hcju/isabelleps huggingface dataset:
MPLS_charles_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'c) fmap) set" shows "MPLS_charles PROB \<equiv> {Inf (PLS_charles (fst p) (snd p) PROB) |p. fst p \<in> valid_states PROB \<and> snd p \<in> valid_plans PROB}"
\<epsilon>_nemp: shows "\<epsilon> \<noteq> {#}"
derivation_zero_localI: fixes W :: "'a set"   and p :: "'a"   and X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and f :: "'a \<Rightarrow> real" assumes "open W"   and "p \<in> W"   and "W \<subseteq> carrier"   and "X \<in> tangent_space p"   and "f \<in> diff_fun_space"   and "\<And>x. x \<in> W \<Longrightarrow> f x = 0" shows "X f = 0"
diff_iff: fixes f :: "'b \<Rightarrow> 'c" shows "diff_fun k charts2 f = (\<forall>x\<in>dest.carrier. \<exists>c1\<in>dest.atlas. \<exists>c2\<in>src.manifold_eucl.dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
simps_1: fixes \<psi> :: "'b ltln"   and Y :: "'b ltln set" shows "(X\<^sub>n \<psi>)[Y]\<^sub>\<mu> = X\<^sub>n \<psi>[Y]\<^sub>\<mu>"
compl_sup: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (sup x y) = inf (uminus x) (uminus y)"
abs_summable_on_product: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "countable A"   and "countable B"   and "f abs_summable_on A"   and "g abs_summable_on B" shows "(\<lambda>(x, y). f x * g y) abs_summable_on A \<times> B"
Keys_Un: fixes A :: "('c \<Rightarrow>\<^sub>0 'd) set"   and B :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "Keys (A \<union> B) = Keys A \<union> Keys B"
sigma_set_inter_init: fixes n :: "nat"   and A :: "nat \<Rightarrow> 'b set"   and sp :: "'b set"   and B :: "'b set set" assumes "\<And>i. i \<le> n \<Longrightarrow> A i \<in> sigma_sets sp B"   and "B \<subseteq> Pow sp" shows "\<Inter> (A ` {m. m \<le> n}) \<in> sigma_sets sp B"
conjug_sym: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v" shows "v \<sim> u"
HAStates_CompFun_States_ChiRel: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> HAStates A" shows "\<Union> (States ` the (CompFun A S)) = ChiRel A `` {S}"
ide_self_inverse: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.inverse_arrows a a"
ik_cons: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t (x # S) = ik\<^sub>s\<^sub>t [x] \<union> ik\<^sub>s\<^sub>t S"
lappend_strict': shows "lappend\<cdot>\<bottom> = (\<Lambda> a. \<bottom>)"
simps_13: shows "pratt2_max \<infinity>\<^sub>2 \<top>\<^sub>2 = \<top>\<^sub>2"
exp_conj_sound: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (P && Q)"
induct: fixes x1 :: "'b com"   and x2 :: "'b set"   and x3 :: "('b \<times> 'b) set"   and x4 :: "('b \<times> 'b) set"   and x5 :: "'b set"   and P :: "'b com \<Rightarrow> 'b set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> 'b set \<Rightarrow> bool" assumes "\<turnstile> x1 sat [x2, x3, x4, x5]"   and "\<And>pre f post guar rely. \<lbrakk>pre \<subseteq> {s. f s \<in> post}; {(s, t). s \<in> pre \<and> (t = f s \<or> t = s)} \<subseteq> guar; stable pre rely; stable post rely\<rbrakk> \<Longrightarrow> P (Basic f) pre rely guar post"   and "\<And>Pa pre rely guar mid Q post. \<lbrakk>\<turnstile> Pa sat [pre, rely, guar, mid]; P Pa pre rely guar mid; \<turnstile> Q sat [mid, rely, guar, post]; P Q mid rely guar post\<rbrakk> \<Longrightarrow> P (Seq Pa Q) pre rely guar post"   and "\<And>pre rely P1 b guar post P2. \<lbrakk>stable pre rely; \<turnstile> P1 sat [pre \<inter> b, rely, guar, post]; P P1 (pre \<inter> b) rely guar post; \<turnstile> P2 sat [pre \<inter> - b, rely, guar, post]; P P2 (pre \<inter> - b) rely guar post; \<forall>s. (s, s) \<in> guar\<rbrakk> \<Longrightarrow> P (Cond b P1 P2) pre rely guar post"   and "\<And>pre rely b post Pa guar. \<lbrakk>stable pre rely; pre \<inter> - b \<subseteq> post; stable post rely; \<turnstile> Pa sat [pre \<inter> b, rely, guar, pre]; P Pa (pre \<inter> b) rely guar pre; \<forall>s. (s, s) \<in> guar\<rbrakk> \<Longrightarrow> P (While b Pa) pre rely guar post"   and "\<And>pre rely post Pa b guar. \<lbrakk>stable pre rely; stable post rely; \<forall>V. \<turnstile> Pa sat [pre \<inter> b \<inter> {V}, {(x, y). x = y}, UNIV, {s. (V, s) \<in> guar} \<inter> post] \<and> P Pa (pre \<inter> b \<inter> {V}) {(x, y). x = y} UNIV ({s. (V, s) \<in> guar} \<inter> post)\<rbrakk> \<Longrightarrow> P (Await b Pa) pre rely guar post"   and "\<And>pre pre' rely rely' guar' guar post' post Pa. \<lbrakk>pre \<subseteq> pre'; rely \<subseteq> rely'; guar' \<subseteq> guar; post' \<subseteq> post; \<turnstile> Pa sat [pre', rely', guar', post']; P Pa pre' rely' guar' post'\<rbrakk> \<Longrightarrow> P Pa pre rely guar post" shows "P x1 x2 x3 x4 x5"
assms_2: shows "e ; t\<^sup>T \<le> 1'"
wf_nFEx: fixes idx :: "'n"   and b :: "'k"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx (FEx b \<phi>)" shows "local.wf idx (nFEx b \<phi>)"
keys_pm_of_poly: fixes x :: "'a"   and p :: "'b poly" shows "keys (pm_of_poly x p) = Poly_Mapping.single x ` {d. poly.coeff p d \<noteq> (0::'b)}"
z_signs: fixes signs :: "rat list"   and I :: "nat list" assumes "list_all (\<lambda>i. i < length signs) I"   and "list_all (\<lambda>s. s = 1 \<or> s = - 1) signs" shows "z I signs = 1 \<or> z I signs = - 1"
poly_card_roots_greater: fixes a :: "real"   and p :: "real poly" shows "card {x. a < x \<and> poly p x = 0} = count_roots_above p a"
fps_inverse_mult: fixes f :: "'b fps"   and g :: "'b fps" shows "inverse (f * g) = inverse f * inverse g"
hyps: assumes "\<not> c \<le> d"   and "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))" shows "(shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}))) = []) = (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})) = [])"
conversionE: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*" obtains "(a, b) \<in> (A\<^sup>\<leftrightarrow>)\<^sup>*"
cases: fixes x :: "interpolation_algorithm" obtains "x = Newton"   | "x = Lagrange"   | "x = Neville_Aitken"
chart_homeomorphism: fixes c :: "('c, 'd) chart" shows "homeomorphism (domain c) (codomain c) (apply_chart c) (inv_chart c)"
concurrent_operations_commute: fixes xs :: "('a \<times> operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "hb.concurrent_ops_commute (node_deliver_messages xs)"
bij_inv_eq_iff: fixes p :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "bij p" shows "(x = inv p y) = (p x = y)"
part_equiv_trans_sym_1: fixes R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "part_equiv R"   and "(a, b) \<in> R"   and "(c, b) \<in> R" shows "(a, c) \<in> R"
assms: fixes x :: "'a" assumes "x \<in> X" shows "f x \<le> \<Down> R M"
sel_1: fixes x1 :: "'a" shows "terminal (TNil x1) = x1"
intros_9: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset"   and G :: "'a formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>"
power_one_right: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "power.power one times a 1 = a"
mult_eq_implies_one_or_zero_step: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mult_eq r" shows "\<exists>I J K. N = I + J \<and> M = I + K \<and> set_mset K \<subseteq> r \<down>m J"
hyps: shows "finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>m. f i = 0)}"
equiv_sym: fixes c :: "com"   and c' :: "com" assumes "c \<sim> c'" shows "c' \<sim> c"
induct: fixes P :: "'b tree list \<Rightarrow> bool"   and a0 :: "'b tree list" assumes "\<And>t. P [t]"   and "\<And>t v va. P (v # va) \<Longrightarrow> P (t # v # va)"   and "P []" shows "P a0"
inject_2: fixes x3 :: "int"   and y3 :: "int" shows "(IntVal x3 = IntVal y3) = (x3 = y3)"
splitFace_nonFinal_face: fixes g :: "graph"   and i :: "nat"   and j :: "nat"   and f' :: "face"   and ns :: "nat list" shows "\<not> final (fst (snd (splitFace g i j f' ns)))"
ide_dom: fixes f :: "'a2" assumes "C2.arr f" shows "C2.ide (C2.dom f)"
assms_3: shows "\<forall>m. sigma_finite_subalgebra M (F m)"
