hcju/isabelleps huggingface dataset:
l_zero: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" assumes "Ring R"   and "a \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> a = a"
Comp_def: fixes G :: "'a set"   and F :: "'a set" shows "Comp G F \<equiv> {h. B.is_iso_class F \<and> B.is_iso_class G \<and> (\<exists>f g. f \<in> F \<and> g \<in> G \<and> g \<star> f \<cong>\<^sub>B h)}"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom \<mu>)"
in_HomD_6: fixes F :: "'a set"   and a :: "'a"   and b :: "'a"   and f :: "'a" assumes "F \<in> Hom a b"   and "f \<in> F" shows "F = \<lbrakk>f\<rbrakk>\<^sub>B"
bop_ueval: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'e"   and x :: "('g, 'f) uexpr"   and y :: "('h, 'f) uexpr"   and b :: "'f" shows "\<lbrakk>bop f x y\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b)"
c: shows "path_image (complex_of \<circ> c) = complex_of ` path_image c"
tagged_partial_division_of_def: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" shows "(s tagged_partial_division_of i) = (finite s \<and> (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> interior K1 \<inter> interior K2 = {}))"
refine: fixes n :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and bi :: "real list"   and b :: "'b"   and yi :: "real"   and y :: "real" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lvivl_rel"   and "(bi, b) \<in> lv_rel"   and "(yi, y) \<in> rnv_rel" shows "(nres_of (project_set_ivl_impl n Xi bi yi), project_set_ivl X b y) \<in> \<langle>lvivl_rel\<rangle>nres_rel"
simplex_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "x \<in> X" shows "f ` x \<in> Y"
h1: shows "h1 = \<langle>x21_, x22_, x23_\<rangle>"
ok3_fin1_final1: fixes r :: "'b mrexp3" assumes "ok3 r" shows "fin1 r = Collect (final1 (strip3 r))"
le_nat_floor: fixes x :: "nat"   and a :: "real" assumes "real x \<le> a" shows "x \<le> nat \<lfloor>a\<rfloor>"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "inj_idom_hom hom" shows "map_poly_inj_idom_hom hom"
conseq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and k :: "nat"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>b {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, k * n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n div k)"   and "0 < k" shows "\<turnstile>\<^sub>3\<^sub>b {P'} c { Q'}"
inducts: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set" assumes "separating_gen G A B S" obtains "\<And>x y p. \<lbrakk>x \<in> A; y \<in> B; path G x p y\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
endofunctor_axioms: shows "endofunctor (\<cdot>) R"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>n. P (Bound n)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Mul a b)"   and "\<And>p n. P p \<Longrightarrow> P (Pw p n)"   and "\<And>c n p. \<lbrakk>P c; P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (C v)" shows "P a0"
iso_iff_mono_and_retraction: fixes f :: "'a \<times> 'a" shows "VV.iso f = (VV.mono f \<and> VV.retraction f)"
trace_smult: fixes A :: "'a mat"   and n :: "nat"   and c :: "'a" assumes "A \<in> carrier_mat n n" shows "trace (c \<cdot>\<^sub>m A) = c * trace A"
hv_props_3: shows "one_chain = \<H> \<union> \<V>"
rows_def: fixes A :: "'b mat" shows "Matrix.rows A = map (Matrix.row A) [0..<dim_row A]"
