hcju/isabelleps huggingface dataset:
indRelTEQ_impl_TRel_weakly_reflects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelTEQ TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
ring_simprules_20: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
reach_Suc_succ: fixes w :: "'c stream"   and n :: "nat"   and A :: "('c, 'd) nba" assumes "w !! n \<in> alphabet A" shows "reach A w (Suc n) = \<Union> (transition A (w !! n) ` reach A w n)"
sum_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "FGModule G smult V"   and "finite A"   and "f ` A \<subseteq> V" shows "sum f A \<in> V"
roots2: fixes z :: "complex" assumes "z \<in> sphere 0 (1 / R)"   and "poly q z = 0" shows "order z q \<le> Suc k"
liftT_substT_strange: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmark.type"   and U :: "POPLmark.type" shows "\<up>\<^sub>\<tau> n k T[n + k \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau> = \<up>\<^sub>\<tau> n (Suc k) T[k \<mapsto>\<^sub>\<tau> \<up>\<^sub>\<tau> n 0 U]\<^sub>\<tau>"
filter2_length: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and as :: "'c list"   and bs :: "'d list" shows "length (fst (filter2 P as bs)) \<equiv> length (snd (filter2 P as bs))"
distinct_11: fixes x1 :: "agent"   and x71 :: "nat"   and x72 :: "msg" shows "Agent x1 \<noteq> Crypt x71 x72"
"case": fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and x1 :: "'g"   and x2 :: "'h" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
prems_1: shows "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : T"
right_zero_vec: fixes v :: "'b Matrix.vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v + 0\<^sub>v n = v"
in_set_selsort_eq: fixes x :: "'b"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "(x \<in> set (linorder_list0.selsort le xs)) = (x \<in> set xs)"
cp_OclImplies: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" shows "(X implies Y) \<tau> = ((\<lambda>_. X \<tau>) implies (\<lambda>_. Y \<tau>)) \<tau>"
fbox_demodalisation3: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(d y \<le> antidomain_semiringl.fbox a (\<cdot>) x (d z)) = (d y \<cdot> x \<le> x \<cdot> d z)"
keys_monom_mult_subset: fixes c :: "'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (MPoly_Type_Class.punit.monom_mult c t p) \<subseteq> MPoly_Type_Class.punit.splus t ` keys p"
rep_eq: fixes x :: "((real, 'a) vec, 'c) vec" shows "blinfun_apply (blinfun_of_vmatrix x) = (*v) x"
comp_galcon_def: fixes G :: "('c, 'b, unit, unit) galcon"   and F :: "('a, 'c, unit, unit) galcon" shows "G \<circ>\<^sub>g F = \<lparr>orderA = \<X>\<^bsub>F\<^esub>, orderB = \<Y>\<^bsub>G\<^esub>, lower = \<pi>\<^sup>*\<^bsub>G\<^esub> \<circ> \<pi>\<^sup>*\<^bsub>F\<^esub>, upper = \<pi>\<^sub>*\<^bsub>F\<^esub> \<circ> \<pi>\<^sub>*\<^bsub>G\<^esub>\<rparr>"
fn_not_fst: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm"   and P :: "'b trm" shows "Fn x T A \<noteq> Fst P"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (tVAR list)"   and "P tBOOL"   and "P tINT"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P ty1 \<rightarrow> ty2" shows "P ty"
factorize_complex_main: fixes p :: "complex poly"   and fctrs :: "complex \<times> (complex \<times> nat) list" assumes "factorize_complex_main p = Some fctrs" shows "is_factorization_of fctrs p"
span_scale: fixes x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "x \<in> span S" shows "c *s x \<in> span S"
uadmit_dterm_adjointS: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<theta> :: "('sf, 'sz) trm" assumes "ssafe \<sigma>"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"   and "dsafe \<theta>" shows "dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = dterm_sem (local.adjoint I \<sigma> \<omega>) \<theta>"
