hcju/isabelleps huggingface dataset:
indep_setsD: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set"   and J :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "indep_sets F I"   and "J \<subseteq> I"   and "J \<noteq> {}"   and "finite J"   and "\<forall>j\<in>J. A j \<in> F j" shows "prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))"
VcompNml_Nml_Dom: fixes t :: "'a local.term" assumes "Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Dom t = t"
axioms_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplexEndomorphism_axioms X f"
monad_fail_alt_axioms_def: fixes fail :: "'d"   and alt :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "monad_fail_alt_axioms fail alt \<equiv> (\<forall>m. alt fail m = m) \<and> (\<forall>m. alt m fail = m)"
lcoset_rel_trans: fixes G :: "'a set" assumes "Group G" shows "trans (lcoset_rel G)"
abs_eq: fixes g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "Applicative.ap (pure g) (pure x) = pure (g x)"
interior_convex_hull_explicit_minimal: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "interior (convex hull S) = (if card S \<le> DIM('a) then {} else {y. \<exists>u. (\<forall>x\<in>S. 0 < u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"
finite_interval_int4: fixes a :: "int"   and b :: "int" shows "finite {i. a < i \<and> i < b}"
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
compl_le_swap2: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and y :: "'e"   and x :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq (uminus y) x" shows "less_eq (uminus x) y"
strictly_preserves_\<iota>: shows "F \<iota>\<^sub>B = \<iota>\<^sub>C"
that: fixes S' :: "ty" assumes "\<Gamma>\<turnstile>U<:S"   and "VarB x S # \<Gamma> \<turnstile> s : S'"   and "\<Gamma>\<turnstile>S'<:U'" shows "thesis"
Max_wrt_among_subset_2: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "Order_Predicates.preorder_on carrier le" shows "Max_wrt_among le A \<subseteq> A"
WT_converter_comp: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and \<I>'' :: "('e, 'f) \<I>"   and conv' :: "('c, 'd, 'e, 'f) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>', \<I>'' \<turnstile>\<^sub>C conv' \<surd>" shows "\<I>, \<I>'' \<turnstile>\<^sub>C conv \<odot> conv' \<surd>"
inflation: fixes x :: "'c" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>" shows "x \<sqsubseteq>\<^bsub>\<X>\<^bsub>F\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x)"
term_variants_Var: fixes P :: "'c \<Rightarrow> 'c list"   and x :: "'d" shows "term_variants_pred P (Var x) (Var x)"
left_unique_rel_pred: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_total A" shows "left_unique (rel_pred A)"
r: shows "reduced_row_echelon_form_upt_k A k"
inject: fixes x :: "char list"   and ya :: "char list" shows "(Name x = Name ya) = (x = ya)"
trans_refl_imp_rtrancl_id: fixes r :: "('b \<times> 'b) set" assumes "trans r"   and "refl r" shows "r\<^sup>* = r"
prems_3: shows "x \<notin> set (take n \<Gamma>)"
Arg_eq_iff: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "(Arg w = Arg z) = (\<exists>x>0. w = complex_of_real x * z)"
bKleisli_def: shows "bKleisli \<equiv> \<Lambda> f g x. f\<cdot>x >>= g"
prems_3: shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) (y # ys)"
app4: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and vs :: "v list"   and op0 :: "op0"   and refs' :: "v store_v list"   and ffi' :: "'b ffi_state"   and res :: "(v, v) result" assumes "evaluate_list ck env s1 (rev es) (s2, Rval vs) \<and> do_app (refs s2, ffi s2) op0 (rev vs) = Some ((refs', ffi'), res) \<and> op0 \<noteq> Opapp" shows "evaluate ck env s1 (App op0 es) (update_ffi (\<lambda>_. ffi') (update_refs (\<lambda>_. refs') s2), res)"
finite_nat_set_bounded_iff: fixes N :: "nat set" shows "finite N = (\<exists>n. \<forall>i\<in>N. i < n)"
composites_are_identities_2: shows "ide ((\<mu> \<star> \<nu>) \<star> \<pi>)"
