hcju/isabelleps huggingface dataset:
induce_subgraph_head: fixes G :: "('d, 'c) pre_digraph"   and vs :: "'d set" shows "head (G \<restriction> vs) = head G"
vsubset_vunion_leftD_2: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<union>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> C" shows "B \<subseteq>\<^sub>\<circ> C"
f: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set" assumes "comp_fun_commute_on f A" shows "f \<in> A \<rightarrow> A \<rightarrow> A"
comp_eq_elim: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'f \<Rightarrow> 'd"   and d :: "'c \<Rightarrow> 'f" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
cempty_is_cimage: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d cset" shows "({}\<^sub>c = f `\<^sub>c A) = (A = {}\<^sub>c)"
ring_id_def: shows "ring_id \<equiv> \<lambda>x\<in>carrier R. x"
lnull_lprefix: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lnull xs" shows "lprefix xs ys"
UN_subset_iff: fixes A :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and B :: "'c set" shows "(\<mu> (\<P> A I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<in> AX0"
lsnd_ins_def: fixes x :: "'a"   and l :: "'a list" shows "lsnd_ins x l \<equiv> x # l"
mHom_Z_M: fixes M :: "('f, 'a, 'g) Module_scheme"   and e :: "'h" assumes "R module M" shows "mHom R (Zm R e) M = {mzeromap (Zm R e) M}"
assms_3: shows "(\<lambda>x. sum_upto f (sqrt x) * sum_upto g (sqrt x) - T x) \<in> O(b)"
power_add: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a\<^bsup>m + n\<^esup> = a\<^bsup>m\<^esup> \<cdot> a\<^bsup>n\<^esup>"
partition_in_shuffles: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "xs \<in> shuffles (filter P xs) (filter (\<lambda>x. \<not> P x) xs)"
parts_insert: fixes X :: "msg"   and H :: "msg set" shows "parts (insert X H) = parts_msg X \<union> parts H"
member_inD: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_in C" shows "\<exists>provC. G\<turnstile>C\<preceq>\<^sub>C provC \<and> G \<turnstile> m member_of provC"
cross3_simps_8: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
finite_Int: fixes F :: "'g set"   and G :: "'g set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
subst_apply_strand_def: fixes S :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
vassn_dest_1: fixes \<Gamma>\<^sub>1 :: "assn"   and \<Gamma>\<^sub>2 :: "assn" assumes "vassn_tag (\<Gamma>\<^sub>1 * \<Gamma>\<^sub>2)" shows "vassn_tag \<Gamma>\<^sub>1 \<and> vassn_tag \<Gamma>\<^sub>2"
zero_le_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
rel_conversep: fixes R1 :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_sum R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_sum R1 R2)\<inverse>\<inverse>"
pow_per: fixes y :: "'b list"   and k :: "nat" assumes "y \<noteq> \<epsilon>"   and "k \<noteq> 0" shows "periodN (y \<^sup>@ k) \<^bold>|y\<^bold>|"
subst_fv_imgI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "\<sigma> v \<noteq> Var v" shows "fv (\<sigma> v) \<subseteq> range_vars \<sigma>"
cont2cont_LAM: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. \<Lambda> y. f x y)"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case EQ of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f2"
Units_inv_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
resultant_le_prod_sq_norm: fixes f :: "int poly"   and g :: "int poly" shows "real_of_int \<bar>resultant f g\<bar> \<le> sqrt (real_of_int (\<parallel>f\<parallel>\<^sup>2 ^ degree g * \<parallel>g\<parallel>\<^sup>2 ^ degree f))"
assms_2: shows "\<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r"
arbitrary_intersection_of_Inter: fixes \<U> :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "\<And>S. S \<in> \<U> \<Longrightarrow> (arbitrary intersection_of P) S" shows "(arbitrary intersection_of P) (\<Inter> \<U>)"