imp_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.imp \<gamma> \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<longrightarrow> \<gamma>' t n"
\<psi>_1: fixes i :: "nat" assumes "i \<in> N" shows "\<psi> i \<in> atlas"
1: fixes y :: "('fun, 'atom) Term.term \<times> nat" assumes "Var y \<in> set U" shows "the_i y < length U \<and> U ! the_i y = Var y"
groupWIs_not_empty_elems: fixes V :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list"   and w :: "'b wordinterval" assumes "V \<in> set (groupWIs c rs)"   and "w \<in> set V" shows "\<not> wordinterval_empty w"
set_iterator_rule_P: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "set_iterator it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
finfun_default_const: fixes b :: "'c" shows "finfun_default (K$ b) = (if finite UNIV then undefined else b)"
acquired_all_shared_in: fixes a :: "nat"   and sb :: "'b memref list"   and A :: "nat set" assumes "a \<in> acquired True sb A" shows "a \<in> acquired True sb {} \<or> a \<in> A \<and> a \<notin> all_shared sb"
S_ForallE_left: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and S\<^sub>2 :: "ty"   and S\<^sub>1 :: "ty"   and T :: "ty" assumes "\<Gamma>\<turnstile>(\<forall>X<:S\<^sub>1. S\<^sub>2)<:T"   and "X \<sharp> \<Gamma>"   and "X \<sharp> S\<^sub>1"   and "X \<sharp> T" shows "T = Top \<or> (\<exists>T\<^sub>1 T\<^sub>2. T = (\<forall>X<:T\<^sub>1. T\<^sub>2) \<and> \<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1 \<and> (TVarB X T\<^sub>1 # \<Gamma>)\<turnstile>S\<^sub>2<:T\<^sub>2)"
LIM_offset_zero_cancel: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d" assumes "(\<lambda>h. f (a + h)) \<midarrow>0::'c\<rightarrow> L" shows "f \<midarrow>a\<rightarrow> L"
space_interval_measure: fixes F :: "real \<Rightarrow> real" shows "space (interval_measure F) = UNIV"
gen_length_cons_def: fixes n :: "nat"   and s :: "'s" shows "local.gen_length_cons n s = n + length (unstream g s)"
catalan_closed_form_gbinomial: fixes n :: "nat" shows "real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)"
boxI: fixes x :: "'a" shows "Box\<cdot>(up\<cdot>x) = box\<cdot>x"
inf_continuous_at_right: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "inf_continuous f" shows "continuous (at_right x) f"
transpose_vector: fixes x :: "('b, 'd) vec"   and A :: "(('b, 'c) vec, 'd) vec" shows "x v* A = Finite_Cartesian_Product.transpose A *v x"
prems_2: shows "pre ({A} WHILE b DO C) P l s"
INF_UNIV_bool_expand: fixes A :: "bool \<Rightarrow> 'c" shows "\<Sqinter> (range A) = A True \<sqinter> A False"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and agent :: "agent" shows "P (case agent of Friend x \<Rightarrow> f x) = (\<forall>x. agent = Friend x \<longrightarrow> P (f x))"
chooseNext_all: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" assumes "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs u p var"   and "CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs u p g" shows "chooseNext_all (u g) (p g) g \<in> dom (p g) \<and> CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs u p g (snd (chooseNext_all (u g) (p g) g))"
lfp_ordinal_induct: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; S \<le> lfp f\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)" shows "P (lfp f)"
iso_map\<^sub>1_ide: fixes f :: "'c" assumes "C.ide f" shows "D.iso (\<rho>\<^sub>1 f)"
induct: fixes P :: "char list list \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> bool"   and a0 :: "char list list"   and a1 :: "nat \<Rightarrow> nat option" assumes "\<And>s. P [] s"   and "\<And>\<pi> \<pi>s s. (\<And>x. \<lbrakk>enabled \<pi> s; x = execute \<pi> s\<rbrakk> \<Longrightarrow> P \<pi>s x) \<Longrightarrow> P (\<pi> # \<pi>s) s" shows "P a0 a1"
