hcju/isabelleps huggingface dataset:
commit_correct: fixes \<eta> :: "nat" shows "okamoto_\<Sigma>_commit.abstract_com.correct \<eta>"
order_one_closed: shows "(coreflexive (1::'a) \<and> transitive (1::'a)) \<and> antisymmetric (1::'a)"
eq: fixes S :: "'a set"   and T :: "'b set" assumes "homeomorphism S T f g" shows "P S = Q T"
uint32_simps_4: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x * y) = int_of_uint32 x * int_of_uint32 y mod 4294967296"
full_poly_assn_def: shows "full_poly_assn = hr_comp poly_assn (fully_unsorted_poly_rel O mset_poly_rel)"
trivial_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and inEdges' :: "'c \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) list"   and defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'f set"   and "uses" :: "'c \<Rightarrow> 'd \<Rightarrow> 'f set"   and phis :: "'c \<Rightarrow> 'd \<times> 'f \<Rightarrow> 'f list option"   and g :: "'c"   and v :: "'f" shows "CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g v \<equiv> \<exists>v'\<in>CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g. CFG_SSA_wf_base.isTrivialPhi \<alpha>n defs phis g v v'"
assms_3: shows "\<forall>x. S2 x \<le> S1 x"
inject: fixes x1 :: "'g"   and y1 :: "'g" shows "(Node x1 = Node y1) = (x1 = y1)"
lS: shows "\<forall>a\<in>S. linear (*a) (*b) (f a)"
section_retraction_of_iso_1: fixes f :: "'a1"   and g :: "'a1" assumes "C1.iso f"   and "C1.ide (g \<cdot>\<^sub>1 f)" shows "C1.inverse_arrows f g"
elims_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x = y" obtains "x = []" and "y"   | xa :: "'c" and ys :: "'c list" where "x = xa # ys" and "y = (Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys)"
clauseFalseIffAllLiteralsAreFalse: fixes clause :: "Literal list"   and valuation :: "Literal list" shows "clauseFalse clause valuation = (\<forall>literal. literal el clause \<longrightarrow> literalFalse literal valuation)"
scaleC_left: fixes r :: "complex"   and a :: "'a"   and b :: "'b" shows "r *\<^sub>C a ** b = r *\<^sub>C (a ** b)"
fbox_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<le> d y" shows " |z] x \<le>  |z] y"
not_gr_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
permutations_of_setD_1: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<in> permutations_of_set A" shows "set xs = A"
multi_psub_of_add_self: fixes A :: "'a multiset"   and x :: "'a" shows "A \<subset># x, A"
vdomain_vcomp_vsubset: fixes s :: "V"   and r :: "V" assumes "\<R>\<^sub>\<circ> s \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "\<D>\<^sub>\<circ> (r \<circ>\<^sub>\<circ> s) = \<D>\<^sub>\<circ> s"
rel_refl_strong: fixes x :: "'b pmf"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> Ra z z" shows "rel_pmf Ra x x"
below_set_def: fixes A :: "'d set"   and B :: "'d set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
graph_edges_it_axioms: shows "graph_edges_it gbm_\<alpha> gbm_invar gbm_edges_list_it"
lcons_inv: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.lcons a t)"
star_rtc3: fixes x :: "'a ppath set" shows "(pp_one \<union> pp_prod x x = x) = (x = pp_star x)"
SC_WbisT_Seq: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_WbisT (c1 ;; c2) = (SC_WbisT c1 \<and> SC_WbisT c2)"
healthy_bounded_byD2: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "sound P" shows "bounded_by b (t P)"
bv_msb_Nil: shows "bv_msb [] = \<zero>"
MkArr_in_hom: fixes A :: "'a E.term"   and B :: "'a E.term"   and f :: "'a" assumes "A \<in> IDE"   and "B \<in> IDE"   and "f \<in> HOM A B" shows "\<guillemotleft>MkArr A B f : MkIde A \<Rightarrow> MkIde B\<guillemotright>"