derivative_quotient_bound: fixes g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and K :: "real"   and t :: "real" assumes "(g has_vderiv_on g') {a..b}"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> g' t / g t \<le> K"   and "continuous_on {a..b} g'"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> 0 < g t"   and "t \<in> {a..b}" shows "g t \<le> g a * exp (K * (t - a))"
presFreshAbs_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresFreshAbs hA (fromMOD MOD)" shows "presFreshAbs (fromIMorAbs hA) MOD"
simps: fixes M :: "('b, 'a) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t M = \<Union> (fv ` M)"
a: shows "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"
Q2_le_one: shows "Q2 \<le>\<^sub>L 1\<^sub>m d"
pt_list_nil: fixes xs :: "'a list" assumes "pt TYPE('a) TYPE('b)" shows "[] \<bullet> xs = xs"
HC_contrapos_np: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> G \<^bold>\<rightarrow> F" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G"
counit_in_hom_2: shows "\<guillemotleft>D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D trg\<^sub>D (F f)\<guillemotright>"
PiE_eq: fixes I :: "'c set"   and S :: "'c \<Rightarrow> 'd set"   and T :: "'c \<Rightarrow> 'd set" shows "(Pi\<^sub>E I S = Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<and> Pi\<^sub>E I T = {} \<or> (\<forall>i\<in>I. S i = T i))"
l2_zigzag: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and p_p :: "(nat \<times> int) list"   and dir :: "dir"   and p' :: "(nat \<times> int) list"   and ps :: "(nat \<times> int) list" assumes "d < length p"   and "p = child p_p dir d"   and "p' \<in> grid (child p (Grid_Point.inv dir) d) {d}"   and "child p (Grid_Point.inv dir) d = child ps dir d" shows "l2_\<phi> (p' ! d) (p_p ! d) = l2_\<phi> (p' ! d) (ps ! d) + l2_\<phi> (p' ! d) (p ! d) / 2"
int_iffI: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G"   and "\<turnstile> G \<longrightarrow> F" shows "\<turnstile> F = G"
suminf_zero: shows "(\<Sum>n. (0::'b)) = (0::'b)"
inf_commute: fixes x :: "'h multiset"   and y :: "'h multiset" shows "x \<inter># y = y \<inter># x"
that: fixes G1 :: "('v, 'w) graph"   and G2 :: "('v, 'w) graph" assumes "nodes G1 = {n. \<exists>ps. valid_graph.is_path (del_unEdge v w v' G) n ps v}"   and "edges G1 = {(n, e, n'). (n, e, n') \<in> edges (del_unEdge v w v' G) \<and> n \<in> nodes G1 \<and> n' \<in> nodes G1}"   and "nodes G2 = {n. \<exists>ps. valid_graph.is_path (del_unEdge v w v' G) n ps v'}"   and "edges G2 = {(n, e, n'). (n, e, n') \<in> edges (del_unEdge v w v' G) \<and> n \<in> nodes G2 \<and> n' \<in> nodes G2}"   and "edges G1 \<union> edges G2 = edges (del_unEdge v w v' G)"   and "edges G1 \<inter> edges G2 = {}"   and "nodes G1 \<union> nodes G2 = nodes (del_unEdge v w v' G)"   and "nodes G1 \<inter> nodes G2 = {}"   and "valid_unMultigraph G1"   and "valid_unMultigraph G2"   and "valid_unMultigraph.connected G1"   and "valid_unMultigraph.connected G2" shows "thesis"
assms_1: shows "F = ZFC_in_HOL.set {\<langle>j, a\<rangle>}"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_reg w curr_win rd s1)) \<and> t2 = snd (fst (write_reg w curr_win rd s2))"
diagonal_to_Smith_i_PQ_eq: fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and Q :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P' :: "(('a, 'c) vec, 'c) vec"   and B' :: "(('a, 'b) vec, 'c) vec"   and Q' :: "(('a, 'b) vec, 'b) vec"   and xs :: "nat list"   and i :: "nat" assumes "P ** A ** Q = B"   and "invertible P"   and "invertible Q"   and "is_bezout_ext bezout"   and "(P', B', Q') = diagonal_to_Smith_i_PQ xs i bezout (P, B, Q)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "isDiagonal B"   and "i \<notin> set xs"   and "i < min (nrows A) (ncols A)" shows "B' = diagonal_to_Smith_i xs B i bezout"
cblinfun_eq_0_on_canonical_basis: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>u. u \<in> set canonical_basis \<Longrightarrow> f *\<^sub>V u = (0::'b)" shows "f = 0"