leg0_simps_3: shows "trg u\<^sub>0 = src u"
Fin: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and x1 :: "abopt"   and s1 :: "st"   and c2 :: "stmt"   and s2 :: "state"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> (x1, s1)"   and "G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> s2"   and "s3 = (if \<exists>err. x1 = Some (Error err) then (x1, s1) else abupd (abrupt_if (x1 \<noteq> None) x1) s2)" shows "G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<rightarrow> s3"
axioms_1: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_prob return bind sample" shows "monad return bind"
gd_term_axioms: shows "gd_term to_pair_unit fst (\<preceq>) (\<prec>) (\<preceq>) (\<prec>)"
ex1_poincare_line: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<noteq> v"   and "u \<in> unit_disc"   and "v \<in> unit_disc" shows "\<exists>!l. is_poincare_line l \<and> u \<in> circline_set l \<and> v \<in> circline_set l"
i_ab: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<rho> (a \<cdot> b) = \<rho> b \<cdot> \<rho> a"
S_M_grounding_of_clss: fixes C :: "'a literal multiset" assumes "C \<in> grounding_of_clss M" obtains D :: "'a literal multiset" and \<sigma> :: "'s" where "D \<in> M \<and> C = D \<cdot> \<sigma> \<and> S_M C = S D \<cdot> \<sigma> \<and> is_ground_subst \<sigma>"
division_segment_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment one = one"
t1: fixes e :: "real" assumes "0 < e" shows "\<exists>y\<in>S. Sup S - e < y"
comp_fun_commute_funpow: fixes g :: "'a \<Rightarrow> nat" shows "comp_fun_commute (\<lambda>x. f x ^^ g x)"
subKGSiGSi: shows "subgroup (K \<inter> \<GG> ! (i + 1)) (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>)"
clock_sim_f: shows "sim_f Clock.MC clock_simMC clock_sim"
Scons_inject1: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "Scons M N = Scons M' N'" shows "M = M'"
vars_3: shows "rvars (\<T> (l ! j)) = rvars (\<T> (l ! (j - 1))) - {x\<^sub>j} \<union> {x\<^sub>i}"
asym: fixes b :: "'b"   and a :: "'b" assumes "b < a"   and "a < b" shows "False"
restrict_apply': fixes x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "x \<in> A" shows "restrict f A x = f x"
assms_2: shows "dip \<in> kD (rt \<xi>)"
global_primitive: fixes f :: "complex \<Rightarrow> complex" assumes "connected S"   and "f holomorphic_on S"   and "\<And>\<gamma> f. \<lbrakk>valid_path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; f holomorphic_on S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) \<gamma>" shows "\<exists>h. \<forall>z\<in>S. (h has_field_derivative f z) (at z)"
uniform_limit_sequentially_iff: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and S :: "'a set"   and l :: "'a \<Rightarrow> 'b" shows "uniform_limit S f l sequentially = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. dist (f n x) (l x) < e)"
in_hom_char: fixes Obj :: "'a set"   and f :: "('a, 'a) concrete_category.arr"   and a :: "('a, 'a) concrete_category.arr"   and b :: "('a, 'a) concrete_category.arr" shows "partial_magma.in_hom (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f a b = (partial_magma.arr (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f \<and> f = a \<and> f = b)"
MSpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and l :: "Locs" shows "MSpec ch mm rs l \<equiv> Init MInit mm l \<and> \<box>[\<exists>p. Write ch mm rs p l]_(mm!l)"
tendsto: fixes f :: "'d \<Rightarrow> 'a"   and a :: "'a"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
CatchMiss: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
