hcju/isabelleps huggingface dataset:
are_identities_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "partial_magma.ide (span_vertical_category.vcomp C) \<nu>"
countable_INT: fixes A :: "'b \<Rightarrow> 'a set"   and X :: "'b set" assumes "A ` X \<subseteq> M"   and "X \<noteq> {}" shows "\<Inter> (A ` X) \<in> M"
sublist_append_both_I: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list"   and d :: "'b list" assumes "subseq a b"   and "subseq c d" shows "subseq (a @ c) (b @ d)"
sem_5e: shows "\<forall>X Y Z. Y \<sqsubseteq> X \<and> ob X Z \<and> \<I>Y \<sqinter> Z \<longrightarrow> ob Y Z"
remdups_wrt_base: fixes uu :: "'b \<Rightarrow> 'c" shows "remdups_wrt uu [] = []"
lift_mult: fixes lift :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "pfd_homomorphism lift" shows "lift (a * b) = lift a * lift b"
2: fixes t'' :: "('fun, 'var) Term.term" assumes "t'' \<in> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>') \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>'"   and "a = assign" shows "t'' \<in> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>) \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>"
bind_foldl_spmf_return: fixes p :: "'r spmf"   and f :: "'r \<Rightarrow> 's \<Rightarrow> 'r spmf"   and xs :: "'s list" shows "p \<bind> (\<lambda>x. foldl_spmf f (return_spmf x) xs) = foldl_spmf f p xs"
map_eq_Ok_conv: fixes f :: "'n \<Rightarrow> 'l"   and g :: "'o \<Rightarrow> 'm"   and x :: "('n, 'o) result"   and y :: "'m" shows "(map_result f g x = Ok y) = (\<exists>x'. x = Ok x' \<and> y = g x')"
inj_on_fst_map_to_set: fixes m :: "'g \<Rightarrow> 'h option" shows "inj_on fst (map_to_set m)"
msg_id_unique: fixes m1 :: "'a"   and i :: "nat"   and m2 :: "'a"   and j :: "nat" assumes "Broadcast m1 \<in> set (history i)"   and "Broadcast m2 \<in> set (history j)"   and "msg_id m1 = msg_id m2" shows "i = j \<and> m1 = m2"
convex_closed_segment: fixes a :: "'b"   and b :: "'b" shows "convex (closed_segment a b)"
IH: fixes xs :: "'a list" assumes "hb_consistent (xs @ [x])"   and "{x} \<union> set xs = set ys"   and "distinct (x # xs)"   and "distinct ys" shows "\<exists>prefix suffix. ys = prefix @ x # suffix \<and> concurrent_set x suffix"
that: assumes "\<not> is_exn \<sigma>"   and "\<sigma>' = \<sigma>"   and "r = Ref.get (the_state \<sigma>) p" shows "thesis"
repeat_subst_fmla: fixes A :: "fmla"   and i :: "name"   and t :: "trm"   and u :: "trm" shows "(A(i::=t))(i::=u) = A(i::=subst i u t)"
conf_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "P \<turnstile> T \<le> T'" shows "P,h \<turnstile> v :\<le> T'"
A2: shows "\<Sum> {1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095, 12285} = 26880"
simps_4: fixes n :: "nat"   and b :: "'a atom" shows "lang n (Atom b) = {[x] |x. lookup b x \<and> x \<in> (set \<circ> \<sigma> \<Sigma>) n}"
permutes_surj: fixes p :: "'c \<Rightarrow> 'c"   and s :: "'c set" assumes "p permutes s" shows "surj p"
intros_3: fixes s :: "s"   and i :: "nat"   and c :: "i32"   and cl :: "cl"   and j :: "nat"   and tf :: "tf"   and vs :: "v list" assumes "stab s i (nat_of_int c) = Some cl"   and "stypes s i j = tf"   and "cl_type cl = tf" shows "\<lparr>s;vs;[$C ConstInt32 c, $Call_indirect j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Callcl cl]\<rparr>"
cod_char: fixes f :: "'a \<times> 'a" shows "C.VV.cod f = (if C.VV.arr f then C.VxV.cod f else C.VxV.null)"
gallery_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "domain.gallery (x # xs)" shows "domain.gallery xs"
BRANCH_def: fixes P :: "'c" shows "BRANCH P \<equiv> P"
span_union_is_sum: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> carrier V"   and "B \<subseteq> carrier V" shows "local.module.span (A \<union> B) = subspace_sum (local.module.span A) (local.module.span B)"
primeI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero x"   and "normalize x = x" shows "normalization_semidom.prime zero times one normalize x"