negatively_invariant_restrict_dom: fixes M :: "'a set" shows "negatively_invariant M = negatively_invariant (M \<inter> X)"
iso_iff_section_and_retraction: fixes f :: "'a \<times> 'a" shows "VoV.iso f = (VoV.section f \<and> VoV.retraction f)"
fbd_Disj: fixes f :: "'c \<Rightarrow> 'd set" shows "Sup_pres (bd\<^sub>\<F> f)"
MarkGuardsI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat"   and f :: "'f" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P mark_guards f c Q,A"
t1: shows "flow0 y t1 \<in> {a<--<b}"
domain_subset_carrier: fixes c :: "('a, 'e) chart" assumes "c \<in> charts" shows "domain c \<subseteq> carrier"
assms: shows "set {a, b} \<subseteq>\<^sub>\<circ> A"
feasible_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "feasible t = (\<forall>P b. bounded_by b P \<and> nneg P \<longrightarrow> bounded_by b (t P) \<and> nneg (t P))"
valuate_minus: fixes p1 :: "linear_poly"   and p2 :: "linear_poly"   and v :: "nat \<Rightarrow> 'b" shows "p1 - p2 \<lbrace> v \<rbrace> = p1 \<lbrace> v \<rbrace> - p2 \<lbrace> v \<rbrace>"
fv_assignment_rhs_subset_fv_st': fixes S :: "('b, 'a) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (assignment_rhs\<^sub>s\<^sub>t S) \<subseteq> fv\<^sub>s\<^sub>t S"
interpret_floatarith_eventually_isDERIV: fixes fa :: "floatarith"   and x :: "'b"   and params :: "real list" assumes "\<And>i j. i < DIM('b) \<Longrightarrow> isDERIV i fa (list_of_eucl x @ params)"   and "max_Var_floatarith fa \<le> DIM('b) + length params" shows "\<forall>i<DIM('b). \<forall>\<^sub>F x in at x. isDERIV i fa (list_of_eucl x @ params)"
bound_trans: fixes X :: "'a set"   and b :: "'a"   and c :: "'a" assumes "bound X (\<sqsubseteq>) b"   and "b \<sqsubseteq> c"   and "X \<subseteq> A"   and "b \<in> A"   and "c \<in> A" shows "bound X (\<sqsubseteq>) c"
PLemmas_1: shows "C DenyAll = D\<^sub>U"
inject_5: fixes x61 :: "'a formula"   and x62 :: "'a formula"   and y61 :: "'a formula"   and y62 :: "'a formula" shows "(x61 \<^bold>\<rightarrow> x62 = y61 \<^bold>\<rightarrow> y62) = (x61 = y61 \<and> x62 = y62)"
punit_min_term: shows "punit.min_term = (0::'a)"
main: fixes j :: "nat"   and i :: "nat"   and w :: "'c list"   and T :: "nat \<times> nat \<Rightarrow> 'd list"   and G :: "('d \<times> ('d, 'c) RHS) list"   and j' :: "nat"   and i' :: "nat" assumes "1 < j"   and "i + j \<le> length w"   and "\<And>i' j'. \<lbrakk>j' < j; 1 \<le> j'; i' + j' \<le> length w\<rbrakk> \<Longrightarrow> set (T (i', j')) = CYK G w i' j'"   and "\<And>i'. \<lbrakk>i' < i; i' + j \<le> length w\<rbrakk> \<Longrightarrow> set (T (i', j)) = CYK G w i' j"   and "1 \<le> j'"   and "i' + j' \<le> length w" shows "set (main G T (length w) i j (i', j')) = CYK G w i' j'"
set_mset_single: fixes b :: "'k" shows "set_mset {#b#} = {b}"
uw\<theta>w'\<theta>'_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<iota> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and w' :: "'c"   and \<theta>' :: "'c" shows "uw\<theta>w'\<theta>' V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<equiv> tabulation_data V H \<a> \<iota> src trg r \<rho> f g \<and> uw\<theta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> \<and> uw\<theta> V H \<a> \<iota> src trg r \<rho> f g u w' \<theta>'"
Lower_def: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "Lower L A = {l. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> l \<sqsubseteq>\<^bsub>L\<^esub> x} \<inter> carrier L"
eq: shows "wrap_iso_arcs hom f = wrap_iso_arcs hom g"
ltlc_eq_on_aux: fixes \<phi> :: "'b ltlc"   and w :: "nat \<Rightarrow> 'b set"   and w' :: "nat \<Rightarrow> 'b set" assumes "pw_eq_on (atoms_ltlc \<phi>) w w'"   and "w \<Turnstile>\<^sub>c \<phi>" shows "w' \<Turnstile>\<^sub>c \<phi>"
unft: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau> t $ Fv x \<tau>) t"