get_edge_simps_2: fixes f :: "'b"   and d :: "'a"   and t :: "'b" shows "getData (f, d, t) = d"
converse_rtranclp_induct5: fixes r :: "'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> 'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and m :: "'b"   and sb :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<S> :: "'f"   and m' :: "'b"   and sb' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e"   and \<S>' :: "'f"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (m, sb, \<O>, \<R>, \<S>) (m', sb', \<O>', \<R>', \<S>')"   and "P m' sb' \<O>' \<R>' \<S>'"   and "\<And>a aa ab ac b ad ae af ag ba. \<lbrakk>r (a, aa, ab, ac, b) (ad, ae, af, ag, ba); r\<^sup>*\<^sup>* (ad, ae, af, ag, ba) (m', sb', \<O>', \<R>', \<S>'); P ad ae af ag ba\<rbrakk> \<Longrightarrow> P a aa ab ac b" shows "P m sb \<O> \<R> \<S>"
has_derivative_const: fixes c :: "'b"   and F :: "'a filter" shows "((\<lambda>x. c) has_derivative (\<lambda>x. 0::'b)) F"
cf_pos_def: fixes p :: "int poly" shows "cf_pos p = (content p = 1 \<and> 0 < lead_coeff p)"
Mul_interfree_Collector_Mutator: fixes j :: "nat"   and n :: "nat" assumes "j < n" shows "interfree_aux (Some (Mul_Collector n), {}, Some (Mul_Mutator j n))"
sees_field_idemp: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "P \<turnstile> D sees F,b:T in D"
add_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(a + b = (0::'a)) = (b = - a)"
lm091: fixes P :: "('d \<times> 'c) set"   and X :: "'d set" assumes "runiq (P\<inverse>)" shows "P `` (Domain P - X) \<inter> P `` X = {}"
single_iteration_inv: fixes i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i' \<le> i"   and "j' \<le> j"   and "i \<le> n"   and "j \<le> n" shows "fw m n k i j i' j' = fw m n k i' j' i' j'"
disjnt_subset1: fixes X :: "'b set"   and Y :: "'b set"   and Z :: "'b set" assumes "disjnt X Y"   and "Z \<subseteq> X" shows "disjnt Z Y"
omit_id: fixes ps :: "('b, 'c) fm list" shows "omit {} ps = ps"
terminal_pathI: fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "path v v' pth"   and "hyps_free pth"   and "terminal_vertex v'" shows "terminal_path v v' pth"
topology_def: fixes T :: "'b set set" shows "topology T \<equiv> (\<forall>x y. x open\<^bsub>T\<^esub> \<longrightarrow> y open\<^bsub>T\<^esub> \<longrightarrow> x \<inter> y open\<^bsub>T\<^esub>) \<and> (\<forall>M. (\<forall>m\<in>M. m open\<^bsub>T\<^esub>) \<longrightarrow> \<Union> M open\<^bsub>T\<^esub>)"
assms_2: shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
finfun_All_except_def: fixes A :: "'a list"   and P :: "('a, bool) finfun" shows "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> finfun_apply P a"
chamber_map: fixes C :: "'a set" assumes "folding.chamber C" shows "folding.chamber (f ` C)"
pne23: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid2 \<noteq> pid3"
prems_2: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
Model_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "CU.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP \<equiv> U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP"
order_lesseq_imp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(\<forall>z. x \<preceq> z \<longrightarrow> y \<preceq> z) = (y \<preceq> x)"
terminating_path_root_end_backward_terminating: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e" shows "backward_terminating_path_root r x"
simps_5: fixes v :: "aexp"   and va :: "aexp" shows "not (Less v va) = bexp.Not (Less v va)"
ENR_delete: fixes S :: "'a set"   and a :: "'a" assumes "ENR S" shows "ENR (S - {a})"
is_discrete: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c" assumes "discrete_diagram J C D" shows "partial_magma.arr J = partial_magma.ide J"
empty_eq_union: fixes M :: "'b multiset"   and N :: "'b multiset" shows "({#} = M + N) = (M = {#} \<and> N = {#})"