row_space_eq_col_space_transpose: fixes n :: "nat"   and A :: "'b mat" shows "vec_space.row_space n A = vec_space.col_space n A\<^sup>T"
acyclic_4a_implies_4: fixes x :: "'a" assumes "acyclic_4a x" shows "acyclic_4 x"
fsigma_Union_compact: fixes S :: "'a set" shows "fsigma S = (\<exists>F. range F \<subseteq> Collect compact \<and> S = \<Union> (range F))"
weakly_fair_def: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "weakly_fair enabled taken = \<box>(\<lambda>s. (\<box>enabled) s \<longrightarrow> (\<diamond>taken) s)"
eq: fixes x :: "real" shows "x\<^sup>2 ^ k = (x ^ k)\<^sup>2"
fBexE: fixes A :: "'c fset"   and P :: "'c \<Rightarrow> bool" assumes "fBex A P" obtains x :: "'c" where "x |\<in>| A" and "P x"
scaleR_eq_iff: fixes b :: "'b"   and u :: "real"   and a :: "'b" shows "(b + u *\<^sub>R a = a + u *\<^sub>R b) = (a = b \<or> u = 1)"
lex_comp_alt: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "lex_comp = linorder.comparator_of less"
x_plus_helpers_12: fixes x :: "nat" shows "15 + x = Suc (x + 14)"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c fundef"   and b :: "'d fundef" shows "rel_fundef R a b = (list_all2 R (body a) (body b) \<and> arity a = arity b)"
sep_implI: fixes h :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>h' h''. \<lbrakk>h \<oplus> h' = Some h''; P h'\<rbrakk> \<Longrightarrow> Q h''" shows "(P \<longrightarrow>\<^sup>* Q) h"
insort_insert_insort: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "insort_insert x xs = insort x xs"
simps_2: fixes l :: "('b ivl \<times> 'b) tree"   and a :: "'b ivl"   and m :: "'b"   and r :: "('b ivl \<times> 'b) tree" shows "inv_max_hi \<langle>l, (a, m), r\<rangle> = (m = max3 a (max_hi l) (max_hi r) \<and> inv_max_hi l \<and> inv_max_hi r)"
max_def: fixes a :: "'a"   and b :: "'a" shows "max a b = (if a \<le> b then b else a)"
star_of_approx_iff: fixes x :: "'a"   and y :: "'a" shows "(star_of x \<approx> star_of y) = (x = y)"
assms_2: shows "v \<in> fmdom' (snd a)"
get_document_ptr_simp3: fixes element_ptr :: "(_) element_ptr"   and document_ptr :: "(_) document_ptr"   and f :: "('m, 'n, 'o, 'v option) RDocument_scheme"   and h :: "(_) heap" shows "get element_ptr (put document_ptr f h) = get element_ptr h"
assms_9: shows "\<parallel>f\<parallel>\<^sup>2 ^ k * \<parallel>g\<parallel>\<^sup>2 ^ n < m\<^sup>2"
simulation_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto>[Rel] Q \<equiv> (\<forall>a x Q'. Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel)) \<and> (\<forall>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
preserves_cod: fixes f :: "('a, 'a) CC.arr" assumes "CC.arr f" shows "cod (if CC.arr f then CC.Map f else null) = (if CC.arr (CC.cod f) then CC.Map (CC.cod f) else null)"
coincidence_sterm': fixes \<theta> :: "('d, 'e) trm"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp"   and J :: "('d, 'f, 'e) interp" assumes "dfree \<theta>"   and "Vagree \<nu> \<nu>' (FVT \<theta>)"   and "Iagree I J {Inl x |x. x \<in> SIGT \<theta>}" shows "sterm_sem I \<theta> (fst \<nu>) = sterm_sem J \<theta> (fst \<nu>')"
mem_Collect_eq: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "s2p (Collect P) a = P a"
drlex_pm_strict_def: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "drlex_pm_strict s t = strict drlex_pm s t"
inj_hom_dimension: fixes E :: "'a set"   and n :: "nat" assumes "inj_on h E"   and "R.dimension n K E" shows "S.dimension n (h ` K) (h ` E)"
le_not_equiv_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "\<not> a \<approx> b" shows "a < b"