oalist_inv_list_of_oalist_tc: fixes xs :: "('c, 'd) oalist_tc" shows "tc.oalist_inv_raw (list_of_oalist_tc xs)"
G_simp: fixes f :: "'c" assumes "C.arr f" shows "G f = terminal_arrow_from_functor.the_coext (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go (C.cod f)) (\<epsilon>o (C.cod f)) (Go (C.dom f)) (f \<cdot>\<^sub>C \<epsilon>o (C.dom f))"
of_nat_nat: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus"   and "0 \<le> z" shows "semiring_1.of_nat one plus zero (nat z) = ring_1.of_int one plus zero minus z"
LeastI2_ex: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
HRP_unique: fixes x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{HRP x y, HRP x y'} \<turnstile> y' EQ y"
normalize_simps_9: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (Guard f g c) = Guard f g (Language.normalize c)"
ZHMSet_inject: fixes x :: "hmultiset zmultiset"   and y :: "hmultiset zmultiset" shows "(ZHMSet x = ZHMSet y) = (x = y)"
E': shows "D.internal_equivalence (F f) g' \<phi> \<psi>"
dependent_finite: fixes S :: "'d set" assumes "finite S" shows "cdependent S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>C v) = (0::'d))"
pseudo_divmod_2: fixes g :: "'a poly"   and f :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "g \<noteq> 0"   and "pseudo_divmod f g = (q, r)" shows "r = 0 \<or> degree r < degree g"
assoc_coherence: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
domI: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "m a = \<lfloor>b\<rfloor>" shows "a \<in> dom m"
ternary_lift_1: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv \<noteq> TernaryFalse) = (tv = TernaryFalse \<or> tv = TernaryUnknown)"
eventually_going_to_nhds: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and A :: "'c set" shows "eventually P (f going_to nhds a within A) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>A. f x \<in> S \<longrightarrow> P x))"
IH: fixes m :: "'a match_expr" assumes "\<And>f m m'. \<lbrakk>f \<in> set fs; normalized_nnf_match m; f m = Some m'\<rbrakk> \<Longrightarrow> matches \<gamma> m' a p = matches \<gamma> m a p"   and "\<And>f m. \<lbrakk>f \<in> set fs; normalized_nnf_match m; f m = None\<rbrakk> \<Longrightarrow> \<not> matches \<gamma> m a p"   and "\<And>f m m'. \<lbrakk>f \<in> set fs; normalized_nnf_match m; f m = Some m'\<rbrakk> \<Longrightarrow> normalized_nnf_match m'"   and "normalized_nnf_match m"   and "compress_normalize_primitive_monad fs m = None" shows "\<not> matches \<gamma> m a p"
set_iterator_weaken_R: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R"   and "\<And>x y. \<lbrakk>x \<in> S0; y \<in> S0; R x y\<rbrakk> \<Longrightarrow> R' x y" shows "set_iterator_genord iti S0 R'"
Abs_nat_int_inverse: fixes y :: "nat set" assumes "y \<in> {S. \<exists>m n. {m..n} = S}" shows "Rep_nat_int (Abs_nat_int y) = y"
nnegD2: fixes P :: "'c \<Rightarrow> 'd" assumes "nneg P" shows "(\<lambda>s. 0::'d) \<le> P"
sees_field_def: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C sees F,b:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of (map (\<lambda>((F, D), b, T). (F, D, b, T)) FDTs) F = \<lfloor>(D, b, T)\<rfloor>"
orbit_size: fixes m :: "'a set" assumes "finite (carrier G)"   and "m \<in> subgroups_of_size (p ^ a)" shows "order G = card (conjG.orbit m) * card (conjG.stabilizer m)"
densinI: fixes B :: "'b set"   and T :: "'b set set"   and A :: "'b set" assumes "B \<subseteq> closure\<^bsub>T\<^esub> A" shows "A densein\<^bsub>T\<^esub> B"
prj1_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "cospan f g"   and "a = f \<down>\<down> g"   and "b = local.dom f" shows "\<guillemotleft>\<p>\<^sub>1[f, g] : a \<rightarrow> b\<guillemotright>"
delete_edge_def: fixes v :: "'d"   and e :: "'e"   and v' :: "'d"   and g :: "('d, 'e, 'f) graph_scheme" shows "delete_edge v e v' g \<equiv> \<lparr>nodes = nodes g, edges = edges g - {(v, e, v')}\<rparr>"