rec_7: fixes g :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "g \<in> set gs"   and "p = monom_mult (1::'b) t g" shows "thesis"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "(t \<^bold>\<star> u)\<^bold>\<down> = (if Nml (t \<^bold>\<star> u) then t \<^bold>\<star> u else (\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<Down> \<^bold>\<lfloor>u\<^bold>\<rfloor>) \<^bold>\<cdot> (t\<^bold>\<down> \<^bold>\<star> u\<^bold>\<down>))"
map_list_bind: fixes g :: "'b \<Rightarrow> 'a"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b list" shows "map g (List.bind xs f) = List.bind xs (map g \<circ> f)"
split_n_m_x_abbrev_dims_1: fixes n :: "nat"   and m :: "nat"   and x :: "(nat, 'a) mapping"   and a :: "'a vec"   and b :: "'a vec" assumes "split_n_m_x n m x = (a, b)" shows "dim_vec a = n"
lproject_lnth: fixes i :: "nat"   and A :: "'b set"   and w :: "'b llist" assumes "enat i < llength (lproject A w)" shows "lproject A w ?! i = w ?! nth_least (liset A w) i"
assms_2: shows "S \<subseteq> U\<^sub>M"
map_inv_add: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)"   and "inj_on g (dom g)"   and "dom f \<inter> dom g = {}"   and "ran f \<inter> ran g = {}" shows "map_inv (f ++ g) = map_inv f ++ map_inv g"
subnetOf_lemma2: fixes a :: "int"   and b :: "int" shows "\<forall>x\<in>subnet_of (a, b). (a, b) \<in> x"
vsv_is_VLambda_on_vlrestriction: fixes A :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V" assumes "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)" shows "VLambda A (app (VLambda Aa f)) = VLambda Aa f \<restriction>\<^sup>l\<^sub>\<circ> A"
a_inv: fixes h :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set" assumes "h \<in> generate_field R H" shows "\<ominus>\<^bsub>R\<^esub> h \<in> generate_field R H"
is_constant_constructorsI_11: fixes x :: "'a ltln"   and y :: "'a ltln" assumes "is_constant x" shows "is_constant (mk_release y x)"
67527: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> - x \<sqinter> y) \<sqinter> y = y"
centre_in_cball: fixes x :: "'b"   and e :: "real" shows "(x \<in> cball x e) = (0 \<le> e)"
strong_convergence: fixes events :: "'a \<Rightarrow> ('a, 'b) event list"   and i :: "'a \<times> nat"   and j :: "'a \<times> nat" assumes "dist_execution events"   and "dist_execution_preliminary.is_valid_state_id events i"   and "dist_execution_preliminary.is_valid_state_id events j"   and "set (dist_execution_preliminary.received_messages events i) = set (dist_execution_preliminary.received_messages events j)" shows "dist_execution_preliminary.state events i = dist_execution_preliminary.state events j"
Abs_nonzero_inverse: fixes y :: "'b" assumes "y \<in> UNIV - {0::'b}" shows "Rep_nonzero (Abs_nonzero y) = y"
translation_composition_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "compose G (a)\<^sub>L (b)\<^sub>L = (a \<cdot> b)\<^sub>L"
prems_4: shows "h \<in> insert x F \<rightarrow> carrier G"
OutOfMemoryC_def: shows "OutOfMemoryC \<equiv> (OutOfMemory, Object, [], [])"
fixp_induct: fixes Sup :: "'g set \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and P :: "'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'g" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq P"   and "monotone less_eq less_eq f"   and "P (Sup {})"   and "\<And>x. P x \<Longrightarrow> P (f x)" shows "P (ccpo.fixp Sup less_eq f)"
finiteFreshVar: fixes A :: "vbl set" assumes "finite A" shows "freshVar A \<notin> A"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>2\<^sub>x\<^sub>A\<^sub>1) (\<cdot>\<^sub>B) SF.map"
sub_def: fixes k :: "num"   and l :: "num" shows "neg_numeral_class.sub k l = numeral k - numeral l"
124: fixes x :: "'a" shows "- - x \<sqinter> x = x"
