: "String.literal" assumes "\<tau>reds1gr uf P t h (es, xs) (es', xs')" shows "\<tau>red1gr uf P t h (Val v\<bullet>M(es), xs) (Val v\<bullet>M(es'), xs')"
relators_2: fixes rs :: "'g list" assumes "rs \<in> R" shows "sum_list rs = (0::'g)"
assms: shows "N1 \<subseteq>\<^sub>N N2"
EssencesCharacterizeCompletely: shows "\<lfloor>(\<lambda>w. \<forall>x xa. x xa w \<and> (\<forall>xb. (xb xa \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<forall>xa. (existsAt xa \<^bold>\<rightarrow> x xa \<^bold>\<rightarrow> xb xa) v)) w) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<forall>xb. existsAt xb v \<longrightarrow> x xb v \<longrightarrow> xb = xa)) w)\<rfloor>"
intros_3: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G"   and "v \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"   and "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)"   and "\<forall>w. v \<rightarrow>\<^bsub>G\<^esub> w \<longrightarrow> w \<in> ParityGame.attractor_inductive G p W" shows "v \<in> ParityGame.attractor_inductive G p W"
ideal_generated_dvd_subset: fixes S :: "'a set"   and d :: "'a" assumes "\<forall>x\<in>S. d dvd x"   and "finite S" shows "ideal_generated S \<subseteq> ideal_generated {d}"
Inf_bool_def: fixes A :: "bool set" shows "Inf A = (False \<notin> A)"
agree_on_subg_compose: fixes R :: "('d, 'e) labeled_graph"   and g :: "('e \<times> 'f) set"   and h :: "('e \<times> 'f) set"   and F :: "('d, 'e) labeled_graph"   and f :: "('e \<times> 'f) set" assumes "agree_on R g h"   and "agree_on F f g"   and "subgraph F R" shows "agree_on F f h"
normalize_quot_aux_1: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "a = fst (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
ParPres2: fixes S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "Eq S T" shows "Eq (S \<parallel> S) (T \<parallel> T)"
qFresh_preserves_alpha1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qFresh zs z X"   and "X #= X'" shows "qFresh zs z X'"
At_def: fixes x2 :: "nat"   and x1 :: "NominalSequents.var list" shows "At x2 x1 \<equiv> Abs_form (At_Rep x2 x1)"
cl_ord_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.cl_ord trm_ord = {(x, y). (basic_superposition.mset_cl x, basic_superposition.mset_cl y) \<in> mult (mult trm_ord)}"
add_increasing: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq b c" shows "less_eq b (plus a c)"
assms_1: shows "RSD_pareto_eqclass agents alts R A"
SourcesA6_L1: shows "Sources level1 sA6 = {}"
max_Var_floatariths_map_plus: fixes fa1 :: "'e \<Rightarrow> floatarith"   and fa2 :: "'e \<Rightarrow> floatarith"   and xs :: "'e list" shows "max_Var_floatariths (map (\<lambda>i. fa1 i + fa2 i) xs) = max (max_Var_floatariths (map fa1 xs)) (max_Var_floatariths (map fa2 xs))"
resize_vec_dim: fixes v :: "'c vec" shows "resize_vec (dim_vec v) v = v"
wf_induct: fixes r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
seq_comp_ref_union_closed: fixes P :: "'a option process"   and Q :: "'a option process" assumes "weakly_sequential P"   and "ref_union_closed P"   and "ref_union_closed Q" shows "ref_union_closed (P ; Q)"
lenext_mono_strong: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and gts' :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "gts ys xs \<Longrightarrow> gts' ys xs"   and "lenext gts ys xs" shows "lenext gts' ys xs"
elims_1: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "('d, 'e, 'f) FSM"   and xb :: "('d \<times> 'e) list"   and y :: "bool" assumes "sequence_to_failure x xa xb = y" obtains M1 :: "('d, 'e, 'f) FSM" and M2 :: "('d, 'e, 'f) FSM" and xs :: "('d \<times> 'e) list" where "x = M1" and "xa = M2" and "xb = xs" and "y = (butlast xs \<in> L M2 \<inter> L M1 \<and> xs \<in> L M1 - L M2)"
