hcju/isabelleps huggingface dataset:
assms_1: shows "e = minarc (c * - c\<^sup>T \<sqinter> g)"
set_lebesgue_integral_eq_integral_2: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "f absolutely_integrable_on S" shows "set_lebesgue_integral lebesgue S f = integral S f"
trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
h_def: fixes a :: "nat"   and b :: "nat" shows "h a b = g a + b"
greaterThan_iff: fixes i :: "'a Word.word"   and k :: "'a Word.word" shows "(i \<in> signed.greaterThan k) = (k <s i)"
C_in_llast: shows "C \<in> N_of_wstate (llast Sts) \<union> P_of_wstate (llast Sts)"
INF_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (g j) (f i)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (f i) (g j)" shows "Inf (f ` A) = Inf (g ` B)"
range_dom_1: fixes f :: "('c \<times> 'b) set" shows "f `` Domain f = Range f"
AssumeH_2: fixes A1 :: "fm"   and A2 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 H) \<turnstile> A2"
ih: fixes \<Gamma>' :: "(name \<times> ty) list" assumes " (x, T\<^sub>1) # \<Gamma> \<subseteq> \<Gamma>' "   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2"
snd_snd: shows "snd (snd f) = snd (snd g)"
multpw_splitR: fixes X1 :: "'a multiset"   and X2 :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(X1 + X2, Y) \<in> multpw ns" obtains Y1 :: "'a multiset" and Y2 :: "'a multiset" where "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "(X2, Y2) \<in> multpw ns"
case_2: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case deny x2 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f2 x2"
Trans: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel"   and "(\<Psi>', T, U) \<in> Rel" shows "(\<Psi>', S, U) \<in> Rel"
arcs_symmetric: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> parcs G" shows "(b, a) \<in> parcs G"
I_intF_def: fixes f :: "'fsym"   and al :: "univ list" shows "I_intF f al \<equiv> Ik_intF f (map2 proj (arOf f) al)"
happen_implies_can_occur: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'" shows "distributed_system.can_occur channel trans send recv ev c"
butlast2: shows "D # rev Cs' @ [C'] = butlast (D # rev Cs') @ [C''] @ [C']"
the_gpv_parametric': fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_gpv'' A C R) (rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R)))) the_gpv the_gpv"
free_in_quad_fm_helper: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and z :: "nat"   and F :: "atom fm" assumes "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d" shows "freeIn (var + z) (quadratic_sub_fm_helper var a b c d F z)"
fps_shift_to_fls: fixes n :: "nat"   and f :: "'a fps" assumes "n \<le> subdegree f" shows "fps_to_fls (fps_shift n f) = fls_shift (int n) (fps_to_fls f)"
is_proper_prefix_def: fixes u :: "'b list"   and v :: "'b list" shows "is_proper_prefix u v = (\<exists>w. w \<noteq> [] \<and> u @ w = v)"
COMP_def: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "COMP g f \<equiv> if (f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> (g \<noteq> Null \<and> Dom g \<in> Obj \<and> Cod g \<in> Obj \<and> Map g \<in> Hom (Dom g) (Cod g)) \<and> Dom g = Cod f then MkArr (Dom f) (Cod g) (Comp (Cod g) (Dom g) (Dom f) (Map g) (Map f)) else Null"
sat_Since_point: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" assumes "Formula.sat \<sigma> V v i (formula.Since \<phi> I \<psi>)" obtains j :: "nat" where "j \<le> i" and "mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I" and "Formula.sat \<sigma> V v i (formula.Since \<phi> (point (\<tau> \<sigma> i - \<tau> \<sigma> j)) \<psi>)"
unit_not_app: fixes A :: "'b trm"   and B :: "'b trm" shows "Unit \<noteq> App A B"
some_gcd_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" shows "dvd.some_gcd times a b \<equiv> SOME x. dvd.is_gcd times x a b"
P1: shows "UP_pre_univ_prop R R (\<lambda>x. x)"