uminus_class: fixes x :: "'b" shows "\<ominus>\<^bsub>cring_class_ops\<^esub> x = - x"
simps_23: fixes Vs :: "String.literal list"   and e1 :: "'a expr"   and C :: "String.literal"   and V :: "String.literal"   and e2 :: "'a expr" shows "compE1 Vs (try e1 catch(C V) e2) = try compE1 Vs e1 catch(C length Vs) compE1 (Vs @ [V]) e2"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Dom (t \<^bold>\<otimes> u) = Dom t \<^bold>\<otimes> Dom u"
assms_2: shows "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> element_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| element_ptr_kinds h)"
f_def: shows "f \<equiv> \<lambda>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x)))"
dvd_lcm2: 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 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 b :: "'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times b (lcm a b)"
delete_notin_dom: fixes k :: "'d"   and al :: "('d \<times> 'e) list" shows "k \<notin> fst ` set (delete k al)"
locked_in_def: fixes Quorum :: "process set set"   and s :: "v_state"   and r :: "nat"   and v :: "val" assumes "quorum_process Quorum" shows "quorum_process.locked_in Quorum s r v = quorum_process.locked_in_vf Quorum (votes s r) v"
find_to_mem: fixes P :: "cld list"   and ctx :: "ctx"   and fqn :: "fqn"   and ctx' :: "ctx"   and cld :: "cld" assumes "find_cld_f P ctx fqn = Some (ctx', cld)" shows "cld \<in> set P"
x2: shows "\<forall>xa. x * xa mod f = 1 \<longrightarrow> \<not> degree xa < degree f"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "rotate1 (x # xs) = xs @ [x]"
nullable0_satisfies0: fixes \<AA> :: "'i"   and a :: "'a" assumes "Length \<AA> = 0" shows "nullable0 a = (\<AA> \<Turnstile>\<^sub>0 a)"
AxTangentVertex: fixes cone :: "'a Cone"   and e :: "'a Point" shows "inPlane (vertex cone) (tangentPlane e cone)"
set_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "set_category_data S" shows "set_category_data.set S img a = img ` {f. partial_magma.in_hom S f (set_category_data.unity S) a}"
prj_simps_1: shows "C.seq \<nu>\<pi>.prj\<^sub>1 \<p>\<^sub>1[\<pi>.leg0 \<cdot> \<nu>\<pi>.prj\<^sub>0, \<rho>.leg1]"
eq_\<I>_generat_reflI: fixes generat :: "('a, 'b, 'c \<Rightarrow> 'd) generat"   and A :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. x \<in> generat_pures generat \<Longrightarrow> A x x"   and "\<And>out c. generat = IO out c \<Longrightarrow> out \<in> outs_\<I> \<I> \<and> (\<forall>input\<in>responses_\<I> \<I> out. D (c input) (c input))" shows "eq_\<I>_generat A \<I> D generat generat"
neg_less_iff_less: 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 b :: "'a"   and a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus b) (uminus a) = less a b"
gSwapInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) Recursion.model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and inp' :: "'a \<Rightarrow> ('c, 'd, 'e, 'h, 'g) qTerm set option"   and inp :: "'a \<Rightarrow> 'b option" shows "gSwapInp MOD zs z1 z2 inp' inp \<equiv> lift2 (gSwap MOD zs z1 z2) inp' inp"
compile_def: fixes rs :: "(name \<times> pterm) fset" shows "Rewriting_Sterm.compile rs = ordered_fmap (map_prod id pterm_to_sterm |`| rs)"
frameImpEmpty: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<phi> :: "'e" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.FrameImp SImp (\<langle>\<epsilon>, \<Psi>\<rangle>) \<phi> = SImp \<Psi> \<phi>"