rep_eq: fixes x :: "'a reg_lan"   and xa :: "'a reg_lan" shows "(x \<le> xa) = (Rep_reg_lan x \<subseteq> Rep_reg_lan xa)"
S: shows "S \<equiv> {I n |n. n \<in> UNIV}"
numeral_eq_one_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus n = one) = (n = num.One)"
mId_mHom: shows "mId\<^bsub>M\<^esub>  \<in> mHom R M M"
rel_sum_simps_2: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b"   and b2 :: "'e" shows "rel_sum R1 R2 (Inl a1) (Inr b2) = False"
alph: shows "{u\<^sub>0, u\<^sub>1} = {x, y}"
test_comp_test_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> (1::'a)) * y * (z \<sqinter> (1::'a)) = (x \<sqinter> (1::'a)) * y \<sqinter> y * (z \<sqinter> (1::'a))"
inCounterM: fixes v :: "vbl"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "counterAssign v \<in> objects (counterModel f)"
skip: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and t :: "state" shows "\<gamma>,p\<turnstile> \<langle>[], t\<rangle> \<Rightarrow>\<^sub>\<alpha> t"
bv_prepend_drop: fixes x :: "nat"   and y :: "nat"   and b :: "'a" shows "x \<le> y \<longrightarrow> bv_prepend x b (drop x (bv_prepend y b [])) = bv_prepend y b []"
distinct_27: fixes x31 :: "POPLmarkRecord.type"   and x32 :: "trm"   and x41 :: "trm"   and x42 :: "trm" shows "(\<lambda><:x31. x32) \<noteq> x41 \<bullet> x42"
len: shows "e_length (e_update (block i t) (gap i t) 1) = Suc (t + 1)"
coprime_self: 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 a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a a = dvd.dvd times a one"
path_component_of_eq: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "(path_component_of X x = path_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y)"
vector_to_cblinfun_apply_one_dim: fixes \<phi> :: "'a"   and \<gamma> :: "'b" shows "vector_to_cblinfun \<phi> *\<^sub>V \<gamma> = one_dim_iso \<gamma> *\<^sub>C \<phi>"
inject: fixes x2 :: "nat"   and y2 :: "nat" shows "([x2]\<^sub>1 = [y2]\<^sub>1) = (x2 = y2)"
id_lens_def: shows "\<Sigma> = \<lparr>lens_get = id, lens_put = \<lambda>_. id\<rparr>"
fun_value_in_range: fixes f :: "ZF"   and x :: "ZF" assumes "isFun f"   and "x |\<in>| HOLZF.Domain f" shows "f \<acute> x |\<in>| HOLZF.Range f"
assms_2: shows "\<forall>A\<in>F. A \<subseteq> E \<and> card A \<le> k"
eval_cexpr_comp: fixes f :: "cexpr"   and g :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val"   and x :: "val" shows "eval_cexpr (f \<circ>\<^sub>c g) \<sigma> x = eval_cexpr f \<sigma> (cexpr_sem (case_nat x \<sigma>) g)"
direct_lim_is_ring: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I" shows "ring (dlim \<FF> \<rho> I) (direct_lim.add_rel \<FF> \<rho> add_str I) (direct_lim.mult_rel \<FF> \<rho> mult_str I) (direct_lim.class_of \<FF> \<rho> I U (zero_str U)) (direct_lim.class_of \<FF> \<rho> I U (one_str U))"
simps_2: fixes x :: "'a"   and l :: "'a tree"   and r :: "'a tree"   and h :: "nat" shows "hinv (MKT x l r h) = (h = 1 + max (height (erase l)) (height (erase r)) \<and> hinv l \<and> hinv r)"
sorted_iff_nth_mono: fixes xs :: "'a list" shows "sorted xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)"
Sup_le_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and b :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
fun_relE2: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'g"   and Rv :: "('f \<times> 'g) set"   and t :: "'f"   and x :: "'e" assumes "(f, f') \<in> Id \<rightarrow> Rv"   and "t = f x" shows "(t, f' x) \<in> Rv"
mult_zero_right: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.mult_zero times zero" shows "times a zero = zero"
simps_3: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "aval (Plus a1 a2) s = value_plus (aval a1 s) (aval a2 s)"