loeb_aux_bprv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv (\<phi>L \<phi>)" shows "bprv \<phi>"
RedCondT: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>if (true) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<rangle>"
red_Src: fixes t :: "'c local.term" assumes "Ide t" shows "Src t\<^bold>\<down> = Src t"
assms_6: shows "a \<otimes> h [^] e' = \<^bold>g [^] z'"
seq_distrib_left_axioms_def: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" shows "seq_distrib_left_axioms (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<equiv> \<forall>(c::'b::refinement_lattice) (d\<^sub>0::'b::refinement_lattice) d\<^sub>1::'b::refinement_lattice. seq c (d\<^sub>0 \<sqinter> d\<^sub>1) \<sqsubseteq> seq c d\<^sub>0 \<sqinter> seq c d\<^sub>1"
healthy_boundedD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "Expectations.bounded (t P)"
map_upds_fold_map_upd: fixes m :: "'c \<Rightarrow> 'd option"   and ks :: "'c list"   and vs :: "'d list" shows "m(ks [\<mapsto>] vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
simps_4: fixes I :: "('a, 'b, 'c) interp"   and t1 :: "('a, 'c) trm"   and t2 :: "('a, 'c) trm"   and v :: "(real, 'c) vec" shows "sterm_sem I (Times t1 t2) v = sterm_sem I t1 v * sterm_sem I t2 v"
content_times_primitive_part_fract: fixes p :: "'a fract poly" shows "smult (fract_content p) (map_poly to_fract (primitive_part_fract p)) = p"
append_rows_def: fixes A :: "'a mat"   and B :: "'a mat" shows "A @\<^sub>r B = four_block_mat A (0\<^sub>m (dim_row A) 0) B (0\<^sub>m (dim_row B) 0)"
isom_gch_units_transpTr4: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n" shows "finite ({i. i \<le> n \<and> g i \<cong> E} - {i, j})"
filter_M_pure_I: fixes xs :: "'x list"   and P :: "'x \<Rightarrow> ('y, 'z, bool) prog"   and h :: "'y" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "pure (filter_M P xs) h"
big_orbits_def: shows "big_orbits = {N \<in> orbits. 1 < card N}"
assms_1: assumes "\<paragraph>''then''"   and "bval b s" shows "wp \<pi> c\<^sub>1 Q s"
typeI_twoCube_smooth_edges: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeI_twoCube twoC"   and "(k, \<gamma>) \<in> boundary twoC" shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
monic_irreducible_factorization: fixes p :: "'c poly" assumes "monic p" shows "\<exists>as f. finite as \<and> p = (\<Prod>a\<in>as. a ^ Suc (f a)) \<and> as \<subseteq> {q. irreducible q \<and> monic q}"
assms_2: shows "\<forall>x\<in>A. invert (f x) = x \<and> invert (g x) = x"
abs_eq: fixes x :: "'a" shows "blinfun_mult_right x = Blinfun ((*) x)"
case_eq_if: fixes f :: "'e \<Rightarrow> 'd"   and optionT :: "('f, 'e) optionT" shows "(case optionT of OptionT x \<Rightarrow> f x) = f (run_option optionT)"
renaming_preserves_semantic: fixes I :: "'at set"   and L :: "'at Literal"   and Sel :: "'at set" assumes "I \<Turnstile> L" shows "rename_interpretation Sel I \<Turnstile> rename_literal Sel L"
xs_ys_def: shows "xs_ys = map (\<lambda>i. (int i, poly p (int i))) [3..<3 + degree p]"
MapsToObj_2: fixes f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "f maps X to Y" shows "Y \<in> obj"
inf_Assertion_def: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "p \<sqinter> q = abs_wpt ({\<cdot> p } \<sqinter> {\<cdot> q })"
swap_cols_rows_def: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "swap_cols_rows k l A = swaprows k l (swapcols k l A)"
narrow_st_def: fixes F1 :: "'b Abs_State.st"   and F2 :: "'b Abs_State.st" shows "F1 \<triangle> F2 = FunDom (\<lambda>x. fun F1 x \<triangle> fun F2 x) (inter_list (Abs_State.dom F1) (Abs_State.dom F2))"
elims_3: fixes x :: "'b extT"   and xa :: "'b extT" assumes "\<not> x \<le> xa" obtains x :: "'b" and y :: "'b" where "xa__ = extT.Val x" and "xa = extT.Val y" and "\<not> x \<le> y"   | uu_ :: "'b" where "xa__ = extT.Top" and "xa = extT.Val uu_"
simps_2: fixes x :: "variable" shows "Assigno x undeft = undefg"