add_rel_aux_def: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" shows "add_rel_aux r s r' s' \<equiv> (r \<cdot> s' + r' \<cdot> s) / (s \<cdot> s')"
res_to_cong_simps_1: fixes x :: "int"   and y :: "int" shows "x mod m \<oplus> y mod m = (x + y) mod m"
conj_forward: fixes P' :: "bool"   and Q' :: "bool"   and P :: "bool"   and Q :: "bool" assumes "P' \<and> Q'"   and "P' \<Longrightarrow> P"   and "Q' \<Longrightarrow> Q" shows "P \<and> Q"
compose_extensional: fixes A :: "'d set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "compose A f g \<in> extensional A"
intro: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a b c. plus (plus a b) c = plus a (plus b c)" shows "class.semigroup_add plus"
mem_affine_3_minus: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "x + v *\<^sub>R (y - z) \<in> S"
real_affinity_le: fixes m :: "'b"   and x :: "'b"   and c :: "'b"   and y :: "'b" assumes "(0::'b) < m" shows "(m * x + c \<le> y) = (x \<le> inverse m * y + - (c / m))"
U: fixes e :: "real" shows "0 < e \<longrightarrow> (\<exists>U\<in>fmeasurable M. S \<subseteq> U \<and> Sigma_Algebra.measure M U < m + e)"
filter_def: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and xs :: "'x" shows "local.filter P xs = oalist_of_list (filter_raw P (list_of_oalist xs))"
entailment_model_def: fixes C :: "'b set"   and Rs :: "(('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph) set"   and init :: "('b Standard_Constant, nat) labeled_graph" shows "entailment_model C Rs init \<equiv> let L = fst ` \<Union> ((edges \<circ> snd) ` Rs) \<union> {S_Bot, S_Top, S_Idt} \<union> S_Const ` C \<union> fst ` edges init; Rules = Rs \<union> standard_rules C L; sel = non_constructive_selector Rules in the_lcg sel Rules (card (vertices init), edges init)"
minus_zero_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - 0\<^sub>v n = v"
iv1: shows "W * adjoint W = 1\<^sub>m n"
simps_9: fixes x31 :: "'a convnet"   and x32 :: "'a convnet"   and x21 :: "'a"   and x22 :: "'a convnet" shows "Pool x31 x32 \<noteq> Conv x21 x22"
select_convs_2: fixes igWls :: "'c \<Rightarrow> 'i \<Rightarrow> bool"   and igWlsAbs :: "'b \<times> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and igVar :: "'b \<Rightarrow> 'h \<Rightarrow> 'i"   and igAbs :: "'b \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'd"   and igOp :: "'g \<Rightarrow> ('e \<Rightarrow> 'i option) \<Rightarrow> ('f \<Rightarrow> 'd option) \<Rightarrow> 'i"   and igFresh :: "'b \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and igFreshAbs :: "'b \<Rightarrow> 'h \<Rightarrow> 'd \<Rightarrow> bool"   and igSwap :: "'b \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'i"   and igSwapAbs :: "'b \<Rightarrow> 'h \<Rightarrow> 'h \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubst :: "'b \<Rightarrow> 'i \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'i"   and igSubstAbs :: "'b \<Rightarrow> 'i \<Rightarrow> 'h \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'j" shows "igWlsAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igWlsAbs"
bexp_neg_negates: fixes mem :: "'Var \<Rightarrow> 'Val"   and e :: "'BExp" shows "ev\<^sub>B mem (bexp_neg e) = (\<not> ev\<^sub>B mem e)"
reversepath_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "reversepath (reversepath g) = g"
orthogonal_matrix_mul: fixes A :: "((real, 'a) vec, 'a) vec"   and B :: "((real, 'a) vec, 'a) vec" assumes "orthogonal_matrix A"   and "orthogonal_matrix B" shows "orthogonal_matrix (A ** B)"
