hcju/isabelleps huggingface dataset:
cs_append_rule: fixes l :: "'a list"   and p :: "'a node ref option"   and x :: "'a" shows "<cs_list l p> cs_append x p <cs_list (l @ [x])>"
less_eq_V_0_iff: fixes x :: "V" shows "(x \<subseteq>\<^sub>\<circ> 0) = (x = 0)"
basis_in_indep_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X_insert.basis_in \<E> X" shows "X_insert.indep_in \<E> X"
infinite: fixes A :: "'d set" assumes "infinite A" shows "card A = 0"
instInp_instInpP: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> \<subseteq> {inp}"   and "\<tau> \<in> ptrm (Suc 0)"   and "t \<in> trm"   and "FvarsT t = {}" shows "instInp (instInpP \<phi> (Suc 0) \<tau>) t = instInpP \<phi> 0 (instInp \<tau> t)"
inject: fixes x21 :: "'d tree"   and x22 :: "'d"   and x23 :: "'d tree"   and y21 :: "'d tree"   and y22 :: "'d"   and y23 :: "'d tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
unifiert_def2: fixes ts :: "fterm set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "ts \<noteq> {}" shows "unifier\<^sub>t\<^sub>s \<sigma> ts = (\<exists>l. (\<lambda>t. t \<cdot>\<^sub>t \<sigma>) ` ts = {l})"
match_list_None_conv: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list" shows "(match_list d P = None) = (matchers (set P) = {})"
appliedUnitPropagate_def: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "appliedUnitPropagate stateA stateB F0 decisionVars \<equiv> \<exists>uc ul. formulaEntailsClause (getF stateA) uc \<and> var ul \<in> decisionVars \<union> vars F0 \<and> isUnitClause uc ul (elements (getM stateA)) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA @ [(ul, False)] \<and> getConflictFlag stateB = getConflictFlag stateA \<and> getC stateB = getC stateA"
has_density_sets: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "sets M = sets N"
arctan_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "arctan_aform_err p X = (let l = Inf_aform_err p X; u = Sup_aform_err p X in min_range_mono p Arctan (\<lambda>x. 1 / (Num 1 + x * x)) l u X)"
tf: shows "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
subst_apps: fixes \<rho> :: "'b \<Rightarrow> ('a, 'b) tm"   and s :: "('a, 'b) tm"   and ts :: "('a, 'b) tm list" shows "subst \<rho> (apps s ts) = apps (subst \<rho> s) (map (subst \<rho>) ts)"
simps_1: fixes xs :: "'a list"   and oid :: "'a" shows "insert_spec xs (oid, None) = oid # xs"
circ_sup_2: fixes x :: "'b"   and y :: "'b" shows "(x \<squnion> y)\<^sup>\<star> \<le> (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star>"
tranclp_into_tranclp2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
increasing_def: fixes ls :: "'a list" shows "increasing ls \<equiv> \<forall>x y as bs. ls = as @ x # y # bs \<longrightarrow> x \<le> y"
add_edge_correct_1: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and u :: "'b"   and v :: "'b"   and d :: "nat" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"   and "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "invar (add_edge (u, v) d g)"
LIMSEQ_unique: fixes "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.t2_space open"   and "filterlim X (topological_space.nhds open a) sequentially"   and "filterlim X (topological_space.nhds open b) sequentially" shows "a = b"
osafe_OVar_simps: fixes c :: "'b" shows "osafe (OVar c) = True"
ivt_decreasing_component_on_1: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and k :: "'b"   and y :: "real" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "f b \<bullet> k \<le> y"   and "y \<le> f a \<bullet> k" shows "\<exists>x\<in>{a..b}. f x \<bullet> k = y"
find_shortest_augmenting_spec_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'a" shows "find_shortest_augmenting_spec_cf cf \<equiv> assert (RGraph c s t cf) \<bind> (\<lambda>_. spec x. case x of None \<Rightarrow> \<not> Graph.connected cf s t | Some p \<Rightarrow> Graph.isShortestPath cf s p t)"
