hcju/isabelleps huggingface dataset:
exec_all_def1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>' = ((\<sigma>, \<sigma>') \<in> (exec_1 P)\<^sup>*)"
degeneralize_nodes_finite: fixes A :: "('e, 'd) dgbta" shows "finite (DBTA.nodes (degeneralize A)) = finite (DGBTA.nodes A)"
move_stability_clm: fixes ts :: "traffic"   and c :: "cars"   and n :: "nat"   and ts' :: "traffic"   and v :: "view" shows "ts \<^bold>\<midarrow>c( c, n ) \<^bold>\<rightarrow> ts' \<longrightarrow> move ts ts' v = v"
fin: shows "finite {y. x \<noteq> y \<and> y pd\<rightarrow> x}"
fmfilter_false: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f"   and P :: "'e \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> \<not> P x" shows "fmfilter P m = fmempty"
eq_Integer_def: shows "eq = (\<Lambda> (MkI\<cdot>x) (MkI\<cdot>y). if x = y then TT else FF)"
singleDSourceLoop: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet" assumes "DAcc i C = {S}"   and "DAcc i S = {C}" shows "\<forall>Z\<in>AbstrLevel i. Z \<noteq> S \<and> Z \<noteq> C \<longrightarrow> C \<notin> Sources i Z"
is_volatile_Read\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_volatile_Read\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
cmib: shows "m\<inverse> O b \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
lift_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> int) list \<Rightarrow> ((nat \<times> int) list \<Rightarrow> real) \<Rightarrow> (nat \<times> int) list \<Rightarrow> real"   and dm :: "nat"   and lm :: "nat"   and d :: "nat" shows "Grid.lift f dm lm d = foldr (\<lambda>p. f d (lm - level p) p) (gridgen (start dm) ({0..<dm} - {d}) lm)"
sim0: shows "R as\<^sub>0 cs\<^sub>0"
N_below_inf_one_pp: fixes x :: "'a" shows "- (- x \<sqinter> top) \<sqinter> top \<le> - - x \<sqinter> top"
has_integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a"   and j :: "'a" assumes "a \<le> c"   and "c \<le> b"   and "(f has_integral i) {a..c}"   and "(f has_integral j) {c..b}" shows "(f has_integral i + j) {a..b}"
uniformly_continuous_on_def: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "uniformly_continuous_on s f = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
in_mset_neg_in_zmset: fixes x :: "'a"   and M :: "'a zmultiset" assumes "x \<in># mset_neg M" shows "x \<in>#\<^sub>z M"
sorted_rev_iff_nth_mono: fixes xs :: "'a list" shows "linorder.sorted (\<lambda>x y. y \<le> x) (rev xs) = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)"
simps_18: fixes va :: "('e, 'f, 'g) Tree"   and v :: "'f" shows "(tNot va =\<^sub>\<alpha> tPred v) = False"
c: fixes z :: "complex" assumes "z \<in> ball 0 r" shows "g z = c * z"
step_red: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "\<langle>c,s\<rangle> \<rightarrow> \<langle>c',s'\<rangle>"
IH: fixes t :: "char list \<Rightarrow> int option" assumes "approx (t |` (- lvars c2)) s" shows "approx (t |` (- lvars c2)) s'"
la_ex_pres_gs_vars: fixes ls :: "local_state"   and gs :: "global_state"   and a :: "action"   and ls' :: "local_state"   and gs' :: "global_state" assumes "la_ex (ls, gs) a = Some (ls', gs')" shows "gs_vars gs' = gs_vars gs"
matches_induction_case_MatchNot: fixes \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and \<beta> :: "'d \<Rightarrow> 'c \<Rightarrow> ternaryvalue"   and m' :: "'d match_expr"   and p :: "'c"   and m :: "'d match_expr"   and a :: "action" assumes "\<alpha> Drop \<noteq> \<alpha> Accept"   and "packet_independent_\<alpha> \<alpha>"   and "\<forall>a. matches (\<beta>, \<alpha>) m' a p = matches (\<beta>, \<alpha>) m a p" shows "matches (\<beta>, \<alpha>) (MatchNot m') a p = matches (\<beta>, \<alpha>) (MatchNot m) a p"
\<sigma>f_def: shows "\<sigma>f \<equiv> \<sigma> (f, Suc (length bef + length aft))"
pick_proj: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "equivP P \<phi>"   and "P x" shows "\<phi> (pick (proj \<phi> x)) x"