encR: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
Sup_atLeastAtMost: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<Squnion>{x..y} = y"
fractrel_iff_normalize_quot_eq: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" shows "fractrel x y = (normalize_quot x = normalize_quot y \<and> snd x \<noteq> (0::'a) \<and> snd y \<noteq> (0::'a))"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "E.Ide (t \<^bold>\<star> u) = (E.Ide t \<and> E.Ide u \<and> E.Src t = E.Trg u)"
admissible_def: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" shows "ccpo.admissible lub ord P = (\<forall>A. Complete_Partial_Order.chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
det_SPEC: fixes \<Phi> :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<Phi> x" shows "nres_of (dRETURN x) \<le> SPEC \<Phi>"
fps_pow_nth_below_subdegree: fixes k :: "nat"   and n :: "nat"   and f :: "'a fps" assumes "k < n * subdegree f" shows "fps_nth (f ^ n) k = (0::'a)"
rfn_tool12: fixes s :: "nat"   and i :: "nat" assumes "0 < s"   and "i mod s = s - 1" shows "Suc (i div s) = Suc i div s"
snd_add: fixes x :: "'e \<times> 'd"   and y :: "'e \<times> 'd" shows "snd (x + y) = snd x + snd y"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb @ [Ghost\<^sub>s\<^sub>b A L R W], \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
H2_J_2_b: fixes P :: "'b \<times> 'c alpha_d_scheme \<Rightarrow> bool"   and a :: "'b"   and b :: "'c alpha_d_scheme" assumes "\<forall>A. case A of (A, A') \<Rightarrow> P (A, A'\<lparr>ok := False\<rparr>) \<longrightarrow> P (A, A'\<lparr>ok := True\<rparr>)"   and "(P ;; J) (a, b)" shows "P (a, b)"
analz_synth_subset_Un2: fixes X :: "msg set"   and X' :: "msg set"   and Y :: "msg set" assumes "analz X \<subseteq> synth (analz X')" shows "analz (Y \<union> X) \<subseteq> synth (analz (Y \<union> X'))"
bij_extend_act: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "bij (extend_act (\<lambda>(x, u). h x))"
simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.fw plus less_eq m n (Suc k) 0 0 = linordered_ab_monoid_add.fw_upd plus less_eq (linordered_ab_monoid_add.fw plus less_eq m n k n n) (Suc k) 0 0"
bind_transfer: fixes R0 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "(crel_vs R0 ===> (R0 ===>\<^sub>T R1) ===>\<^sub>T R1) (\<lambda>v f. f v) (\<bind>)"
not_cd_impl_ipd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "k < i"   and "\<not> IFC_def.is_cdi prob i \<pi> k"   and "\<pi> i \<noteq> IFC_def.return prob" obtains j :: "nat" where "j \<in> {k..i}" and "\<pi> j = IFC_def.ipd prob (\<pi> k)"
assms_2: shows "\<mu> \<star> \<nu> \<noteq> null"
igWlsInp_defs_1: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'h option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
cong_sym: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "R \<in> congruences"   and "R a b" shows "R b a"
distinct_1: fixes x1 :: "'g"   and x2 :: "'h" shows "Inl x1 \<noteq> Inr x2"
strict_coboundedI2: fixes c :: "'c DBMEntry"   and b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "c \<prec> b" shows "c \<prec> linordered_monoid.max a b"
extr_def: shows "extr \<equiv> \<lambda>T H. {x. extrp (\<lambda>x. x \<in> T) (\<lambda>x. x \<in> H) x}"
aff: fixes X :: "'a set" assumes "X \<in> \<F>'" shows "aff_dim X \<le> d"
SC3Q: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "singleCombinators p" shows "singleCombinators (qsort p l)"
measurable_pmf_measure1: fixes M :: "'c pmf"   and N :: "'d measure" shows "measure_pmf M \<rightarrow>\<^sub>M N = UNIV \<rightarrow> space N"
bot_least: fixes bot :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.order_bot bot less_eq less" shows "less_eq bot a"
well_ord_adjunction: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq>\<^bsub>D\<^esub> a"   and "Worder (Iod D X)" shows "Worder (Iod D (X \<union> {a}))"