alpha: fixes a :: "name"   and x :: "'d"   and b :: "name"   and y :: "'d" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
segments_subset_convex_hull_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "closed_segment a b \<subseteq> convex hull {a, b, c}"
simps_18: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "value" shows "AExp_Lexorder.less_aexp_aux (Minus e1 e2) (L v) = False"
measurable_Shleg: fixes n :: "nat" shows "Shleg n \<in> borel_measurable borel"
vdiff_vsubset: fixes A :: "V"   and B :: "V"   and D :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "D \<subseteq>\<^sub>\<circ> C" shows "A -\<^sub>\<circ> C \<subseteq>\<^sub>\<circ> B -\<^sub>\<circ> D"
G'_def: shows "G' \<equiv> LG (on_triple g `` edges (snd R) \<union> snd G'') {0..<max (fst G'') (nextMax (Range g))}"
eq: shows "density (density M (\<lambda>x. ennreal (f x))) (\<lambda>x. ennreal (g x / f x)) = density M (\<lambda>x. ennreal (g x))"
ide_REP: fixes A :: "('a, 'a set) Maps.arr" assumes "Maps.arr A" shows "ide (Maps.REP A)"
simps: fixes G :: "'b list \<Rightarrow> bool"   and a :: "'b list" shows "hullp G a = (a = \<epsilon> \<or> (\<exists>w1 w2. a = w1 \<cdot> w2 \<and> G w1 \<and> hullp G w2))"
hyperpow_le: fixes x :: "'b star"   and y :: "'b star"   and n :: "nat star" assumes "0 < x"   and "x \<le> y" shows "x pow n \<le> y pow n"
sel_rews_5: fixes nat :: "nat" shows "appF\<cdot>(ValN\<cdot>nat) = \<bottom>"
c_A_def: fixes n :: "nat" shows "c_A n = int (index (swaps (paid_A ! n) (s_A n)) (qs ! n) + 1)"
HFinite_mult: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "x * y \<in> HFinite"
attractor_def: fixes p :: "Player"   and W :: "'a set" shows "attractor p W = lfp (\<lambda>S. W \<union> S \<union> directly_attracted p S)"
distrib_1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
ideD_1: fixes a :: "'b" assumes "D.ide a" shows "D.arr a"
inf_level: fixes l1 :: "level"   and l2 :: "level" shows "inf l1 l2 \<equiv> if Lo \<in> {l1, l2} then Lo else Hi"
assms_1: shows "inv_into M (ordermap M VWF) \<alpha> \<le> inv_into M (ordermap M VWF) \<beta>"
some_third_chamber_def: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set" assumes "ThickChamberComplex X" shows "ThickChamberComplex.some_third_chamber X C D z \<equiv> SOME E. E \<in> X - {C, D} \<and> z \<lhd> E"
WT_rassocl\<^sub>C: fixes \<I>1 :: "('h, 'k) \<I>"   and \<I>2 :: "('i, 'l) \<I>"   and \<I>3 :: "('j, 'm) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3), (\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<oplus>\<^sub>\<I> \<I>3 \<turnstile>\<^sub>C rassocl\<^sub>C \<surd>"
non_empty_cycle_root_var_axioms_1: fixes r :: "'a"   and x :: "'a" shows "non_empty_cycle_root r x = (x\<^sup>T ; 1 \<le> x\<^sup>T\<^sup>+ ; r \<and> is_inj x \<and> is_p_fun x \<and> point r \<and> r \<le> x\<^sup>T ; 1)"
distinct_323: fixes x121 :: "char list"   and x122 :: "char list"   and x123 :: "'a exp list"   and x201 :: "char list"   and x202 :: "'a exp"   and x203 :: "char list list"   and x204 :: "'a exp" shows "x121\<bullet>\<^sub>sx122(x123) \<noteq> RI (x201,x202) ; x203 \<leftarrow> x204"
assms_2: shows "ev \<in> set (take (j - i) (drop i t))"
ceiling_less_one: fixes x :: "'a" shows "(\<lceil>x\<rceil> < 1) = (x \<le> (0::'a))"
hyps_4: shows "Fun f T = t \<cdot> \<delta>"
simps_6: fixes t :: "'c local.term" shows "Inv \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = \<^bold>\<l>\<^bold>[Inv t\<^bold>]"
concurrent_store_store_independent_technical: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e :: "'b"   and mo :: "'a"   and r :: "'a" assumes "xs prefix of j"   and "(i, Store e mo i) \<in> set (node_deliver_messages xs)"   and "(r, Store e i r) \<in> set (node_deliver_messages xs)" shows "hb (i, Store e mo i) (r, Store e i r)"
pause_writer_def: fixes out :: "'o"   and c :: "'i \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.pause_writer out c = WriterT (pause out (\<lambda>input. run_writer (c input)))"