axiom_existseq_def: fixes x :: "String.literal"   and t :: "tm" shows "axiom_existseq x t \<equiv> if \<not> occurs_in x t then Thm (Exists x (Atom (Rl STR ''='' [Var x, t]))) else Thm Truth"
bisimE_1: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "P \<leadsto>[bisim] Q"
mHom_mem: fixes N :: "('i, 'b, 'j) Module_scheme"   and f :: "'a \<Rightarrow> 'i"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f m \<in> carrier N"
that: fixes Z :: "'a set" assumes "Z \<in> P"   and "X = Z - {elem}" shows "thesis"
comp_arr_dom: fixes f :: "'s"   and a :: "'s" assumes "S.arr f"   and "S.dom f = a" shows "f \<cdot> a = f"
WTrt_elim_cases_7: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e\<bullet>F{D} : T" obtains C :: "char list" where "P,E,h,sh \<turnstile> e : Class C" and "P \<turnstile> C has F,NonStatic:T in D"   | "P,E,h,sh \<turnstile> e : NT"
iii: shows "vector_derivative g (at ((v - u) * x + u) within {0..1}) = vector_derivative g (at ((v - u) * x + u))"
cab: shows "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
intros_3: fixes v2 :: "'b val"   and r2 :: "'b rexp"   and r1 :: "'b rexp" assumes "\<turnstile> v2 : r2" shows "\<turnstile> val.Right v2 : Plus r1 r2"
mk_linear_orders_f_MaxR_opt: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and C :: "'a set"   and B :: "'a set" assumes "f_range_on A f"   and "x \<in> f C"   and "consistency_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B" shows "\<exists>r\<in>mk_linear_orders f C B. MaxR.MaxR_opt (linord_of_list r) C = Some x"
is_natural_transformation: shows "natural_transformation A B G G \<tau>'\<tau>.map"
dSM1r: shows "DS (M1 * \<rho> * adjoint M1) \<in> carrier_mat d d"
bitnot_bitnot: fixes b :: "bit" shows "(\<not>\<^sub>b \<not>\<^sub>b b) = b"
assms_1: shows "finite {b. \<exists>a. P a b}"
pmu_mu: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f"   and "isotone f" shows "p\<mu> f = \<mu> f"
subtopology_superset: fixes U :: "'c topology"   and V :: "'c set" assumes "topspace U \<subseteq> V" shows "subtopology U V = U"
cong: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "prod g A = prod h B"
h_1: shows "steps0 (s, l, r) A (Suc stp) = (s', l', r')"
member_empty_empty: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "(RBT_Set2.member t = (\<lambda>_. False)) = (t = RBT_Set2.empty)"
exec_step_ind_cases_18: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "exec_step_ind (StepI Throw) P h stk loc C M pc ics frs sh \<sigma>" obtains "\<sigma> = (\<lfloor>the_Addr (hd stk)\<rfloor>, h, (stk, loc, C, M, pc, ics) # frs, sh)" and "hd stk \<noteq> Null"   | "\<sigma> = (\<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>, h, (stk, loc, C, M, pc, ics) # frs, sh)" and "hd stk = Null"
bounded_linear_scaleR_left: fixes x :: "'d" shows "bounded_linear (\<lambda>r. r *\<^sub>R x)"
gc_initial_state_def: fixes s :: "('e, 'f, 'g, 'h) Model.local_state" shows "gc_initial_state s = (fM s = initial_mark \<and> phase s = ph_Idle \<and> ghost_honorary_grey s = {} \<and> W s = {})"
coprime_approx_main_uint64: fixes f :: "int poly"   and g :: "int poly" assumes "p \<le> 4294967295"   and "coprime_approx_main p (finite_field_ops64 (uint64_of_int p)) f g" shows "coprime_m f g"
run_effectI: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "run c (Some h) (Some h') r" shows "effect c h h' r"
pending_ssE: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "pending s \<subseteq> g_E G"
simps_1: fixes n :: "int"   and a :: "'av"   and S :: "'av st option" shows "inv_aval' (N n) a S = (if test_num' n a then S else None)"
simps_1: fixes \<phi> :: "fml"   and \<psi> :: "fml" shows "Ando (Afml \<phi>) (Afml \<psi>) = Afml (\<phi> && \<psi>)"
simps_13: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a" shows "(v;; va \<le> {vb} WHILE vc DO vd {ve}) = False"
igSwapIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapPresGWls MOD" shows "igSwapIPresIGWls (fromMOD MOD)"