pdivides_zero: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p pdivides []"
face_cycle_set_self: fixes a :: "'b" shows "a \<in> rev.H.face_cycle_set a"
axioms_1: fixes A :: "'b llist set"   and N :: "enat" assumes "GaleStewartDeterminedGames.finite_GSgame A N" shows "GSgame A N"
lm156: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "Graph f `` X = f ` X"
surjective: fixes r :: "('c, 'd) eq_object_scheme" shows "r = \<lparr>carrier = carrier r, eq = (.=\<^bsub>r\<^esub>), \<dots> = eq_object.more r\<rparr>"
mk_trace_def: fixes ioa :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set" shows "mk_trace ioa \<equiv> filter_oseq (\<lambda>a. a \<in> externals (asig_of ioa))"
servTicket_authentic_Tgs: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>authK. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"
assert_index_valid: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "index_valid ats s" shows "index_valid (insert a ats) (assert a s)"
linears_singleI: fixes t :: "term" assumes "linear t" shows "linears [t]"
NSLIM_const: fixes k :: "'b"   and x :: "'a" shows "(\<lambda>x. k) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S k"
omega_comp_fix: fixes x :: "'a"   and y :: "'a" shows "x ^ \<omega> * y = x * x ^ \<omega> * y \<sqinter> y"
encode_problem_serializable_sound_2: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<forall>k<length (\<Phi>\<inverse> \<Pi> \<A> t). are_all_operators_non_interfering ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
add_mono_thms_linordered_semiring_3: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i \<le> j \<and> k = l" shows "i + k \<le> j + l"
older_seniors_card_mono: fixes A :: "nat set"   and x :: "nat"   and n :: "nat" assumes "A \<subseteq> older_seniors x n" shows "card A \<le> card (older_seniors x n)"
emeasure_real: fixes M :: "'c measure"   and A :: "'c set" assumes "finite_measure M" shows "\<exists>r\<ge>0. emeasure M A = ennreal r"
assms_3: shows "q (deg R q) \<otimes> p (deg R p) \<noteq> \<zero>"
distinct_card: fixes xs :: "'k list" assumes "distinct xs" shows "card (set xs) = length xs"
dual_eq_iff: fixes x :: "'d dual"   and y :: "'d dual" shows "(x = y) = (undual x = undual y)"
simps_16: fixes v :: "\<I>"   and va :: "Formula.formula Regex.regex" shows "remove_neg (formula.MatchF v va) = formula.MatchF v va"
in_Solutions_iff: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" shows "((x, y) \<in> Linear_Diophantine_Equations.Solutions a b) = (length x = length a \<and> length y = length b \<and> a \<bullet> x = b \<bullet> y)"
fsubset_fsingletonD: fixes A :: "'b fset"   and x :: "'b" assumes "A |\<subseteq>| {|x|}" shows "A = {||} \<or> A = {|x|}"
right_uvar_def: fixes x :: "'d \<Longrightarrow> 'g" shows "1-x = x ;\<^sub>L mrg_right"
ACI_sym: fixes r :: "'a rexp"   and s :: "'a rexp" assumes "ACI r s" shows "ACI s r"
smult_right_add: fixes a :: "'c"   and x :: "('c, 'd) sq_matrix"   and y :: "('c, 'd) sq_matrix" shows "a *\<^sub>S (x + y) = a *\<^sub>S x + a *\<^sub>S y"
cmp_props_5: shows "tab\<^sub>1 (r \<star> s) \<star> rs.cmp \<cong> tab\<^sub>1 r \<star> rs.p\<^sub>1"
FactRing_iso_set_aux: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h" shows "(\<lambda>X. the_elem (h ` X)) \<in> ring_iso (R Quot a_kernel R S h) (S\<lparr>carrier := h ` carrier R\<rparr>)"
all: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "all x \<phi> \<in> fmla"
B_2: fixes x :: "'a" shows "norm (f x) \<le> B * norm x"
derivative_is_smooth': fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "(k + 1)-smooth_on S f" shows "k-smooth_on S (\<lambda>x. frechet_derivative f (at x) v)"
assms_2: fixes vs :: "nat \<Rightarrow> nat"   and vs' :: "nat \<Rightarrow> nat" assumes "P vs vs'" shows "vs' v = vs v"
seq_inf_par: fixes a :: "'a"   and b :: "'a" shows "a ; b \<le> a * b"