FP: shows "\<Union> (F ` P) = {..<brn d1}"
up_filter: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order less_eq less" shows "ord.filter less_eq {y. less_eq x y}"
conj_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x AND y) AND z = x AND y AND z"
cindexE_cong: fixes s :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "finite s"   and "\<And>x. \<lbrakk>a < x; x < b; x \<notin> s\<rbrakk> \<Longrightarrow> f x = g x" shows "cindexE a b f = cindexE a b g"
two_less_eq_exp_length: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) LENGTH('b))"
InputsOrNi: shows "UNIV = Inputs \<union> {NotAnInput}"
dvdm_def: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.dvdm m f g = (\<exists>h. poly_mod.Mp m g = poly_mod.Mp m (f * h))"
sin_cos_squared_add2: fixes x :: "'a" shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = (1::'a)"
compose_def: fixes A :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" shows "compose A g f = (\<lambda>x\<in>A. g (f x))"
l_def: shows "l = (\<lambda>x. liminf (\<lambda>xa. ereal (u xa x / real xa)))"
Spec: fixes P :: "'d set"   and r :: "('d \<times> 'd) set"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> {s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Spec r Q,A"
list_to_fun_Nil: fixes cs :: "'d list" shows "list_to_fun [] cs = 0"
reduce_idem: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and E :: "'a literal list" shows "reduce Ds C (reduce Ds C E) = reduce Ds C E"
cnf_of_encode_goal_state_set_1: fixes \<Pi> :: "'b strips_problem"   and v :: "'b"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "v \<in> dom (\<Pi>\<^sub>G)" shows "(\<Pi>\<^sub>G) v = Some True \<longrightarrow> (\<exists>!C. C \<in> cnf ((\<Phi>\<^sub>G \<Pi>) t) \<and> C = {(State t (index (\<Pi>\<^sub>\<V>) v))\<^sup>+})"
simple_function_compose: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "simple_function M f" shows "simple_function M (g \<circ> f)"
pnormal_last_length: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times"   and "0 < length p"   and "last p \<noteq> zero" shows "semiring_0.pnormal zero p"
inf_top_right: fixes x :: "'a" shows "inf_class.inf x top_class.top = x"
quasi_comm: shows "y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>"
assms_2: fixes H :: "circline"   and M :: "moebius" assumes "P H" shows "P (moebius_circline M H)"
guha_equal_NoAction: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d" assumes "no_overlaps \<gamma> ft"   and "OF_priority_match \<gamma> ft p = NoAction" shows "guha_table_semantics \<gamma> ft p None"
chine_composite'_2: shows "\<mu>\<nu>_\<pi>.chine = \<langle>\<langle>\<mu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1\<rbrakk> \<nu>.chine \<cdot> doms.Prj\<^sub>0\<^sub>1\<rangle> \<lbrakk>\<nu>.cod.leg0 \<cdot> \<p>\<^sub>0[\<mu>.cod.leg0, \<nu>.cod.leg1], \<pi>.cod.leg1\<rbrakk> \<pi>.chine \<cdot> doms.Prj\<^sub>0\<rangle>"
subst_cls_list_nth: fixes i :: "nat"   and Cs :: "'a literal multiset list"   and \<tau> :: "'s" assumes "i < length Cs" shows "(Cs \<cdot>cl \<tau>) ! i = Cs ! i \<cdot> \<tau>"
adapt_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "adapt_stoch_proc F X N" shows "\<forall>n. random_variable N (X n)"
name_set_fresh_fresh: fixes pi :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and x :: "'e" assumes "set pi \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "pi \<bullet> x = x"
measurable_compose: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and L :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M L"