select_convs_1: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and gbg_F :: "'a set set"   and gba_L :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and more :: "'c" shows "gba_L \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, gbg_F = gbg_F, gba_L = gba_L, \<dots> = more\<rparr> = gba_L"
nonregular_event_induces_snapshot: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration" assumes "\<not> ps c p \<noteq> None"   and "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev = p"   and "\<not> regular_event ev" shows "\<not> regular_event ev \<longrightarrow> ps c' p \<noteq> None"
others: fixes y :: "'a" assumes "y \<in> carrier G"   and "y divides b"   and "y divides c" shows "y divides a"
ext_hd_or_tl_axioms: shows "ext_hd_or_tl msetext_huet"
upred_eq_true: fixes p :: "'a upred" shows "(p =\<^sub>u true) = p"
check_s_def: shows "check_s = Incremental_State_Ops_Simplex.check_s check_code"
OclImplies_def: 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>" shows "X implies Y \<equiv> not X or Y"
prod: shows "\<forall>w. P w (x \<otimes> \<midarrow> y) = (P w x \<and> P w (\<midarrow> y))"
is_zero_row_imp_is_zero_row_upt: fixes i :: "'d"   and A :: "(('e, 'f) vec, 'd) vec"   and k :: "nat" assumes "is_zero_row i A" shows "is_zero_row_upt_k i k A"
simps_1: fixes uu :: "typ"   and uv :: "Core.term"   and uw :: "Core.term" shows "beta_reducible (Abs uu uv $ uw) = True"
Pi_def: fixes A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
elementary_Inter: fixes f :: "'a set set" assumes "finite f"   and "f \<noteq> {}"   and "\<forall>s\<in>f. \<exists>p. p division_of s" shows "\<exists>p. p division_of \<Inter> f"
atrm_num: fixes n :: "'trm" assumes "n \<in> num" shows "n \<in> atrm"
theFT_Un_theNFT: fixes cl :: "('a, 'b, 'c) cmd list" shows "theFT cl \<union> theNFT cl = {..<length cl}"
And_prop_entailment: fixes S :: "'c ltl set"   and xs :: "'c ltl list" shows "S \<Turnstile>\<^sub>P And xs = (\<forall>x\<in>set xs. S \<Turnstile>\<^sub>P x)"
assms_1: shows "(c, s) \<rightarrow>*t s''"
simps_1: fixes b :: "real"   and c :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalLinearUni b c (EqUni p) = allZero' p"
Table0_axioms: shows "Table0 f1 f2 f1' f2' e c"
dagger_unfoldl_distr: fixes y :: "'a ppath set"   and x :: "'a ppath set" shows "y \<union> pp_prod (pp_prod x (pp_star x)) y = pp_prod (pp_star x) y"
tagged_division_of_def: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" shows "(s tagged_division_of i) = (s tagged_partial_division_of i \<and> \<Union> {K. \<exists>x. (x, K) \<in> s} = i)"
simps_4: fixes n :: "nat"   and m :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound n m (F \<^bold>\<and> G) = cut_bound n (cut_bound n m F) G"
good_ruleset_alt: fixes rs :: "'c rule list" shows "good_ruleset rs = (\<forall>r\<in>set rs. get_action r = action.Accept \<or> get_action r = action.Drop \<or> get_action r = Reject \<or> get_action r = Log \<or> get_action r = Empty)"
prems_2: shows "{x. ev x \<in> set (a0 # al)} \<subseteq> A"
tangent_space_dim_eq: fixes X :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c" assumes "X \<subseteq> manifold_eucl.dest.tangent_space p" shows "manifold_eucl.dest.tangent_space.dim p X = dim X"
gcsort_count_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and p :: "nat"   and t' :: "nat \<times> nat list \<times> 'c list"   and f :: "'c \<Rightarrow> nat" assumes "index_less index key"   and "add_inv n t"   and "n \<le> p"   and "t' \<in> gcsort_set index key p t"   and "count_inv f t" shows "count_inv f t'"
xeqI: fixes x :: "Value"   and s :: "Store"   and t :: "Store" assumes "alive x s = alive x t"   and "\<And>l. s\<turnstile> l reachable_from x \<Longrightarrow> s@@l = t@@l" shows "s \<equiv>[x] t"
neutr_eq_iff: fixes a :: "'b"   and b :: "'b" shows "(bot = sup a b) = (a = bot \<and> b = bot)"