Wretr_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "Wretr theta \<equiv> {(c, d). matchC_M theta c d \<and> matchT_M c d}"
le_err_antisym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e1 :: "'b err"   and e2 :: "'b err" assumes "order r"   and "e1 \<sqsubseteq>\<^bsub>le r\<^esub> e2"   and "e2 \<sqsubseteq>\<^bsub>le r\<^esub> e1" shows "e1 = e2"
Nml_HcompNml_1: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Nml (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
sees_field_decl_above: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F:T in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space local.scale Basis_pair"
assms: fixes i :: "nat"   and j :: "nat" shows "i < length xs \<and> j < length xs \<and> i \<noteq> j \<and> xs ! i = xs ! j \<longrightarrow> f (xs ! i) = (0::'a) \<and> f (xs ! j) = (0::'a)"
s_chop: fixes i :: "nat"   and t :: "('p, 's, 'm) event list"   and c :: "('p, 's, 'm) configuration" assumes "i \<le> length t" shows "s c t i = s c (take i t) i"
smult_scalar_prod_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and a :: "'b" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "(a \<cdot>\<^sub>v v\<^sub>1) \<bullet> v\<^sub>2 = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
absolutely_integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" shows "f absolutely_integrable_on {a..b}"
\<Delta>_def: shows "\<Delta> \<equiv> discriminant_cubic_depressed e f"
is_L_bisimulation_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and R :: "'d fs_set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.is_L_bisimulation satisfies transition effect_apply L R \<equiv> \<forall>F. symp (R F) \<and> (\<forall>P Q. R F P Q \<longrightarrow> (\<forall>f. f \<in>\<^sub>f\<^sub>s F \<longrightarrow> (\<forall>\<phi>. satisfies (effect_apply f P) \<phi> \<longrightarrow> satisfies (effect_apply f Q) \<phi>))) \<and> (\<forall>P Q. R F P Q \<longrightarrow> (\<forall>f. f \<in>\<^sub>f\<^sub>s F \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* (effect_apply f Q, F, f) \<longrightarrow> transition (effect_apply f P) \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. transition (effect_apply f Q) \<langle>\<alpha>,Q'\<rangle> \<and> R (L (\<alpha>, F, f)) P' Q'))))"
isomorphic_implies_same_tab_2: fixes r' :: "'a" assumes "r\<^sub>0\<^sup>* \<star> s\<^sub>1 \<cong> r'" shows "r\<^sub>0s\<^sub>1.p\<^sub>1 = tab\<^sub>1 r'"
times_mod_expand: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly" shows "a \<otimes>\<^bsub>R\<^esub> b = a mod f \<otimes>\<^bsub>R\<^esub> (b mod f)"
proj_arcs_G_in_arcs_H: fixes a :: "'b" assumes "a \<in> arcs G" shows "proj_arcs_G a \<in> arcs H"
succ_rel_insert_later: fixes b :: "'b"   and list :: "'b list"   and oid :: "'b"   and ref :: "'b"   and a :: "'b" assumes "succ_rel (insert_spec (b # list) (oid, Some ref)) = insert_alt (succ_rel (b # list)) (oid, ref)"   and "a \<noteq> ref"   and "distinct (a # b # list)" shows "succ_rel (insert_spec (a # b # list) (oid, Some ref)) = insert_alt (succ_rel (a # b # list)) (oid, ref)"
R_subst: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> R" shows "(P x, P y) \<in> R"
R1_of_list: fixes x :: "'c list" shows "list_R1_to_list (list_to_list_R1 x) = x"
isotone_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c" shows "ord.isotone less_eq f \<equiv> \<forall>x y. less_eq x y \<longrightarrow> less_eq (f x) (f y)"
outer_transitivity_between2: 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 "Bet A B C"   and "Bet B C D"   and "B \<noteq> C" shows "Bet A C D"
last_cons_def: fixes x :: "'a option"   and s :: "'s" shows "local.last_cons x s = (if unstream g s = [] then the x else last (unstream g s))"
prems: shows "coincide_on \<sigma> \<eta> (vars_of (t1 \<cdot> t2))"
invertible_matrix_change_of_basis: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "invertible (matrix_change_of_basis X Y)"