cond_unit_T: fixes P :: "('b, 'c) uexpr"   and Q :: "('b, 'c) uexpr" shows "P \<triangleleft> true \<triangleright> Q = P"
nonneg_escape_rate: fixes x :: "'a" shows "0 \<le> escape_rate x"
generator_cases: fixes y :: "'e \<Rightarrow> ('f, 'e) step" assumes "y \<in> {g. terminates g}" obtains x :: "('f, 'e) generator" where "y = generator x"
Collect_conj_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "\<lbrace>\<acute>P \<and> \<acute>Q\<rbrace> = \<lbrace>\<acute>P\<rbrace> \<inter> \<lbrace>\<acute>Q\<rbrace>"
wf_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "iG.digraph_isomorphism hom" shows "wf_digraph (app_iso hom (app_iso hom G))"
ptrm_nprv_exi: fixes \<sigma> :: "'fmla"   and n :: "nat"   and \<phi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "\<phi> \<in> fmla" shows "nprv {\<sigma>, exi out (cnj \<sigma> \<phi>)} \<phi>"
subset_refl: fixes A :: "'d set" shows "A \<subseteq> A"
\<xi>_3: shows "Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) \<xi>"
Derives1_nonterminal: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 a i r b"   and "splits_at a i \<alpha> N \<beta>" shows "fst r = N \<and> is_nonterminal N"
extended_Gromov_product_le_dist: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y \<le> extended_Gromov_distance (to_Gromov_completion e) x"
Lim_in_closed_set: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and l :: "'a" assumes "closed S"   and "\<forall>\<^sub>F x in F. f x \<in> S"   and "F \<noteq> bot"   and "(f \<longlongrightarrow> l) F" shows "l \<in> S"
S3: shows "b_least2 f x (Suc y) < Suc y"
exec_Normal_elim_cases_1: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and f :: "'c"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> t" obtains "t = Fault f"
simps_2: fixes n :: "nat"   and ns :: "nat list"   and f :: "'a \<Rightarrow> 'a"   and as :: "'a list" shows "mapAt (n # ns) f as = (if n < |as| then mapAt ns f (as[n := f (as ! n)]) else mapAt ns f as)"
N_def: shows "N = Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})"
idom_ops_poly: shows "idom_ops (poly_ops ff_ops) poly_rel"
e: shows "0 < \<bar>k1 - k2\<bar> / 2"
MinkowskiPrimitive_axioms: shows "MinkowskiPrimitive \<E> \<P>"
assms_2: shows "Deliver (i1, Create i2 e) \<in> set xs"
valid_refs_invI: fixes s :: "'a process_name \<Rightarrow> ('c, 'a, 'd, 'b) Model.local_state" assumes "\<And>m x y. \<lbrakk>(x reaches y) s; x \<in> roots (s (mutator m)) \<union> ghost_honorary_root (s (mutator m)) \<union> mut_m.tso_store_refs m s \<or> grey x s\<rbrakk> \<Longrightarrow> obj_at (\<lambda>s. True) y s" shows "valid_refs_inv s"
exhaust: fixes y :: "xname" obtains "y = Throwable"   | "y = NullPointer"   | "y = OutOfMemory"   | "y = ClassCast"   | "y = NegArrSize"   | "y = IndOutBound"   | "y = ArrStore"
small_vunion_set: fixes A :: "V"   and B :: "V" shows "small {x. x \<in>\<^sub>\<circ> A \<or> x \<in>\<^sub>\<circ> B}"
bisim1: shows "rel_spmf (\<lambda>(x, uu_, s) (y, s'). x = y \<and> s = s') (exec_gpv (\<lambda>(s', s) y. map_spmf (\<lambda>((x, s'), s). (x, (), s)) (exec_gpv prf.random_oracle (prf_encrypt_oracle () y) s)) \<A>1 ((), Map.empty)) (exec_gpv rf_encrypt \<A>1 Map.empty)"
qj': shows "qj \<notin> p ` {0..<nr}"
invar_2Nodes: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and lv :: "nat"   and rl :: "('b \<times> nat) tree"   and rx :: "'b"   and rlv :: "nat"   and rr :: "('b \<times> nat) tree" shows "invar \<langle>l, (x, lv), \<langle>rl, (rx, rlv), rr\<rangle>\<rangle> = (invar l \<and> invar \<langle>rl, (rx, rlv), rr\<rangle> \<and> lv = Suc (lvl l) \<and> (lv = Suc rlv \<or> rlv = lv \<and> lv = Suc (lvl rr)))"
bezout1_nat: fixes a :: "nat"   and b :: "nat" shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
ModelIkTpartG_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "ModelIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl intT intF intP \<equiv> ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl \<and> ModelIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw intT intF intP"