map_snd_lessI: fixes xs :: "(nat \<times> 'b) list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "(nat \<times> 'b) list" assumes "set xs {\<sqsubseteq>\<^bsub>r\<^esub>} set ys" shows "set (map_snd OK xs) {\<sqsubseteq>\<^bsub>Err.le r\<^esub>} set (map_snd OK ys)"
length_split: shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> 0 < \<parallel>ext va\<parallel> \<and> 0 < \<parallel>ext u\<parallel>)"
IH: fixes y :: "V" assumes "y \<in> elts ya__" shows "TC (x + y) = TC x \<squnion> lift x (TC y)"
ccache_approx_def: shows "approx = (\<subseteq>)"
F_def: fixes X :: "'a set" shows "F X \<equiv> {y \<in> A. \<exists>x\<in>X. y \<sim> f x} \<union> f ` X"
induct: fixes P :: "('v \<times> 'v) list \<Rightarrow> ('v \<times> 'v) list \<Rightarrow> 'v graph \<Rightarrow> ('v \<Rightarrow> 'a) \<Rightarrow> bool"   and a0 :: "('v \<times> 'v) list"   and a1 :: "('v \<times> 'v) list"   and a2 :: "'v graph"   and a3 :: "'v \<Rightarrow> 'a" assumes "\<And>keep uu_ uv_. P [] keep uu_ uv_"   and "\<And>f fs keep G nP. \<lbrakk>sinvar (delete_edges_list G (fs @ keep)) nP \<Longrightarrow> P fs keep G nP; \<not> sinvar (delete_edges_list G (fs @ keep)) nP \<Longrightarrow> P fs (f # keep) G nP\<rbrakk> \<Longrightarrow> P (f # fs) keep G nP" shows "P a0 a1 a2 a3"
merge_prod_cong: fixes p1 :: "'c \<times> 'd"   and p2 :: "'c \<times> 'd"   and ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and ma' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and mb :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option"   and mb' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option" assumes "\<And>a b. \<lbrakk>a \<in> fsts p1; b \<in> fsts p2\<rbrakk> \<Longrightarrow> ma a b = ma' a b"   and "\<And>a b. \<lbrakk>a \<in> snds p1; b \<in> snds p2\<rbrakk> \<Longrightarrow> mb a b = mb' a b" shows "merge_prod ma mb p1 p2 = merge_prod ma' mb' p1 p2"
abs_eq: fixes xb :: "('sf, 'sc, 'sz) interp"   and xa :: "('sf, 'sz) trm"   and x :: "(real, 'sz) vec" assumes "eq_onp is_interp xb xb"   and "eq_onp dfree xa xa" shows "blin_frechet (good_interp xb) (simple_term xa) x = Blinfun (frechet xb xa x)"
circline_type_neg_card_gt3: fixes H :: "circline" assumes "circline_type H < 0" shows "\<exists>A B C. A \<noteq> B \<and> A \<noteq> C \<and> B \<noteq> C \<and> {A, B, C} \<subseteq> circline_set H"
nn_integral_mono: fixes M :: "'k measure"   and u :: "'k \<Rightarrow> ennreal"   and v :: "'k \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
in_mono: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> B" shows "s2p A x \<longrightarrow> s2p B x"
simps_1: fixes x :: "'b ext" shows "ext.Bot + x = x"
IH_2: shows "continuous_on UNIV (blin_frechet (good_interp I) (simple_term \<theta>\<^sub>2))"
pull_up_configuration_state_rank: fixes q :: "'b"   and n :: "nat" assumes "configuration q n = {}" shows "state_rank q n = None"
map_tpOfV_getVars: fixes \<sigma>l :: "'tp list" shows "map GE.tpOfV (GE.getVars \<sigma>l) = \<sigma>l"
id_2: shows "length (berlekamp_basis f) = length [] + length (berlekamp_basis f)"
intro: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool" assumes "Map M_empty M_update M_delete M_lookup M_invar" shows "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"
A_mult_def: shows "A_mult = mult_row A_interchange (mod_type_class.from_nat i) ((1::'a) / A_interchange $ mod_type_class.from_nat i $ mod_type_class.from_nat k)"
float_cases: fixes a :: "('a, 'b) IEEE.float" shows "is_nan a \<or> is_infinity a \<or> is_normal a \<or> is_denormal a \<or> is_zero a"
rep_in_Hom: fixes f :: "'c local.term set" assumes "arr f" shows "rep f \<in> Hom (DOM f) (COD f)"
subtopology_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
inject_1: fixes x1 :: "'a literal_expr"   and y1 :: "'a literal_expr" shows "(Literal x1 = Literal y1) = (x1 = y1)"
a_comm': fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y = ad y \<cdot> ad x"
eclose_singleton: fixes a :: "hf" shows "eclose \<lbrace>a\<rbrace> = eclose a \<triangleleft> a"