card_domain_partitions: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "finite B" shows "card ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) = card (f ` A)"
ordinal_le_timesL: fixes x :: "ordinal"   and y :: "ordinal" assumes "0 < x" shows "y \<le> x * y"
combine_infinity_invariant_3: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_until (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"
filter_mset_cong: fixes M :: "'b multiset"   and M' :: "'b multiset"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "M = M'"   and "\<And>a. a \<in># M \<Longrightarrow> P a = Q a" shows "filter_mset P M = filter_mset Q M"
FormP_fresh_iff: fixes a :: "atom"   and y :: "tm" shows "a \<sharp> FormP y = a \<sharp> y"
exp_conj_mono_right: fixes Q :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "Q \<tturnstile> R" shows "P && Q \<tturnstile> P && R"
norm0': shows "\<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 \<noteq> 0"
A3b: fixes y :: "i" assumes "AxiomsSet6" shows "cod y \<cdot> y \<cong> y"
\<phi>\<psi>: shows "\<phi> \<circ> \<psi> = J'.map"
rep_eq: fixes x :: "real_alg_3"   and xa :: "real_alg_3" shows "equal_3 x xa = equal_2 (rep_real_alg_3 x) (rep_real_alg_3 xa)"
sliding_window_def: shows "local.sliding_window = local.iterate Leaf ws"
LIM_zero_cancel: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" assumes "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'d)) F" shows "(f \<longlongrightarrow> l) F"
induct: fixes x1 :: "'a option"   and x2 :: "('a \<times> 'a option) list"   and P :: "'a option \<Rightarrow> ('a \<times> 'a option) list \<Rightarrow> bool" assumes "insert_seq x1 x2"   and "\<And>start oid. P start [(oid, start)]"   and "\<And>start list prev ref oid. \<lbrakk>insert_seq start (list @ [(prev, ref)]); P start (list @ [(prev, ref)])\<rbrakk> \<Longrightarrow> P start (list @ [(prev, ref), (oid, Some prev)])" shows "P x1 x2"
simple_distributed_simple_function: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real" assumes "simple_distributed M X Px" shows "simple_function M X"
wf_syscls_is_class_Thread: fixes P :: "'b prog" assumes "wf_syscls P" shows "is_class P Thread"
le_supI2: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x b" shows "less_eq x (sup a b)"
beh_in_reach: fixes b :: "'e list"   and T :: "('e, 'f) TS_scheme"   and s :: "'e" assumes "b \<in> beh T"   and "s \<in> set b" shows "s \<in> reach T"
simps_1: fixes s1 :: "'d stream"   and s2 :: "'e stream" shows "shd (szip s1 s2) = (shd s1, shd s2)"
maxchain_imp_chain: fixes C :: "'a set" assumes "B.maxchain C" shows "chain_on P C {x \<in> A. \<not> Q x}"
noteq_irrefl: fixes r :: "'a array" assumes "r =!!= r" shows "False"
pdevs_apply_msum_pdevs: fixes n :: "nat"   and f :: "'d pdevs"   and g :: "'d pdevs"   and i :: "nat" shows "pdevs_apply (msum_pdevs n f g) i = (if i < n then pdevs_apply f i else pdevs_apply g (i - n))"
coeff_le_sq_norm: fixes f :: "'b poly"   and i :: "nat" shows "\<bar>coeff f i\<bar>\<^sup>2 \<le> \<parallel>f\<parallel>\<^sup>2"
If_andalso: fixes t :: "'a"   and e :: "'a"   and p :: "tr"   and q :: "tr" shows "If p andalso q then t else e = If p then If q then t else e else e"
satisfies_bounds_consistent: fixes v :: "nat \<Rightarrow> 'a"   and s :: "('b, 'a) state" shows "v \<Turnstile>\<^sub>b \<B> s \<longrightarrow> \<diamond> s"