less_eq_dual_def: shows "(\<le>) = rel_dual (\<lambda>x y. y \<le> x)"
adopt_node_child_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr" assumes "l_adopt_node type_wf known_ptr known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document"   and "h \<turnstile> ok adopt_node document_ptr child" shows "child |\<in>| node_ptr_kinds h"
convergent_prod_to_zero_iff: fixes f :: "nat \<Rightarrow> 'e" assumes "convergent_prod f" shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> (0::'e) = (\<exists>i. f i = (0::'e))"
OB_def: shows "OB = (\<lambda>(k, ms). map (observe k) ms)"
Int_Collect_mono: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x" shows "A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
WT_gpv_bindD1: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv" assumes "\<Gamma> \<turnstile>g gpv \<bind> f \<surd>" shows "\<Gamma> \<turnstile>g gpv \<surd>"
assms_1: fixes y :: "'a"   and z :: "'a sort_key" assumes "x = \<lbrakk>NonFinal (y, sort_dir.Left) z\<rbrakk>" shows "p"
smallomegaI_filterlim_at_infinity: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "LIM x F. f x / g x :> at_infinity" shows "f \<in> \<omega>[F](g)"
identity: shows "map_prod (\<lambda>x. x) (\<lambda>x. x) = id"
permute_pure: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "perm"   and x :: "'c" assumes "class.pure permute" shows "permute p x = x"
cr_wn': fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and m' :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob n \<pi> m"   and "\<not> IFC_def.is_cdi prob m' \<pi> m"   and "m < m'" shows "n < m'"
simps_2: fixes uu :: "'b"   and a :: "('a, 'b) Node"   and b :: "('a, 'b) Node" shows "FingerTreeStruc.nodeToDigit (Node2 uu a b) = Two a b"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', P \<oplus>\<^sub>\<top> \<tau>.P, \<tau>.P) \<in> Rel"
finite_measure_finite_Union: fixes S :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and p :: "'b spmf" assumes "finite S"   and "A ` S \<subseteq> sets (measure_spmf p)"   and "disjoint_family_on A S" shows "Sigma_Algebra.measure (measure_spmf p) (\<Union> (A ` S)) = (\<Sum>i\<in>S. Sigma_Algebra.measure (measure_spmf p) (A i))"
arcs_add_arc: fixes a :: "'b" shows "arcs (add_arc a) = insert a (arcs G)"
basic: fixes e :: "e list"   and e' :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<lparr>e\<rparr> \<leadsto> \<lparr>e'\<rparr>" shows "\<lparr>s;vs;e\<rparr> \<leadsto>_ i \<lparr>s;vs;e'\<rparr>"
cmf: shows "m O f \<subseteq> s \<union> ov \<union> d"
right_cancel: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "(y + x = z + x) = (y = z)"
map_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and p :: "'d spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
adjoint_eqI: fixes F :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and G :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "\<And>x y. (F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)" shows "F = G*"
pbij_lens_axioms_def: fixes x :: "'d \<Longrightarrow> 'e" shows "pbij_lens_axioms x \<equiv> \<forall>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
lex: assumes "coll a b c"   and "coll a b d" shows "Counterclockwise_2D_Arbitrary.lex b a"
card_cartesian_product: fixes A :: "'a set"   and B :: "'b set" shows "card (A \<times> B) = card A * card B"
4: shows "length (nths (dims Aw) (Collect even)) = N_half"
fd_def: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> y = d (x \<cdot> y)"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "strict_bicategory V H \<a> \<i> src trg"   and "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "equivalence_in_strict_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
id_lens_def: shows "1\<^sub>L = \<lparr>lens_get = id, lens_put = \<lambda>_. id\<rparr>"