of_int_dirichlet_prod: fixes f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and n :: "nat" shows "of_int (dirichlet_prod f g n) = dirichlet_prod (\<lambda>n. of_int (f n)) (\<lambda>n. of_int (g n)) n"
stails_strict: shows "stails\<cdot>\<bottom> = \<bottom>"
simps_1: shows "subcomp sA1 = {sA11, sA12}"
onode_sos_dest_is_net_state'': fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and p :: "'e"   and R :: "nat set"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, NodeS i p R), a, \<sigma>', s') \<in> onode_sos S" shows "\<exists>\<zeta>' R'. s' = NodeS i \<zeta>' R'"
rk_ABPRa: fixes A :: "Points"   and B :: "Points"   and P :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and Q :: "Points"   and R :: "Points"   and a :: "Points" assumes "rk {A, B, P} = 3"   and "rk {A, B, C, A', B', C', P} = 3"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {A', B', P, Q} = 4" shows "4 \<le> rk {A, B, P, R, a}"
HPhase1or2Write_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "p \<noteq> q"   and "HInv4a s p"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
ord_spmf_reflI: fixes p :: "'c spmf"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord x x" shows "ord_spmf ord p p"
prefix_fininfE: fixes u :: "'a list"   and v :: "'a stream" assumes "u \<le>\<^sub>F\<^sub>I v" obtains w :: "'a stream" where "v = u @- w"
irrefl: shows "\<forall>x\<in>set xs. \<not> gt x x"
eexp_strict_mono2: fixes x :: "ereal"   and y :: "ereal" assumes "x < y" shows "eexp x < eexp y"
primorial'_conv_primorial: fixes n :: "nat" assumes "0 < n" shows "primorial' n = primorial (real (nth_prime (n - 1)))"
leftmost_cons_less: fixes i :: "nat"   and u :: "symbol list"   and v :: "symbol list" assumes "i < length u" shows "leftmost i (u @ v) = leftmost i u"
\<iota>_def: shows "\<F>\<^sub>SC.\<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<equiv> \<I>"
manipulable_def: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt" shows "manipulable f \<equiv> \<exists>P\<in>Prof. \<exists>i. \<exists>L\<in>Lin. f P <\<^bsub>P i\<^esub> f (P(i := L))"
lossless_R2_xor: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "lossless_spmf (R2_xor A B)"
ref_two_step': fixes A :: "'a nres"   and B :: "'a nres"   and R :: "('b \<times> 'a) set" assumes "A \<le> B" shows "\<Down> R A \<le> \<Down> R B"
get_ancestors_si_parent_child_a_host_shadow_root_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors" shows "((ptr, child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*) = (ptr \<in> set ancestors)"
disc_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "p a" shows "is_TNil (unfold_tllist p g1 g21 g22 a)"
add_edge_ind: fixes a :: "nat"   and w :: "int"   and b :: "nat"   and L :: "(nat \<times> int \<times> nat) list"   and F :: "(nat \<times> int \<times> nat) set" assumes "(a, w, b) \<in> set L" shows "add_edge a w b \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = F\<rparr> = \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = insert (a, w, b) F\<rparr>"
hidden_def: fixes xs :: "'a list"   and i :: "nat" shows "hidden xs i \<equiv> i < length xs \<and> xs ! i \<in> set (drop (i + 1) xs)"
c_manifold_def: fixes charts :: "('b, 'c) chart set"   and k :: "enat" shows "c_manifold charts k \<equiv> \<forall>c1 c2. c1 \<in> charts \<longrightarrow> c2 \<in> charts \<longrightarrow> k-smooth_compat c1 c2"
dbl_inc_def: fixes minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and uminus :: "'f \<Rightarrow> 'f"   and one :: "'f"   and x :: "'f" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one x = plus (plus x x) one"
lfp_ordinal_induct: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; S \<le> lfp f\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (\<Squnion>M)" shows "P (lfp f)"