assms_1: shows "F : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B"
measurable_linear_image: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set" assumes "linear f"   and "S \<in> lmeasurable" shows "f ` S \<in> lmeasurable"
remove_imps: fixes p :: "('a, 'b) form"   and z' :: "('a, 'b) form list"   and z :: "('a, 'b) form list" assumes "infinite (- params p)"   and "z' \<turnstile> put_imps p z" shows "rev z @ z' \<turnstile> p"
h': fixes x :: "'a" shows "h' x = (case f x of Terminal_corec a \<Rightarrow> TERMINAL a | Linear_corec b x' \<Rightarrow> LINEAR b (h' x') | Branch_corec c xs \<Rightarrow> BRANCH c (\<lambda>d. h' (xs d)) | Resumption_corec r \<Rightarrow> Rep_resumption r)"
divides_mult: fixes a :: "'a list"   and c :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)"   and "a divides\<^bsub>poly_ring R\<^esub> b" shows "c \<otimes>\<^bsub>poly_ring R\<^esub> a divides\<^bsub>poly_ring R\<^esub> c \<otimes>\<^bsub>poly_ring R\<^esub> b"
Sup_insert_finite: fixes S :: "'c set"   and x :: "'c" assumes "finite S" shows "Sup (insert x S) = (if S = {} then x else max x (Sup S))"
is_span: shows "C.span trg_trg.leg0 trg_trg.leg1"
vc_mono: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and C :: "acom" assumes "\<forall>l s. P l s \<longrightarrow> P' l s"   and "vc C P" shows "vc C P'"
enum_def: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s" shows "kuhn_simplex.enum base upd i j = (if j \<in> upd ` {..<i} then Suc (base j) else base j)"
hd_append2: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
assms: shows "\<turnstile> p \<^bold>\<longrightarrow> q"
simps: fixes a :: "'a word"   and b :: "'a word" shows "word_upto a b = (if a = b then [a] else word_upto a (b - 1) @ [b])"
that_4: fixes CTa :: "nat \<Rightarrow> classDef option"   and e0 :: "exp"   and f :: "nat"   and e0' :: "exp" assumes "x = CTa"   and "xa = FieldProj e0 f"   and "xb = FieldProj e0' f"   and "CTa \<turnstile> e0 \<rightarrow> e0'" shows "thesis"
sel: fixes x :: "'a \<Rightarrow> 'b \<times> 'a" shows "run_state (State x) = x"
le_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
in_nlists_UNIV: fixes xs :: "'h list"   and n :: "nat" shows "(xs \<in> nlists UNIV n) = (length xs = n)"
swapAbs_simp: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "Abs xs x X $[y1 \<and> y2]_ys = Abs xs (x @xs[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys)"
integrable_continuous: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on (cbox a b) f" shows "f integrable_on cbox a b"
filt_equiv_prob_space_subalgebra: fixes N :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and M :: "'c measure"   and G :: "'c measure" assumes "prob_space N"   and "filt_equiv F M N"   and "sigma_finite_subalgebra M G" shows "sigma_finite_subalgebra N G"
rbt_min_opt_is_min: fixes t :: "('b, unit) RBT_Impl.rbt"   and y :: "'b" assumes "rbt_sorted t"   and "t \<noteq> rbt.Empty"   and "y \<in> set (RBT_Impl.keys t)" shows "rbt_min_opt t \<le> y"
poly_mapping_eq_zeroI: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" assumes "keys s = {}" shows "s = 0"
replicate_mset_plus: fixes a :: "nat"   and b :: "nat"   and C :: "'a" shows "replicate_mset (a + b) C = replicate_mset a C + replicate_mset b C"
SigmaD2: fixes a :: "'e"   and b :: "'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
Rep_Abs_prop_entailment: fixes A :: "'a ltl set"   and \<phi> :: "'a ltl" shows "A \<Turnstile>\<^sub>P Rep (Abs \<phi>) = A \<Turnstile>\<^sub>P \<phi>"
is_median_altdef: fixes x :: "'b"   and xs :: "'b list" shows "is_median x xs = (size {#y \<in># mset xs. y < x#} \<le> length xs div 2 \<and> size (filter_mset ((<) x) (mset xs)) \<le> length xs div 2)"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b" shows "(case <> of <> \<Rightarrow> f1 | x % xa \<Rightarrow> f2 x xa) = f1"