identity_rules: fixes G :: "('b Standard_Constant, 'c) labeled_graph"   and L :: "'b Standard_Constant set" assumes "graph G"   and "maintainedA (identity_rules L) G"   and "fst ` edges G \<subseteq> L" shows "\<exists>f. f \<circ> f = f \<and> ident_rel S_Idt (map_graph_fn G f) \<and> subgraph (map_graph_fn G f) G \<and> (\<forall>l x y. ((l, x, y) \<in> edges G) = ((l, f x, f y) \<in> edges G))"
vertex_cover_def: fixes E :: "'b set set"   and C :: "'b set" shows "vertex_cover E C = (\<forall>e\<in>E. e \<inter> C \<noteq> {})"
f\<^sub>2uv: shows "between (vertices f\<^sub>2) u v = vs"
Id_in_AllowedActs: fixes F :: "'c program" shows "Id \<in> AllowedActs F"
topo_open: fixes B :: "'b set set"   and t :: "'b set" shows "(t open\<^bsub>topo B\<^esub>) = (t \<in> topo B)"
enabled_def: shows "enabled \<equiv> gfp (\<lambda>p x1 x2. \<exists>\<omega> s. x1 = s \<and> x2 = \<omega> \<and> p (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
oreturn_bind_cong_apply: fixes x :: "'d"   and x' :: "'d"   and m :: "'d \<Rightarrow> 'f \<Rightarrow> 'e option"   and s :: "'f"   and m' :: "'d \<Rightarrow> 'f \<Rightarrow> 'e option" assumes "x = x'"   and "m x' s = m' x' s" shows "(oreturn x |>> m) s = (oreturn x' |>> m') s"
constVal_Some_induct: fixes e :: "expr"   and v :: "val"   and P :: "expr \<Rightarrow> bool" assumes "constVal e = Some v"   and "\<And>v. P (Lit v)"   and "\<And>unop e'. P e' \<Longrightarrow> P (UnOp unop e')"   and "\<And>binop e1 e2. \<lbrakk>P e1; P e2\<rbrakk> \<Longrightarrow> P (BinOp binop e1 e2)"   and "\<And>b bv e1 e2. \<lbrakk>constVal b = Some bv; the_Bool bv; P b; P e1\<rbrakk> \<Longrightarrow> P (b ? e1 : e2)"   and "\<And>b bv e1 e2. \<lbrakk>constVal b = Some bv; \<not> the_Bool bv; P b; P e2\<rbrakk> \<Longrightarrow> P (b ? e1 : e2)" shows "P e"
stellar_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set" shows "stellar slices W \<equiv> \<forall>p. p \<in> W \<longrightarrow> slices p \<noteq> {}"
rw_tail: fixes w :: "'b list" assumes "set w \<subseteq> L" shows "w = [] \<or> set (tl w) \<subseteq> L"
sup_absorb1: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "x \<sqinter> y = x"
comp_orD_def: fixes p2 :: "'f \<Rightarrow> 'e decision option"   and p1 :: "'d \<Rightarrow> 'f decision option" shows "p2 \<circ>\<^sub>orD p1 \<equiv> map_option flat_orD \<circ> (lift p2 \<circ>\<^sub>m p1)"
elims: fixes x :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'c"   and y :: "'b" assumes "someFunc x xa = y" obtains P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'c" where "x = P" and "xa = xb" and "y = Eps (P xb)"
errMOD_igSwapAbsIPresIGWlsAbsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igSwapAbsIPresIGWlsAbs MOD"   and "igWlsAbsDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igSwapAbsIPresIGWlsAbsSTR (errMOD MOD)"
simps_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
mod_star_conv: fixes h :: "heap"   and as :: "nat set"   and A :: "assn"   and B :: "assn" shows "(pHeap h as \<Turnstile> A * B) = (\<exists>as1 as2. as = as1 \<union> as2 \<and> as1 \<inter> as2 = {} \<and> pHeap h as1 \<Turnstile> A \<and> pHeap h as2 \<Turnstile> B)"
map_fst_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
simps_5: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a"   and y51 :: "nat"   and y52 :: "'a"   and y53 :: "'a" shows "(RecvMarker x51 x52 x53 = RecvMarker y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
shortest_path_pos_cost_pred_axioms_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat" shows "shortest_path_pos_cost_pred_axioms G dist c s num \<equiv> (s \<in> verts G \<and> dist s = 0) \<and> (\<forall>v. v \<in> verts G \<longrightarrow> (dist v = \<infinity>) = (num v = \<infinity>)) \<and> (\<forall>e. e \<in> arcs G \<longrightarrow> 0 \<le> c e)"