crename_OrR1': fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = OrR1 <c>.N d"   and "c \<sharp> (R, a, b)"   and "d \<sharp> a" shows "(\<exists>N'. R = OrR1 <c>.N' d \<and> N'[a\<turnstile>c>b] = N) \<or> (\<exists>N'. R = OrR1 <c>.N' a \<and> b = d \<and> N'[a\<turnstile>c>b] = N)"
elims_3: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a stream" assumes "\<not> holds x xa" obtains P :: "'a \<Rightarrow> bool" and xs :: "'a stream" where "x = P" and "xa = xs" and "\<not> P (shd xs)"
sorted_distinct_set_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
simps_7: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> (\<phi> M\<^sub>n \<psi>) w = \<F>\<G> \<phi> w \<union> \<F>\<G> \<psi> w"
True_implies_equals: fixes P :: "prop" shows "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
limit_def: fixes x :: "nat \<Rightarrow> 'a" shows "limit x \<equiv> {a. \<exists>\<^sub>\<infinity>n. x n = a}"
4: shows "t \<squnion> e \<le> - - ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g)"
abs_eq: fixes x :: "bool list" shows "of_bl x = Word.Word (bl_to_bin x)"
sigma_finite_measure_axioms: shows "sigma_finite_measure (distr M T Y)"
TC_add': fixes z :: "V"   and x :: "V"   and y :: "V" shows "(z \<sqsubset> x + y) = (z \<sqsubset> x \<or> (\<exists>v. v \<sqsubset> y \<and> z = x + v))"
is_nth_powerI: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x = y ^ n" shows "is_nth_power n x"
boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> min b c" obtains "a \<le> b" and "a \<le> c"
inverse_arrowsE: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "inverse_arrows f g" obtains "ide (g \<cdot> f)" and "ide (f \<cdot> g)"
smult_conv_map_poly: fixes c :: "'b"   and p :: "'b poly" shows "smult c p = map_poly ((*) c) p"
Spec_wf_conv: fixes \<tau> :: "'i"   and p :: "'j"   and r :: "(('e \<times> 'f) \<times> 'i \<times> 'j) set"   and P :: "'f \<Rightarrow> 'k \<Rightarrow> 'e set"   and Q :: "'f \<Rightarrow> 'k \<Rightarrow> 'g"   and A :: "'f \<Rightarrow> 'k \<Rightarrow> 'h"   and Procs :: "'f set" shows "(\<lambda>(P, q, Q, A). (P \<inter> {s. ((s, q), \<tau>, p) \<in> r}, q, Q, A)) ` (\<Union>p\<in>Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)}) = (\<Union>q\<in>Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<tau>, p) \<in> r}, q, Q q Z, A q Z)})"
assms_2: shows "f \<in> restr_to_subalg M F \<rightarrow>\<^sub>M N"
endomorph: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleEnd G smult V T" shows "T ` V \<subseteq> V"
chine_simps_3: shows "C.cod HfHgh.chine = HfHgh.cod.apex"
integer_of_int_int_of_integer: fixes k :: "integer" shows "integer_of_int (int_of_integer k) = k"
snapshot_state_must_have_been_reached: fixes t :: "('a, 'b, 'c) event list"   and p :: "'a"   and u :: "'b"   and i :: "nat" assumes "trace init t final"   and "ps final p = Some u"   and "\<not> ps (S t i) p \<noteq> None"   and "ps (S t (i + 1)) p \<noteq> None"   and "i < length t" shows "states (S t i) p = u"
ThrowThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw (throw e),s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
left_moving_Bk_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_goon_left_moving_O m rs (c, Oc # list)"   and "hd c = Bk" shows "wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
sum_2: fixes f :: "2 \<Rightarrow> 'a" shows "sum f UNIV = f 1 + f 2"
f_subdist: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "f (x \<cdot> y) \<le> f x"
mcontI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c" assumes "monotone orda ordb f"   and "cont luba orda lubb ordb f" shows "mcont luba orda lubb ordb f"
assms_5: shows "\<forall>n>n\<^sub>0. f n = f (nat \<lceil>a * real n + b\<rceil>) + f (nat \<lceil>c * real n + d\<rceil>) + C\<^sub>1 * real n + C\<^sub>2"
If_cases: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "finite A" shows "(\<Sum>x\<in>A. if P x then h x else g x) = sum h (A \<inter> {x. P x}) + sum g (A \<inter> - {x. P x})"