finfun_weak_induct: fixes P :: "('b, 'a) finfun \<Rightarrow> bool"   and x :: "('b, 'a) finfun" assumes "\<And>b. P (finfun_const b)"   and "\<And>f a b. P f \<Longrightarrow> P (finfun_update f a b)" shows "P x"
semiring_norm_81: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m < num.Bit0 n) = (m < n)"
heap_step_admissible: fixes P :: "'d \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'d" shows "option.admissible (\<lambda>f. \<forall>h h' r. f h = Some (r, h') \<longrightarrow> P x h h' r)"
SN_on_trancl: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r A" shows "SN_on (r\<^sup>+) A"
subgroup_axioms: shows "subgroup (left.translation ` G) left.transformation.Sym (compose G) (identity G)"
coll_add_cancel: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "coll 0 a (a + b)" shows "coll 0 a b"
ARR_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "free_monoidal_category.ARR C (free_monoidal_category.mkarr C t)"
integrable_real_density: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M" shows "integrable (density M (\<lambda>x. ennreal (f x))) g = integrable M (\<lambda>x. f x * g x)"
check_all_combinations_def: fixes c :: "'d \<Rightarrow> 'd \<Rightarrow> 'c + unit"   and xs :: "'d list" shows "check_all_combinations c xs = (try forallM (\<lambda>x. try forallM (c x) xs catch (\<lambda>x. Inl (snd x))) xs catch (\<lambda>x. Inl (snd x)))"
connected_components_of_subset_sing: fixes X :: "'b topology"   and S :: "'b set" shows "(connected_components_of X \<subseteq> {S}) = (connected_space X \<and> (topspace X = {} \<or> topspace X = S))"
cts_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "bd_cts t" shows "bd_cts (wp (Embed t))"
machin': shows "pi / 4 = 12 * arctan (1 / 18) + 8 * arctan (1 / 57) - 5 * arctan (1 / 239)"
rel_G_conversep: fixes L1 :: "'j \<Rightarrow> 'b \<Rightarrow> bool"   and L2 :: "'k \<Rightarrow> 'd \<Rightarrow> bool"   and Co1 :: "'l \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'m \<Rightarrow> 'f \<Rightarrow> bool"   and Contra1 :: "'n \<Rightarrow> 'g \<Rightarrow> bool"   and Contra2 :: "'o \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_G L1\<inverse>\<inverse> L2\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> = (rel_G L1 L2 Co1 Co2 Contra1 Contra2)\<inverse>\<inverse>"
P1: shows "\<zero>\<^bsub>residue_ring 1\<^esub> \<in> carrier (residue_ring 1)"
dsafe: fixes f :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> f = Some f'" shows "dsafe f'"
Mapping_lookup_delete: fixes k :: "'b"   and m :: "('b, 'a) mapping"   and k' :: "'b" shows "Mapping.lookup (Mapping.delete k m) k' = (if k = k' then None else Mapping.lookup m k')"
least_consequence_graph_def: fixes t :: "'d itself"   and Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and S :: "('e, 'g) labeled_graph"   and G :: "('e, 'g) labeled_graph" shows "least_consequence_graph t Rs S G \<equiv> consequence_graph Rs G \<and> least t Rs S G"
ext_inject: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e"   and sprFst' :: "('b, 'c, 'd) BEState"   and sprLst' :: "('b, 'c, 'd) BEState"   and sprCRel' :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more' :: "'e" shows "(\<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = \<lparr>sprFst = sprFst', sprLst = sprLst', sprCRel = sprCRel', \<dots> = more'\<rparr>) = (sprFst = sprFst' \<and> sprLst = sprLst' \<and> sprCRel = sprCRel' \<and> more = more')"
is_ubI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and u :: "'b" assumes "class.po below"   and "\<And>x. x \<in> S \<Longrightarrow> below x u" shows "po.is_ub below S u"
lincomb_as_lincomb_list: fixes ws :: "'a vec list"   and f :: "'a vec \<Rightarrow> 'a" assumes "set ws \<subseteq> carrier_vec n" shows "lincomb f (set ws) = lincomb_list (\<lambda>i. if \<exists>j<i. ws ! i = ws ! j then 0::'a else f (ws ! i)) ws"
cases: fixes x :: "('h + 'i) + 'j" obtains a :: "'h" where "x = Inl (Inl a)"   | b :: "'i" where "x = Inl (Inr b)"   | c :: "'j" where "x = Inr c"