Lcm_insert: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm (insert a A) = lcm a (Lcm A)"
star_of_inverse: fixes x :: "'a" shows "star_of (inverse x) = inverse (star_of x)"
mirror_elem_def: fixes n :: "nat"   and I :: "nat set" shows "mirror_elem n I \<equiv> nat_mirror n (iMin I) (Max I)"
stable_pair_on_allocation: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" shows "inj_on Xd (fst XD_XH \<inter> snd XD_XH)"
L0: fixes w :: "letter list"   and v :: "letter list"   and ws :: "letter list list" assumes "emb w v" shows "L v (w # ws)"
inducts_1: fixes x1 :: "('exp, 'id) MWLsCom"   and P1 :: "('exp, 'id) MWLsCom \<Rightarrow> bool"   and P2 :: "('exp, 'id) MWLsCom list \<Rightarrow> bool" assumes "\<turnstile>\<^bsub>\<C>\<^esub> x1"   and "\<And>\<iota>. P1 (skip\<^bsub>\<iota>\<^esub>)"   and "\<And>x e \<iota>. AssignSideCondition x e \<iota> \<Longrightarrow> P1 (x:=\<^bsub>\<iota>\<^esub> e)"   and "\<And>V \<iota>. \<lbrakk>\<turnstile>\<^bsub>\<V>\<^esub> V; P2 V\<rbrakk> \<Longrightarrow> P1 (spawn\<^bsub>\<iota>\<^esub> V)"   and "\<And>c1 c2. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c1; P1 c1; \<turnstile>\<^bsub>\<C>\<^esub> c2; P1 c2\<rbrakk> \<Longrightarrow> P1 (c1;c2)"   and "\<And>c b \<iota>. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c; P1 c; WhileSideCondition b\<rbrakk> \<Longrightarrow> P1 (while\<^bsub>\<iota>\<^esub> b do c od)"   and "\<And>c1 c2 b \<iota>. \<lbrakk>\<turnstile>\<^bsub>\<C>\<^esub> c1; P1 c1; \<turnstile>\<^bsub>\<C>\<^esub> c2; P1 c2; IfSideCondition b c1 c2\<rbrakk> \<Longrightarrow> P1 (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi)"   and "\<And>V. \<forall>i<length V. \<turnstile>\<^bsub>\<C>\<^esub> V ! i \<and> P1 (V ! i) \<Longrightarrow> P2 V" shows "P1 x1"
numeral_Bit0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
singleton_conv2: fixes a :: "'c" shows "{x. a = x} = \<eta> a"
flowD_outside: fixes g :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and e :: "'v vertex \<times> 'v vertex" assumes "flow \<Psi> g"   and "e \<notin> \<^bold>E\<^bsub>\<Psi>\<^esub>" shows "g e = 0"
D: shows "cfs ! Suc k = (size (acomp a\<^sub>1 @ acomp a\<^sub>2 @ (if f then [JMPLESS i] else [JMPGE i])) + (if bval (Less a\<^sub>1 a\<^sub>2) s = f then i else 0), s, stk)"
assms: fixes x :: "'a"   and y :: "'a" shows "R x y = (R x = R y)"
indep_sets_cong: fixes I :: "'e set"   and J :: "'e set"   and F :: "'e \<Rightarrow> 'a set set"   and G :: "'e \<Rightarrow> 'a set set" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> F i = G i" shows "indep_sets F I = indep_sets G J"
commutative: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" assumes "x \<in> carrier_local_ring_at"   and "y \<in> carrier_local_ring_at" shows "add_local_ring_at x y = add_local_ring_at y x"
det_echelon_form_of_euclidean_iarrays: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det A = (let A' = echelon_form_of_det_iarrays (matrix_to_iarray A) euclid_ext2 in (1::'b) div fst A' * (\<Prod>i\<leftarrow>[0..<nrows_iarray (matrix_to_iarray A)]. snd A' !! i !! i))"
item_nonterminal_def: fixes x :: "item" shows "item_nonterminal x = fst (item_rule x)"
divides_refl: fixes a :: "'a list" assumes "a \<in> carrier (poly_ring R)" shows "a divides\<^bsub>poly_ring R\<^esub> a"
fmul2fadd1fold: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat"   and x :: "'c" shows "fmul2 (foldseq fadd1 s n) x = foldseq fadd1 (\<lambda>k. fmul2 (s k) x) n"
tso_lock_invL_def: fixes m :: "'d" shows "mut_m.tso_lock_invL m = (\<lambda>s. (atS (mutator m) mut_m.tso_lock_locs s \<longrightarrow> sys_mem_lock s\<down> = Some (mutator m)) \<and> (atS (mutator m) (- mut_m.tso_lock_locs) s \<longrightarrow> sys_mem_lock s\<down> \<noteq> Some (mutator m)))"
B'_def: fixes x :: "'a" shows "B' x = {y \<in> B x. (0::'c) \<noteq> f x y}"
assms: shows "H = (\<lambda>s. G s \<and> Q s)"