is_nan_copysign: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" shows "IEEE.is_nan (copysign x y) = IEEE.is_nan x"
lladd: shows "matrix_seq.lowner_lub (\<lambda>n. matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>1) n + matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>2) n) = matrix_seq.lowner_lub (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>1)) + matrix_seq.lowner_lub (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>2))"
linorder_rank_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b" shows "linorder_rank R A x = card {y \<in> A - {x}. (y, x) \<in> R}"
l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_val create_character_data known_ptr \<equiv> known_ptr = DocumentClass.known_ptr \<and> create_character_data = l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data set_val get_disconnected_nodes set_disconnected_nodes"
reduced_GB_monomial_lt_reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.reduced_GB (monomial (1::'c) ` lpp ` punit.reduced_GB F) = monomial (1::'c) ` lpp ` punit.reduced_GB F"
obj_sumE: fixes s :: "'a + 'b"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
suc_nth_d_is: shows "lnth Ns (Suc n) = N \<union> {(C0, active)} \<union> M"
fact_plus1: shows "fact (n + 1) = (n + 1) * fact n"
rcoset_replist_Hrep: fixes H :: "'g set"   and gs :: "'g list" assumes "Subgroup H"   and "is_rcoset_replist H gs" shows "\<exists>g\<in>set gs. g \<in> H"
flip_commute: fixes a :: "'c"   and b :: "'c" shows "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
wf_update_until: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ne :: "nat"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set" assumes "wf_until_aux \<sigma> n R pos \<phi> I \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne + length aux) \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne + length aux) \<psi>) rel2"   and "fv \<phi> \<subseteq> fv \<psi>" shows "wf_until_aux \<sigma> n R pos \<phi> I \<psi> (update_until I pos rel1 rel2 (\<tau> \<sigma> (ne + length aux)) aux) ne"
c_pair_inj1: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "c_pair x1 y1 = c_pair x2 y2" shows "x1 = x2"
walk_in_V: fixes xs :: "nat list" assumes "walk xs" shows "set xs \<subseteq> V"
cases: fixes a1 :: "nat"   and a2 :: "hmultiset" assumes "local.aligned\<^sub>d a1 a2" obtains M :: "hmultiset" and e :: "nat" where "a1 = e" and "a2 = M" and "\<forall>m\<in>#hmsetmset M. e \<le> local.decode 0 m"
n_dist_var2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!(!x \<cdot> !y \<oplus> t x \<cdot> !z) = !x \<cdot> t y \<oplus> t x \<cdot> t z"
scalar_prod_append: fixes v1 :: "'b vec"   and n1 :: "nat"   and v2 :: "'b vec"   and n2 :: "nat"   and w1 :: "'b vec"   and w2 :: "'b vec" assumes "v1 \<in> carrier_vec n1"   and "v2 \<in> carrier_vec n2"   and "w1 \<in> carrier_vec n1"   and "w2 \<in> carrier_vec n2" shows "(v1 @\<^sub>v v2) \<bullet> (w1 @\<^sub>v w2) = v1 \<bullet> w1 + v2 \<bullet> w2"
intro: assumes "\<And>x. \<midarrow> x = (THE z. \<forall>w. P w z = (\<not> O w x))"   and "\<And>x. \<exists>w. \<not> O w x \<Longrightarrow> \<exists>z. \<forall>w. P w z = (\<not> O w x)"   and "\<And>x y. x \<ominus> y = (THE z. \<forall>w. P w z = (P w x \<and> \<not> O w y))" shows "CMC_axioms"
l2_defs_5: shows "l2_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signals = \<lambda>x. 0, chan = {}, bad = bad_init\<rparr>}"
mem_restr_upd_None: fixes R :: "'a list set"   and xs :: "'a option list"   and i :: "nat" assumes "mem_restr R xs" shows "mem_restr R (xs[i := None])"
sets_pair_measure_cong: fixes M1 :: "'c measure"   and M1' :: "'c measure"   and M2 :: "'d measure"   and M2' :: "'d measure" assumes "sets M1 = sets M1'"   and "sets M2 = sets M2'" shows "sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