rSym: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "(S, R) \<in> X"
alw_next_imp_propagate_next: fixes s :: "('b, 't, 'loc) Combined.configuration stream"   and p :: "'b" assumes "alw (holds inv_init_imp_prop_safe) s"   and "alw (holds InvGlobPointstampsEq) s"   and "alw (holds cri.InvMsgInGlob) (smap exchange_config s)"   and "alw Next s" shows "alw (relates local.next'\<^sup>+\<^sup>+) (smap (\<lambda>s. prop_config s p) s)"
sp_F_eq_sp_F': shows "Subx.sp (Fubx_get (Fubx_add F f (Subx.generalize_fundef fd))) = Subx.sp (Fubx_get F)"
linear_injective_isomorphism: fixes f :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "inj f"   and "dim UNIV = dim UNIV" shows "\<exists>f'. linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
defUse_path_dominated: fixes g :: "'g"   and v :: "'val"   and ns :: "'node list"   and n :: "'node"   and n' :: "'node" assumes "g \<turnstile> defNode g v-ns\<rightarrow>n"   and "defNode g v \<notin> set (tl ns)"   and "v \<in> allUses g n"   and "n' \<in> set ns" shows "old.dominates g (defNode g v) n'"
set_val_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_set_val_get_tag_name type_wf set_val set_val_locs get_tag_name get_tag_name_locs" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
assms_3: shows "D r [\<beta>] [\<alpha>] \<sigma> \<tau>"
env_nrelated_closed: fixes \<Gamma> :: "name list"   and env :: "(name, Term_Class.term) fmap"   and nenv :: "(name, nterm) fmap"   and \<Gamma>' :: "name list" assumes "nrelated.P_env \<Gamma> env nenv"   and "closed_env nenv" shows "nrelated.P_env \<Gamma>' env nenv"
ball_imageD: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and P :: "'e \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
simps_1: fixes x1 :: "'b trm"   and x2 :: "'b trm"   and y1 :: "'b trm"   and y2 :: "'b trm" shows "(Eq x1 x2 = Eq y1 y2) = (x1 = y1 \<and> x2 = y2)"
code: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "siterate f x = x ## siterate f (f x)"
lift_character: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "character (G\<lparr>carrier := H\<rparr>) \<chi>"   and "z ^ subgroup_indicator G H a = \<chi> (a [^] subgroup_indicator G H a)" shows "character (G\<lparr>carrier := adjoin G H a\<rparr>) (lift_character (\<chi>, z))"
tendsto_complex_of_real_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter" shows "((\<lambda>x. complex_of_real (f x)) \<longlongrightarrow> complex_of_real c) F = (f \<longlongrightarrow> c) F"
\<phi>: shows "local.iso \<phi> \<and> \<guillemotleft>\<phi> : a \<Rightarrow> a'\<guillemotright>"
mix_pmf_subset_of_original: fixes a :: "real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "a \<in> {0..1}" shows "set_pmf (mix_pmf a p q) \<subseteq> set_pmf p \<union> set_pmf q"
has_typ_imp_closed: fixes t :: "Core.term"   and ty :: "typ" assumes "\<turnstile>\<^sub>\<tau> t : ty" shows "is_closed t"
cascadingoutputlaw: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and s :: "'b" shows "a \<le>\<^sub>\<K> c \<or> b = (1::'a) \<or> s \<circ> a ; \<lambda> (s, c) \<circ> b = (0::'a)"
not_authKeys_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<notin> authKeys evs"   and "K \<notin> range shrK"   and "evs \<in> kerbIV" shows "\<forall>SK. \<not> AKcryptSK K SK evs"
wf_chain_process_ret: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list" assumes "wf_chain \<Gamma> rs" shows "wf_chain \<Gamma> (process_ret rs)"
inverse_dimvec_def: fixes x :: "('a, 'b) dimvec" shows "inverse x = dim_lambda (\<lambda>i. - dim_nth x i)"
stack_exhaust: fixes k :: "stack"   and c :: "'a" shows "k = stack.Id \<or> (\<exists>y n l. y \<sharp> l \<and> y \<sharp> c \<and> k = [y]n\<ggreater>l)"