type_definition_dlist: shows "type_definition list_of_dlist Abs_dlist {xs. distinct xs}"
birkhoff_sum_small_asymp_pos_nonergodic: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "integrable M f"   and "0 < e" shows "AE x in M. infinite {n. birkhoff_sum f n x \<in> {real n * real_cond_exp M Invariants f x..real n * real_cond_exp M Invariants f x + e}}"
ALPHA: shows "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"
PO_m1_inv4_rfresh_trans: shows "{m1_inv4_rfresh \<inter> m1_inv3r_init \<inter> m1_inv2r_serv \<inter> m1_inv3_ifresh \<inter> m1_secrecy} TS.trans m1 {> m1_inv4_rfresh}"
resr_distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<rightarrow> z = (x \<rightarrow> z) \<sqinter> (y \<rightarrow> z)"
lossless_spmf_bind_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "lossless_spmf (p \<bind> f) = (\<forall>x\<in>set_pmf p. lossless_spmf (f x))"
n_de_morgan_var2: fixes x :: "'a"   and y :: "'a" shows "!(!x + !y) = \<tt>\<tt> x \<cdot> \<tt>\<tt> y"
mojmir_accept_iff_token_set_accept2: shows "accept\<^sub>M = (\<exists>i<max_rank. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i) \<and> (\<forall>j<i. finite (local.merge j) \<and> finite (succeed j)))"
compute_\<ff>s_correct: fixes s :: "'b list" shows "compute_\<ff>s s \<le> compute_\<ff>s_SPEC s"
rd_ref: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and P' :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q' :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "R (P \<turnstile> Q) \<in> {p. is_CSP_process p}"   and "R (P' \<turnstile> Q') \<in> {p. is_CSP_process p}"   and "\<And>a b. P (a, b) \<Longrightarrow> P' (a, b)"   and "\<And>a b. Q' (a, b) \<Longrightarrow> Q (a, b)" shows "action_of (R (P \<turnstile> Q)) \<sqsubseteq> action_of (R (P' \<turnstile> Q'))"
assms_2: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<in> P[X]"   and "p \<notin> ideal F" shows "1 \<in> ideal (insert p F)"
assms_8: shows "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"
IH_2: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "(F2, F2, \<Gamma> \<Rightarrow> \<Delta> \<longrightarrow> F2, \<Gamma> \<Rightarrow> \<Delta>) \<and> (\<Gamma> \<Rightarrow> F2, F2, \<Delta> \<longrightarrow> \<Gamma> \<Rightarrow> F2, \<Delta>)"
prv_eqv_eqv_neg_prv_fls: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (eqv \<phi> \<chi>)"   and "prv (eqv \<phi> (neg \<chi>))" shows "prv fls"
bits_ident: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "plus (push_bit n (drop_bit n a)) (take_bit n a) = a"
star_simulation_right_equal: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z * x = y * z" shows "z * x\<^sup>\<star> = y\<^sup>\<star> * z"
simps_1: fixes bs :: "'a list"   and c :: "'a" shows "Ipoly bs (c)\<^sub>p = c"
subset_open_segment: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(open_segment a b \<subseteq> open_segment c d) = (a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
ik_cons: fixes x :: "('c, 'd) strand_step"   and S :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t (x # S) = ik\<^sub>s\<^sub>t [x] \<union> ik\<^sub>s\<^sub>t S"
G_def: shows "G = restrict_indets ` extended_ord.punit.reduced_GB (homogenize None ` extend_indets ` F)"
continuous_imp_closed_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "compact_space X"   and "Hausdorff_space Y" shows "closed_map X Y f"
other: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "U \<sigma> \<sigma>'" shows "P (\<sigma>', l)"
inv_solve_left': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(inv b \<otimes> c = a) = (c = b \<otimes> a)"
max_rank_lowerbound: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_FG_to_rabin \<Sigma> \<G> w" shows "0 < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>)"
Inter_anti_mono: fixes B :: "'a set set"   and A :: "'a set set" assumes "B \<subseteq> A" shows "\<Inter> A \<subseteq> \<Inter> B"