set_cases: fixes e :: "'b"   and a :: "'b hp" assumes "e \<in> set_hp a" obtains z2 :: "'b hp list" where "a = Hp e z2__"   | z1 :: "'b" and z2 :: "'b hp list" and x :: "'b hp" where "a = Hp z1 z2" and "x \<in> set z2" and "e \<in> set_hp x"
those_eq_None_set_iff: fixes xs :: "'a option list" shows "(those xs = None) = (None \<in> set xs)"
RELATESI: fixes R :: "('c \<times> 'd) set" shows "RELATES R"
simps_1: fixes a :: "nat" shows "digit_encode [] a = []"
linear_compose_sum: fixes S :: "'c set"   and f :: "'c \<Rightarrow> ('d, 'e) vec \<Rightarrow> ('d, 'f) vec" assumes "\<forall>a\<in>S. Vector_Spaces.linear (*s) (*s) (f a)" shows "Vector_Spaces.linear (*s) (*s) (\<lambda>x. \<Sum>a\<in>S. f a x)"
CFG_axioms: shows "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"
nc_finished_eq_reachable: assumes "\<not> cond s"   and "\<not> is_break param s" shows "dom (finished s) = reachable"
gen_gens: fixes x :: "'e"   and gens :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "x \<in> gens" shows "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
red_subset: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.red G p q"   and "G \<subseteq> F" shows "punit.red F p q"
assms: shows "Fun f T \<sqsubseteq> t \<cdot> \<theta>"
reach_init_valid: fixes s :: "'state" assumes "reach s" shows "s = istate \<or> (\<exists>tr. valid tr \<and> srcOf (hd tr) = istate \<and> tgtOf (last tr) = s)"
simps_66: fixes x7 :: "CTypeId"   and x6 :: "Arraytype" shows "CClassT x7 \<noteq> ArrT x6"
seq_char: fixes g :: "J.Arr"   and f :: "J.Arr" shows "J.seq g f = (f = J.AA \<and> (g = J.AA \<or> g = J.AT) \<or> f = J.BB \<and> (g = J.BB \<or> g = J.BT) \<or> f = J.AT \<and> g = J.TT \<or> f = J.BT \<and> g = J.TT \<or> f = J.TT \<and> g = J.TT)"
set_bset_inject: fixes x :: "'d bset"   and y :: "'d bset" shows "(set_bset x = set_bset y) = (x = y)"
omax_ordLeq_elim: fixes R :: "('c \<times> 'c) set set"   and p :: "('d \<times> 'd) set"   and r :: "('c \<times> 'c) set" assumes "finite R"   and "\<forall>r\<in>R. Well_order r"   and "omax R \<le>o p"   and "r \<in> R" shows "r \<le>o p"
fut_invariant: fixes \<phi>\<Phi> :: "('a, 'k) aformula \<times> ('a, 'k) aformula list" assumes "fut_invariant \<phi>\<Phi>"   and "case \<phi>\<Phi> of (\<phi>, \<Phi>) \<Rightarrow> \<phi> \<notin> set \<Phi>" shows "fut_invariant (fut_step \<phi>\<Phi>)"
round_max_less: fixes t :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "fst (snd t) \<noteq> []"   and "Suc 0 < Max (set (fst (snd t)))" shows "Max (set (fst (snd (round index key p q r t)))) < Max (set (fst (snd t)))"
fpairs_vunion: fixes A :: "V"   and B :: "V" shows "fpairs (A \<union>\<^sub>\<circ> B) = fpairs A \<union>\<^sub>\<circ> fpairs B"
simps_1: fixes x2 :: "'b"   and y2 :: "'b" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
right_hom_axioms: shows "right_hom (\<cdot>) (\<star>) \<omega>"
emeasure_notin_sets: fixes A :: "'d set"   and M :: "'d measure" assumes "A \<notin> sets M" shows "emeasure M A = 0"
ltln_Until_to_StrongRelease: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> M\<^sub>n (\<phi> or\<^sub>n \<psi>)"
of_nat_of_int_mod_ring: shows "of_nat = of_int_mod_ring \<circ> int"
simps_10: fixes I :: "\<I>"   and r :: "formula Regex.regex" shows "convert_multiway (MatchP I r) = MatchP I (convert_multiway_regex r)"
hypnat_of_nat_Suc: fixes n :: "nat" shows "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1"
continuous_inner: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x \<bullet> g x)"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c. P (c)\<^sub>p"   and "\<And>c n p. \<lbrakk>\<And>x. P c; \<And>x. P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"