intercept_upf_enc_def: fixes k :: "'prf_key"   and b :: "bool" shows "intercept_upf_enc k b = (\<lambda>(L, D) (m1, m0). case length m1 = prf_clen \<and> length m0 = prf_clen of True \<Rightarrow> lift_spmf (spmf_of_set prf_domain) \<bind> (\<lambda>x. let c = prf_fun k x [\<oplus>] (if b then m1 else m0) in Generative_Probabilistic_Value.Pause (Inl (x @ c)) Generative_Probabilistic_Value.Done \<bind> (\<lambda>t. Generative_Probabilistic_Value.Done (Some (x, c, projl t), insert (x, c, projl t) L, D))) | False \<Rightarrow> Generative_Probabilistic_Value.Done (None, L, D))"
some_img_def: shows "S.some_img = (SOME img. set_category_given_img (\<cdot>) img \<AA>)"
cont_def: fixes f :: "'g \<Rightarrow> 'h" shows "cont f \<equiv> \<forall>C. C \<noteq> {} \<longrightarrow> f (Sup C) = Sup (f ` C)"
ang_vec_opposite_opposite: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "\<angle> (- z1) (- z2) = \<angle> z1 z2"
ses_outputs_are_events_def: fixes SES :: "('c, 'd) SES_rec" shows "ses_outputs_are_events SES \<equiv> O\<^bsub>SES\<^esub> \<subseteq> E\<^bsub>SES\<^esub>"
HaddP_Zero_D1: fixes x :: "tm"   and y :: "tm"   and H :: "fm set" shows "insert (HaddP Zero x y) H \<turnstile> x EQ y"
config_n_init3: fixes init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs). snd (snd x) = init"
pointbase_def: fixes F :: "'c set \<Rightarrow> 'b set"   and I :: "'c set" shows "pointbase F I = \<Union> {F X |X. finite X \<and> X \<subseteq> I}"
bilinear_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "bilinear f = ((\<forall>x. linear (f x)) \<and> (\<forall>y. linear (\<lambda>x. f x y)))"
both_above_arg1: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "t \<sqsubseteq> t \<otimes>\<otimes> t'"
Poincare_recurrence_thm_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "A - recurrent_subset A \<in> null_sets M"
elims_3: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<not> \<guillemotleft>x\<guillemotright> xa" obtains p :: "'b \<Rightarrow> bool" where "x = pa__" and "xa = []"   | p :: "'b \<Rightarrow> bool" and a :: "'b" and S :: "'b list" where "x = p" and "xa = a # S" and "\<not> p a"
bl_shiftr1': fixes w :: "'b word" shows "to_bl (shiftr1 w) = butlast (False # to_bl w)"
count_mset: fixes xs :: "'a list"   and x :: "'a" shows "count (mset xs) x = length (filter ((=) x) xs)"
min_Some_not_None: fixes t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.min (Some t # xs) \<noteq> None"
keys_to_list_def: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "pp_pm.extended_ord.punit.keys_to_list p = pp_pm.extended_ord.punit.pps_to_list (keys p)"
ss: fixes x0 :: "'g"   and x1 :: "'a"   and a0 :: "'j"   and a1 :: "'d"   and s :: "'e"   and e0 :: "'e"   and z0 :: "'k"   and e1 :: "'e"   and z1 :: "'f"   and s' :: "'e"   and e0' :: "'e"   and z0' :: "'k"   and e1' :: "'e"   and z1' :: "'f" assumes "(x0, x1) \<in> valid_pub_OR"   and "check_OR (x0, x1) (a0, a1) s ((e0, z0), e1, z1)"   and "check_OR (x0, x1) (a0, a1) s' ((e0', z0'), e1', z1')"   and "s \<noteq> s'"   and "s \<in> challenge_space"   and "s' \<in> challenge_space" shows "lossless_spmf (\<A>ss_OR (x0, x1) ((a0, a1), s, (e0, z0), e1, z1) ((a0, a1), s', (e0', z0'), e1', z1')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss_OR (x0, x1) ((a0, a1), s, (e0, z0), e1, z1) ((a0, a1), s', (e0', z0'), e1', z1')). ((x0, x1), w') \<in> Rel_OR)"
get_heap_is_OK: fixes h :: "'a"   and f :: "'a \<Rightarrow> ('a, 'b, 'c) prog" shows "h \<turnstile> ok (get_heap \<bind> f) = h \<turnstile> ok f h"
knows2b: fixes A :: "specID"   and e :: "Expression list"   and e1 :: "Expression list" shows "knows A (e @ e1) \<longrightarrow> knows A e"