dX_range': fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> Singleton_for_d.dX X d" shows "x \<in> X \<and> Xd x = d"
tensor_mat_def: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" shows "A \<Otimes> B \<equiv> Tensor.mat_of_cols_list (dim_row A * dim_row B) (mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B))"
set_of_paths_finite: fixes M :: "('d, 'e, 'f) FSM"   and q1 :: "'f"   and q2 :: "'f"   and k :: "nat" assumes "well_formed M"   and "q1 \<in> nodes M" shows "finite {p. path M p q1 \<and> target p q1 = q2 \<and> length p \<le> k}"
Lcm_subset: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "A \<subseteq> B" shows "dvd.dvd times (Lcm A) (Lcm B)"
trie_odlist_update_with_def: shows "trie_odlist_update_with = (\<lambda>k. trie_update_with' (toList k))"
matcher_agree_on_exact_matches_alt: fixes exact :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and approx :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue" shows "matcher_agree_on_exact_matches exact approx = (\<forall>p m. approx m p \<noteq> TernaryUnknown \<longrightarrow> bool_to_ternary (exact m p) = approx m p)"
Fix_lsl_lsu: shows "Fix \<nu> = Fix \<nu>\<^sup>\<natural>"
prob_le_1: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M" shows "Sigma_Algebra.measure M A \<le> 1"
nabla_noether_iff: fixes x :: "'a" shows "Noetherian x = (\<nabla>x = (zero_class.zero::'a))"
matrix_vector_mult_diff_distrib: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "('b, 'd) Finite_Cartesian_Product.vec"   and y :: "('b, 'd) Finite_Cartesian_Product.vec" shows "A *v (x - y) = A *v x - A *v y"
intros: fixes C :: "sequent"   and A :: "form"   and Ps :: "sequent list" assumes "C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>)" shows "rightPrincipal (Ps, C) A"
AnnConseq: fixes c :: "'b ann_com"   and q :: "'b set"   and q' :: "'b set" assumes "\<turnstile> c  q"   and "q \<subseteq> q'" shows "\<turnstile> c  q'"
set_minus_filter_out: fixes xs :: "'g list"   and y :: "'g" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
assms: fixes a :: "real"   and b :: "real" shows "{x \<in> S. a \<le> f x \<and> f x < b} \<in> sets (lebesgue_on S)"
epm_12: fixes v :: "'b"   and w :: "'b"   and e :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.vector v" shows "w \<sqinter> - stone_algebra_class.comp_inf.prim_EP w v e \<sqinter> stone_algebra_class.comp_inf.prim_E w v e = bot"
hmem_ne: fixes x :: "hf"   and y :: "hf" assumes "x \<^bold>\<in> y" shows "x \<noteq> y"
get_owner_document_def: shows "Core_DOM_Functions.get_owner_document \<equiv> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document Core_DOM_Functions.get_root_node get_disconnected_nodes"
rt: shows "s \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U"
unfold_congs_3: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "l r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "l_update f r = l_update f' r'"
IH: fixes k :: "nat"   and m :: "nat" assumes "k < K" shows "dyad_rec_dom (b, l, r, real m / 2 ^ k)"
is_root_poly_mult_imp_is_root: fixes p :: "'a list"   and q :: "'a list"   and x :: "'a" assumes "p \<in> carrier (poly_ring R)"   and "q \<in> carrier (poly_ring R)"   and "is_root (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x" shows "is_root p x \<or> is_root q x"
power_add: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a +^ (m + n) = a +^ m + a +^ n"
prime_elem_multiplicity_mult_distrib: fixes p :: "'a"   and x :: "'a"   and y :: "'a" assumes "prime_elem p"   and "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "multiplicity p (x * y) = multiplicity p x + multiplicity p y"