in_bound_step: fixes c :: "nat \<times> integer"   and bp :: "instr array"   and s0 :: "integer"   and c' :: "nat \<times> integer" assumes "c \<in> config_bound bp s0"   and "c' \<in> set (nexts bp c)" shows "c' \<in> config_bound bp s0"
mult_one_right: fixes n :: "eint" shows "n * 1 = n"
rel_mono: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_bset R \<le> rel_bset Ra"
filter_trms_inclusion: 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"   and C :: "'b literal set"   and E :: "'b trm set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "filter_trms C E \<subseteq> E"
one_add_floor: fixes x :: "'d" shows "\<lfloor>x\<rfloor> + 1 = \<lfloor>x + (1::'d)\<rfloor>"
frac_le_1I: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "x \<le> (1::'a)"   and "x \<le> y" shows "frac x \<le> y"
fresh_star_subset: fixes B :: "'b set"   and x :: "atom set"   and A :: "'b set" assumes "finite B"   and "x \<sharp>* B"   and "A \<subseteq> B" shows "x \<sharp>* A"
degree_m_le: fixes m :: "int"   and f :: "int poly" shows "degree (poly_mod.Mp m f) \<le> degree f"
assms_5: shows "E.Src (Dom F) = E.Src (Cod F)"
assms: shows "\<forall>b\<^sub>\<sigma>. local.etp.HCP_adv \<A> \<sigma> b\<^sub>\<sigma> D \<le> HCP_ad"
common_abs: fixes t :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a :: "'a" assumes "t \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a t" shows "toSet (fst ec) = common_abs t"
adjoint_dim_row: fixes A :: "'b mat" shows "dim_row (adjoint A) = dim_col A"
mat2: fixes env :: "v sem_env"   and s :: "'a state"   and e :: "exp0"   and s' :: "'a state"   and err :: "v error_result"   and pes :: "(pat \<times> exp0) list" assumes "Big_Step_Unclocked_Single.evaluate env s e (s', Rerr err)" shows "Big_Step_Unclocked_Single.evaluate env s (Mat e pes) (s', Rerr err)"
simplex_imp_polyhedron: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "polyhedron S"
measurable_gfp_coinduct: fixes P :: "'c measure \<Rightarrow> bool"   and M :: "'c measure"   and F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "P M"   and "inf_continuous F"   and "\<And>M A. \<lbrakk>P M; \<And>N. P N \<Longrightarrow> A \<in> N \<rightarrow>\<^sub>M count_space UNIV\<rbrakk> \<Longrightarrow> F A \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "gfp F \<in> M \<rightarrow>\<^sub>M count_space UNIV"
closed_segment_subset_domain: fixes t0 :: "real"   and t :: "real" assumes "t0 \<in> cball t__ et"   and "t \<in> cball t__ et" shows "{t0--t} \<subseteq> cball t__ et"
ereal_complete_Sup: fixes S :: "ereal set" shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
cases: fixes x :: "role_t" obtains "x = Init"   | "x = Resp"
sup_ge1_ObjectType: fixes \<C> :: "'a"   and \<sigma> :: "'a basic_type" shows "\<langle>\<C>\<rangle>\<^sub>\<T> \<le> \<langle>\<C>\<rangle>\<^sub>\<T> \<squnion> \<sigma>"
two_cubes_trace_vertical_boundaries: shows "two_chain_vertical_boundary two_chain \<subseteq> \<gamma>"
rep_eq: fixes x :: "'a" shows "blinfun_apply (prod_right x) = (**) x"
arr_char: fixes f :: "'a" shows "hhom\<^sub>C.arr f = \<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright>"
cgenideal_is_principalideal: fixes i :: "'a" assumes "i \<in> carrier R" shows "principalideal (PIdl i) R"
ring_iso_def: fixes R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme" shows "ring_iso R S = {h \<in> ring_hom R S. bij_betw h (carrier R) (carrier S)}"
\<B>\<^sub>i\<^sub>u_def: fixes state :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>u state = (case state of State x1 x2 x3 x4 x5 x6 \<Rightarrow> x3)"
HLD_def: fixes s :: "'b set" shows "HLD s = holds (\<lambda>x. x \<in> s)"
subst_fv_bounded_if_img_bounded': fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and M :: "('b, 'a) Term.term set" assumes "range_vars \<theta> \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M" shows "fv\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
hyperDiff1_def: fixes A :: "'f set option"   and a :: "'f" shows "A \<ominus> a \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> \<lfloor>A - {a}\<rfloor>"