abs_eq: fixes xa :: "'b ltl"   and x :: "'b ltl" shows "ltl_prop_equiv_quotient_eq_test (Abs xa) (Abs x) = xa \<equiv>\<^sub>P x"
A3: shows "weak_reduction_contrasimulation (Rel\<inverse>) (STCal Source Target)"
complete_infl_mono_imp_ex_qfp: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "f ` A \<subseteq> A"   and "UNIV-complete A less_eq"   and "\<forall>a\<in>A. less_eq a (f a) \<or> (\<forall>b\<in>A. less_eq b a \<longrightarrow> less_eq (f b) (f a))" shows "\<exists>s\<in>A. sympartp less_eq (f s) s"
Ab_vec: shows "vec_to_iarray A !! b \<noteq> (0::'b)"
InvIncomingInfoUpright_imp_InvGlobalIncomingInfoUpright: fixes s :: "('c, 'd) configuration stream" assumes "holds InvIncomingInfoUpright s" shows "holds InvGlobalIncomingInfoUpright s"
eq_fin_concat: fixes u :: "'item list"   and w\<^sub>1 :: "'item list"   and v :: "'item list"   and w\<^sub>2 :: "'item list" shows "(u @ w\<^sub>1 @ v =\<^sub>F u @ w\<^sub>2 @ v) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
dirichlet_prod'_def: fixes f :: "nat \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and x :: "real" shows "dirichlet_prod' f g x = sum_upto (\<lambda>m. f m * g (x / real m)) x"
Re_winding: fixes h :: "real \<Rightarrow> complex" assumes "\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z"   and "Re (h 0) = Re z"   and "Re (h 1) = Re z"   and "valid_path h"   and "z \<notin> path_image h" shows "2 * Re (winding_number h z) = jumpF_pathfinish h z - jumpF_pathstart h z"
zero_le_ereal: fixes z :: "real" assumes "0 \<le> z" shows "0 \<le> ereal z"
gen_length_fusion_def: fixes g :: "('a, 'b) generator"   and n :: "nat"   and s :: "'b" shows "gen_length_fusion g n s = n + length (list.unfoldr g s)"
ab_group_add_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.ab_group_add plus zero minus uminus \<equiv> class.comm_monoid_add plus zero \<and> class.ab_group_add_axioms plus zero minus uminus"
elem: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> vcard xs" shows "xs\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i"
pos_deriv_imp_strict_mono: fixes f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "\<And>x. (f has_real_derivative f' x) (at x)"   and "\<And>x. 0 < f' x" shows "strict_mono f"
subset_iff_psubset_eq: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = (A \<subset> B \<or> A = B)"
halfchsys_decomp_2: shows "f \<turnstile> folding_g.\<C> \<inter> (g \<turnstile> folding_g.\<C>) = {}"
extended_ord_lp: fixes p :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "None \<notin> indets p" shows "restrict_indets_pp (extended_ord.lpp p) = lpp (restrict_indets p)"
lookup_override_on_eq: fixes m1 :: "'b \<Rightarrow> 'a"   and m2 :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and x :: "'b" shows "(m1 ++\<^bsub>S\<^esub> m2) x = (if x \<in> S then m2 x else m1 x)"
r1: shows "right (Abs_real_int (left r4, right r2)) = right r2"
iso_iff: shows "(x \<in> Group.iso (G\<lparr>carrier := generate G {a}\<rparr>) (H\<lparr>carrier := generate H {b}\<rparr>)) = (carrier (H\<lparr>carrier := generate H {b}\<rparr>) \<subseteq> x ` carrier (G\<lparr>carrier := generate G {a}\<rparr>) \<and> (\<forall>xa\<in>carrier (G\<lparr>carrier := generate G {a}\<rparr>). x xa = \<one>\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub> \<longrightarrow> xa = \<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>))"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "norm (f x) \<le> g x"
poss3: shows "squash (rpoly q x2) = 0 \<longrightarrow> (\<exists>r<x1. x2 \<le> r \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
r_singleton_encode_prim: shows "prim_recfn 1 r_singleton_encode"
residuated_strict: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "f \<bottom> = \<bottom>"