epi_implies_arr: fixes f :: "'b" assumes "B.epi f" shows "B.arr f"
binomial_Suc_Suc_eq_times: fixes n :: "nat"   and k :: "nat" shows "Suc n choose Suc k = Suc n * (n choose k) div Suc k"
st_1: shows "f C1_differentiable_on {a..c} - S"
measure_eq_emeasure_eq_ennreal: fixes x :: "real"   and M :: "'b measure"   and A :: "'b set" assumes "0 \<le> x"   and "emeasure M A = ennreal x" shows "Sigma_Algebra.measure M A = x"
RedFAcc: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and v :: "val"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "hp s a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>addr a\<bullet>F{D},s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
t_mult_closure: fixes x :: "'a"   and y :: "'a" assumes "\<tt>\<tt> x = x"   and "\<tt>\<tt> y = y" shows "\<tt>\<tt> (x \<cdot> y) = x \<cdot> y"
mset_subset_eq_single: fixes a :: "'c"   and B :: "'c multiset" assumes "a \<in># B" shows "{#a#} \<subseteq># B"
ABs: shows "set ABs \<subseteq> carrier_mat n n \<times> carrier_mat n n"
i_def: shows "i \<equiv> SOME i. i \<in> Basis"
nat_div_as_int: shows "(div) = (\<lambda>a b. nat (int a div int b))"
assms_3: shows "Measurement_System.more x = Measurement_System.more y"
continuous_at_csqrt: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont csqrt z"
ivl_disj_un_one_1: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{..l} \<union> {l<..<u} = {..<u}"
len_of_finite_3_def: fixes x :: "Enum.finite_3 itself" shows "len_of x \<equiv> 4"
fin_digraphI_app_iso: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "fin_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "fin_digraph (app_iso hom G)"
cod_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.cod C (partial_magma.dom C f) = partial_magma.dom C f"
assms_3: shows "mat (row_length M1) (length M1) M1"
hom_Min_commute: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'b"   and N :: "'b set" assumes "class.linorder less_eq less"   and "\<And>x y. h (ord.min less_eq x y) = ord.min less_eq (h x) (h y)"   and "finite N"   and "N \<noteq> {}" shows "h (linorder.Min less_eq N) = linorder.Min less_eq (h ` N)"
vimage_UN: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "f -` \<Union> (B ` A) = (\<Union>x\<in>A. f -` B x)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('a, 'b) atm" shows "F.satL \<xi> (Pos a) = F.satA \<xi> a"
weakSimTauChain: fixes Q :: "pi"   and Q' :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S" shows "\<exists>P'. P \<Longrightarrow>\<^sub>\<tau> P' \<and> (P', Q') \<in> Rel"
size_mset_removeAll_mset_le_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(size (removeAll_mset x M) < size M) = (x \<in># M)"
\<delta>_prod_finite: fixes \<delta>1 :: "('a, 'b) ta_rule set"   and \<delta>2 :: "('c, 'b) ta_rule set" assumes "finite \<delta>1"   and "finite \<delta>2" shows "finite (\<delta>_prod \<delta>1 \<delta>2)"
cwiseext_imp_len_lexext: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "cwiseext gt ys xs" shows "len_lexext gt ys xs"
maxVnameLen_aux_newname: fixes e :: "expr"   and V :: "char list" assumes "maxVnameLen_aux e < length V" shows "V \<notin> rhs_aux e"
hop_count_positive: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, uu_). \<forall>ip\<in>kD (rt \<xi>). 1 \<le> the (dhops (rt \<xi>) ip))"
assms_1: shows "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"
S_bij_betw_chamber0: shows "bij_betw (\<lambda>s. \<langle>S - {s}\<rangle>) S (dual_order.poset_simplex_map \<P> 0)"
drop_bit_drop_bit: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit m (drop_bit n a) = drop_bit (m + n) a"
balance_tree_def: fixes t :: "'a tree" shows "balance_tree t = bal_tree (size t) t"