neville_aitken_merge: fixes xs :: "('a \<times> 'a \<times> 'a poly) list"   and j :: "nat"   and l :: "nat"   and k :: "nat" assumes "xs = map (\<lambda>i. (x i, x (i + j), local.neville_aitken_main i (i + j))) [l..<Suc (l + k)]" shows "local.neville_aitken_merge xs = map (\<lambda>i. (x i, x (i + Suc j), local.neville_aitken_main i (i + Suc j))) [l..<l + k]"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2 # cs, css, Normal s) \<rightarrow> (c\<^sub>1 # c\<^sub>2 # cs, css, Normal s)"
iMODb_Suc_pred_insert_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "insert r [ r + m, mod m, c ] = [ r, mod m, Suc c ]"
wf_induct: fixes r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
distr_state_measure_Suc: fixes t :: "pdf_type" shows "distr (state_measure (Suc ` V) (case_nat t \<Gamma>)) (state_measure V \<Gamma>) (\<lambda>\<sigma>. \<sigma> \<circ> Suc) = state_measure V \<Gamma>"
alg_dvd_triv_left: fixes y :: "'b"   and x :: "'b" assumes "algebraic_int y" shows "x alg_dvd x * y"
that: fixes y :: "'a set" assumes "y \<lhd> x"   and "f ` y = z" shows "thesis"
IH: fixes x :: "('sf + 'a, 'sz) trm" assumes "x \<in> range x2a"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT x}. FVT (\<sigma> i))" shows "sterm_sem (adjointFO I \<sigma> \<nu>) x = sterm_sem (adjointFO I \<sigma> \<omega>) x"
abs_eq: fixes x :: "'a \<Rightarrow> 'a" assumes "eq_onp (\<lambda>f. bij f \<and> finite {a. f a \<noteq> a}) x x" shows "affected (Perm x) = {a. x a \<noteq> a}"
VEnd_composite_left: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and S :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T"   and "VectorSpaceEnd smult V S" shows "VectorSpaceEnd smult V (S \<circ> T)"
complete_upd: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c"   and y :: "'d" assumes "complete t" shows "complete (treeI (upd x y t)) \<and> hI (upd x y t) = height t"
sup_bot_right: fixes x :: "'c" shows "x + (0::'c) = x"
xvecFreshyvec: shows "xvec \<sharp>* yvec"
Lim_norm_ubound: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and e :: "real" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net"   and "\<forall>\<^sub>F x in net. norm (f x) \<le> e" shows "norm l \<le> e"
simps_5: fixes g :: "safety"   and r :: "'a regex"   and s :: "'a regex" shows "local.safe_regex Past g (Times r s) = ((g = Lax \<or> fv_regex fv s \<subseteq> fv_regex fv r) \<and> local.safe_regex Past g r \<and> local.safe_regex Past Lax s)"
has_products_def: fixes I :: "'a set" shows "J.has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J J.comp D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. J.has_as_product J D a))"
has_vector_derivative_transform_at: fixes d :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "0 < d"   and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"   and "(f has_vector_derivative f') (at x)" shows "(g has_vector_derivative f') (at x)"
Reflect_def: fixes P' :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" shows "P' P Reflect A B \<equiv> A \<noteq> B \<and> P' P ReflectL A B \<or> A = B \<and> A Midpoint P P'"
ring_iso_set_refl: fixes R :: "('c, 'd) ring_scheme" shows "id \<in> ring_iso R R"
lemma_termdiff4: fixes k :: "real"   and f :: "'a \<Rightarrow> 'b"   and K :: "real" assumes "0 < k"   and "\<And>h. \<lbrakk>h \<noteq> (0::'a); norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h" shows "f \<midarrow>0::'a\<rightarrow> (0::'b)"
senior_monotonic_Suc: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "senior x (Suc n) \<le> senior x n"
inf_right_idem: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># y \<inter># y = x \<inter># y"
sum_prod: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "(\<Sum>x\<in>A. (f x, g x)) = (sum f A, sum g A)"
object_ptr_kinds_eq_h: shows "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h2 \<turnstile> object_ptr_kinds_M|\<^sub>r"
leg1_simps_3: shows "src (tab\<^sub>1 h) = src (tab\<^sub>0 h)"