map_component_def: fixes f :: "'k \<Rightarrow> 'k"   and v :: "'t" shows "map_component f v = term_of_pair (pp_of_term v, f (component_of_term v))"
bounded_measure: fixes A :: "'a set" shows "prob A \<le> prob (space M)"
noDAsort: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "noDenyAll1 p" shows "noDenyAll1 (FWNormalisationCore.sort p l)"
chain_not_lnull: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R xs" shows "\<not> lnull xs"
simps_7: fixes P :: "'b ptrm" shows "ptrm_fvs (PSnd P) = ptrm_fvs P"
dnso2: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
hb_consistent_prefix: fixes history :: "nat \<Rightarrow> ('c \<times> 'd) event list"   and xs :: "('c \<times> 'd) event list"   and i :: "nat" assumes "network_with_ops history"   and "node_histories.prefix_of_node_history history xs i" shows "happens_before.hb_consistent (network.hb history) (trivial_network_with_ops.node_deliver_messages xs)"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "local.filter_min_aux (x # xs) ys = (if \<exists>y\<in>set xs \<union> set ys. rel y x then local.filter_min_aux xs ys else local.filter_min_aux xs (x # ys))"
H: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset"   and D :: "'a multiset" assumes "A + B = C + D" shows "A = A \<inter># C + (A - C) \<inter># D"
hyps: fixes a :: "symbol list" assumes "Derivation a E b" shows "count_terminals a \<le> count_terminals b"
collapse: fixes prod :: "'b \<times> 'c" shows "(fst prod, final_b prod) = prod"
dagger_denest2: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "(x \<union> y)\<^sup>* = (x\<^sup>* ; y)\<^sup>* ; x\<^sup>*"
Im3_sum: fixes f :: "'a \<Rightarrow> quat"   and S :: "'a set" shows "Im3 (sum f S) = (\<Sum>x\<in>S. Im3 (f x))"
int_simps_12: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> #True) = #True"
surj_def: fixes f :: "'r \<Rightarrow> 'q" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
distinct_map_entry: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "distinct (list.map fst xs)" shows "distinct (list.map fst (AList.map_entry k f xs))"
simps_3: fixes n :: "nat" shows "fib (Suc (Suc n)) = fib (Suc n) + fib n"
p\<^sub>0_simps: shows "trg t\<^sub>0u\<^sub>1.p\<^sub>0 = src u\<^sub>1"
effective_well_defined_Tensor: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and M1 :: "'a list list"   and M2 :: "'a list list" assumes "Matrix_Tensor.mult id f"   and "Matrix_Legacy.mat (mult.row_length M1) (length M1) M1"   and "Matrix_Legacy.mat (mult.row_length M2) (length M2) M2" shows "Matrix_Legacy.mat (mult.row_length M1 * mult.row_length M2) (length M1 * length M2) (mult.Tensor f M1 M2)"
hom_init: fixes r :: "'b rexp" shows "hom_ab (init_a r) = init_b r"
ssubst_fresh: fixes y :: "name"   and x :: "name"   and v :: "trm"   and k :: "stack" assumes "y \<sharp> (x, v, k)" shows "y \<sharp> ssubst x v k"
le_numeral_extra_1: shows "(0::'a) \<le> (1::'a)"
f_def: shows "f = (\<lambda>x. poly q x / poly p x)"
Cons: fixes ys :: "'a list"   and x :: "'a" assumes "\<forall>y\<in>set ys. le x y"   and "sortedP ys" shows "sortedP (x # ys)"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)" shows "\<exists>a\<in>elts (\<AA>\<lparr>Arr\<rparr>). \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
rsa_base_axioms: fixes n :: "nat" shows "rsa_base (prime_set n)"
hyps: assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "\<exists>n j. n < length al \<and> j < f (al ! n) \<and> i = lsum f (take n al) + j"
list_all2_antisym: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y"   and "list_all2 P xs ys"   and "list_all2 Q ys xs" shows "xs = ys"
r_into_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "r x y" shows "equivclp r x y"
lincomb_scalar_prod_left: fixes W :: "'b vec set"   and n :: "nat"   and v :: "'b vec"   and a :: "'b vec \<Rightarrow> 'b" assumes "W \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n" shows "module.lincomb (module_vec TYPE('b) n) a W \<bullet> v = (\<Sum>w\<in>W. a w * (w \<bullet> v))"
AVar: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e1 :: "expr"   and Q :: "'b assn"   and e2 :: "expr"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} e1-\<succ> {Q}"   and "\<forall>a. G,A\<turnstile>{Q\<leftarrow>In1 a} e2-\<succ> {\<lambda>Val:i:. avar G i a ..; R}" shows "G,A\<turnstile>{Normal P} e1.[e2]=\<succ> {R}"
ws_emb_u: shows "\<forall>i. Suc i < length ws \<longrightarrow> ws ! i \<noteq> u \<and> ws ! i \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
Prefixes_is_prefix: fixes q :: "'a list"   and p :: "'a list" shows "(q \<in> Prefixes p) = is_prefix q p"
schroeder_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z = 0) = (y \<cdot> x\<^sup>T ; z = 0)"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> F.D.CCC.arr f" shows "F.D.T.ToTC f = F.D.null"
subs''_def: shows "subs'' \<equiv> \<lambda>x xvec Tvec. psi_input_psiCase_rec_3 (\<zero>) (\<lambda>M N P. Output (substTerm M xvec Tvec) (substTerm N xvec Tvec)) (\<lambda>M I. Input (substTerm M xvec Tvec)) (\<lambda>C. Case) (\<lambda>P Q. (\<parallel>)) (\<lambda>y P. Res y) (\<lambda>\<Psi>. \<lbrace>substAssert \<Psi> xvec Tvec\<rbrace>) (\<lambda>P. Bang) (\<lambda>M P. Trm (substTerm M xvec Tvec)) (\<lambda>y I. Bind y) (\<bottom>\<^sub>c) (\<lambda>\<Phi> P C. Cond (substCond \<Phi> xvec Tvec)) x"
pred_sexpE: fixes p :: "('c, unit) Old_Datatype.node set \<times> ('c, unit) Old_Datatype.node set" assumes "p \<in> pred_sexp" obtains M :: "('c, unit) Old_Datatype.node set" and N :: "('c, unit) Old_Datatype.node set" where "p = (Ma__, Scons Ma__ Na__)" and "Ma__ \<in> sexp" and "Na__ \<in> sexp"   | M :: "('c, unit) Old_Datatype.node set" and N :: "('c, unit) Old_Datatype.node set" where "p = (N, Scons M N)" and "M \<in> sexp" and "N \<in> sexp"
lfilledk_def_1: shows "Lfilled j lholed' (vs' @ [$Br (j + (1 + k))]) lfilledk"
pref_prod_root: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<^sup>@ k" shows "u \<le>p r \<cdot> u"
stut_before: fixes F :: "'a \<Rightarrow> 'b" shows "STUTINV $F"
h3b: shows "length (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) \<le> length subsets"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P P' A\<^sub>P \<Psi>\<^sub>P"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (Cases Cs) P' [] (\<one>)"
simps: fixes x :: "'b extC"   and y :: "'b extC" shows "x + y = extC.Val const"
Int_iff: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
1: fixes i :: "'c" assumes "i \<in> B - A" shows "h i = \<one>"
discrim_respects: shows "(\<lambda>U. {freediscrim U}) respects exprel"
B: shows "tLast t \<in> set (toList (snd ec))"
assms: shows "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y"
w0_lset_P: shows "w0 \<in> lset P"
assoc: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" shows "(a @ b) @ c = a @ b @ c"
RSpan_add_closed: fixes ms :: "'m list"   and n :: "'m"   and n' :: "'m" assumes "set ms \<subseteq> M"   and "n \<in> RSpan ms"   and "n' \<in> RSpan ms" shows "n + n' \<in> RSpan ms"
weakNonInput: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a<x> \<prec> P'" shows "False"
symcl_def: fixes R :: "('b \<times> 'b) set" shows "symcl R \<equiv> R \<union> R\<inverse>"
sum_linear: fixes x :: "'a"   and y :: "'a" shows "sum_class.sum x \<le> sum_class.sum y \<or> sum_class.sum y \<le> sum_class.sum x"
poly_gcd_monic: fixes p :: "'a poly"   and q :: "'a poly" assumes "p \<noteq> 0 \<or> q \<noteq> 0" shows "monic (gcd p q)"
run_owner_ii_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_owner_ii n s)"
integrable_on_def: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a set" shows "(f integrable_on i) = (\<exists>y. (f has_integral y) i)"