add_pow_rdistr: fixes a :: "'a"   and b :: "'a"   and k :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<otimes> [k] \<cdot> b = [k] \<cdot> (a \<otimes> b)"
iso_def: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
exec_app: shows "\<Gamma>\<turnstile> \<langle>sequence Seq ((x # xs) @ ys),Normal s\<rangle> =n\<Rightarrow> t"
rv1_Inr_last: fixes s1 :: "'e \<Rightarrow> ('f, 'e, 'd) F1"   and s2 :: "'d \<Rightarrow> ('g, 'e, 'd) F2"   and kl :: "(bd_type_F + bd_type_F) list"   and xb :: "bd_type_F"   and xa :: "'e" shows "\<exists>b1'. rv1 s1 s2 (kl @ [Inr xb]) xa = Inr b1'"
setbydecision_all_def: fixes iface :: "iface"   and rs :: "'a common_primitive rule list"   and dec :: "final_decision" shows "setbydecision_all TYPE('b) iface rs dec = {ip. \<forall>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs Undecided = Decision dec}"
largest_def: fixes x :: "('a, 'b) IEEE.float itself" shows "largest x = 2 ^ (emax x - 1) / 2 ^ bias x * (2 - 1 / 2 ^ LENGTH('b))"
rank_None_before: fixes n :: "nat"   and x :: "nat" assumes "n < x" shows "rank x n = None"
cod_simp: fixes f :: "'a" assumes "S.arr f" shows "S.cod f = C.cod f"
rtrancl_is_UN_relpow: fixes R :: "('a \<times> 'a) set" shows "R\<^sup>* = \<Union> (range ((^^) R))"
a_all_ptrs_in_heap_def: fixes h :: "(_) heap" shows "local.a_all_ptrs_in_heap h = ((\<forall>ptr\<in>fset (object_ptr_kinds h). set |h \<turnstile> get_child_nodes ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)) \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)))"
assms_2: shows "diagram ars (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')"
run_return: fixes x :: "'b"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b" assumes "run (return x) \<sigma> \<sigma>' r" obtains "r = x" and "\<sigma>' = \<sigma>" and "\<not> is_exn \<sigma>"   | "\<sigma> = None"
hypnat_one_not_prime: shows "1 \<notin> starprime"
map_graph_def: fixes f :: "('c \<times> 'b) set"   and G :: "('a, 'c) labeled_graph" shows "map_graph f G = LG (on_triple f `` edges G) (f `` vertices G)"
diff_add_eq_diff_diff_swap: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
fset_from_list_of_fset: fixes s :: "'b fset" shows "fset_from_list (list_of_fset s) = s"
older_seniors_Max_in: fixes x :: "nat"   and n :: "nat" assumes "older_seniors x n \<noteq> {}" shows "Max (older_seniors x n) \<in> older_seniors x n"
hyps_5: shows "valid_path_aux cs' (as' @ [a])"
mult_strict_left_mono_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times c a) (times c b)"
less_imp_le: fixes x :: "'b zmultiset"   and y :: "'b zmultiset" assumes "x \<subset>#\<^sub>z y" shows "x \<subseteq>#\<^sub>z y"
orthogonal_def: fixes x :: "'a"   and y :: "'a" shows "orthogonal x y = (x \<bullet> y = 0)"
add_neg_numeral_special_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) (numeral.numeral one plus n) = neg_numeral.sub minus plus one n num.One"
ax4: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<box>[P \<longrightarrow> Q]_v \<longrightarrow> \<box>[P]_v \<longrightarrow> \<box>[Q]_v"
neg_D: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> neg A"   and "H \<turnstile> A" shows "H \<turnstile> B"
is_nth_power_0_iff: fixes n :: "nat" shows "is_nth_power n (0::'a) = (0 < n)"
Base_foo_member_in_Base: shows "tprg \<turnstile> (Base, mdecl Base_foo) member_in Base"
finite_measure_eq_sum_singleton: fixes S :: "'a set" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "Sigma_Algebra.measure M S = (\<Sum>x\<in>S. Sigma_Algebra.measure M {x})"
y: shows "P y (\<midarrow> x) = (\<not> O y x)"
nfoldli_shift: shows "nfoldli [a..<b] c f = nfoldli (map (\<lambda>i. i + ofs) [a..<b]) c (\<lambda>x s. ASSERT (ofs \<le> x) \<bind> (\<lambda>_. f (x - ofs) s))"
Cs: shows "SimplicialComplex.pmaxsimpchain (f \<turnstile> X) (C0 # Cs)"
in_results'_rpv: fixes x :: "'d"   and rpv :: "('d, 'f, 'e) rpv" shows "(x \<in> results'_rpv rpv) = (\<exists>input. x \<in> results'_gpv (rpv input))"
cong: fixes M :: "'b measure" shows "prob_space.indep M \<equiv> prob_space.indep M"