hcju/isabelleps huggingface dataset:
atoms_dnf: fixes \<phi> :: "'c fm"   and as :: "'c list"   and a :: "'c" assumes "nqfree \<phi>"   and "as \<in> set (dnf \<phi>)"   and "a \<in> set as" shows "a \<in> atoms \<phi>"
src_hcomp: fixes \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.hseq \<mu> \<nu>" shows "src\<^sub>D (\<mu> \<star>\<^sub>D \<nu>) = src\<^sub>D \<nu>"
sup_ge1_basic_type: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" shows "\<tau> \<le> \<tau> \<squnion> \<sigma>"
valid_Not: fixes P :: "('state, 'effect) L_state"   and x :: "('a, 'pred, ('act, 'effect) L_action) Formula.formula" shows "L_transform.valid P (Formula.Not x) = (\<not> L_transform.valid P x)"
measurable_Pair: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and M1 :: "'b measure"   and g :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M M1"   and "g \<in> M \<rightarrow>\<^sub>M M2" shows "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
K_def: fixes i :: "int"   and j :: "int" shows "K i j \<equiv> if i = j then i else Max (mins i j)"
keys_atomize_poly: fixes p :: "nat \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "keys (atomize_poly p) = (\<Union>k\<in>keys p. (\<lambda>t. term_of_pair (t, k)) ` keys (lookup p k))"
conjugate_vec_sprod_comm: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "v \<bullet>c w = conjugate w \<bullet> v"
prim_rec: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3" shows "PrimRecOp g h \<in> PrimRec2"
add_le_same_cancel2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
subspace_hyperplane2: fixes a :: "'a" shows "subspace {x. x \<bullet> a = 0}"
is_lub_upper: fixes P :: "'b prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" assumes "P \<turnstile> lub(U, V) = T" shows "P \<turnstile> U \<le> T \<and> P \<turnstile> V \<le> T"
y_2: shows "range (flow0 y) = \<omega>_limit_set x"
lm089: fixes x :: "'c"   and X :: "'c set"   and Y :: "'d set" assumes "x \<notin> X" shows "(\<Union>f\<in>injections X Y. {f +< {(x, y)} |y. y \<in> Y - Range f}) = (\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f})"
green_typeII_chain_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" shows "green_typeII_chain i j F two_chain s \<equiv> R2 i j \<and> green_typeII_chain_axioms i j F two_chain s"
cons_rule: fixes P :: "assn"   and P' :: "assn"   and Q :: "'b \<Rightarrow> assn"   and Q' :: "'b \<Rightarrow> assn"   and c :: "'b Heap" assumes "P \<Longrightarrow>\<^sub>A P'"   and "\<And>x. Q x \<Longrightarrow>\<^sub>A Q' x"   and "<P'> c <Q>" shows "<P> c <Q'>"
sylow_thm: shows "\<exists>H. subgroup H G \<and> card H = p ^ a"
delta': fixes a :: "'c"   and g :: "'c \<Rightarrow> 'b" shows "(\<Prod>b. if a = b then g b else (1::'b)) = g a"
intras_same_level_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "\<forall>a\<in>set as. intra_kind (kind a)" shows "CFG.same_level_path kind get_return_edges as"
iterate_sel_def: fixes it :: "('c option \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c option \<Rightarrow> 'c option) \<Rightarrow> 'c option \<Rightarrow> 'c option"   and f :: "'d \<Rightarrow> 'c option" shows "iterate_sel it f = it (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x \<sigma>. f x) None"
abs_eq: fixes xa :: "'a ltl"   and x :: "'b list" shows "f_foldl_abs (Abs xa) x = Abs (foldl f xa x)"
sums_diff: fixes a :: "'a"   and g :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n - g n) sums (a - b)"
adjoined_in_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H'\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := H'\<rparr>)" shows "a \<in> adjoin (G\<lparr>carrier := H'\<rparr>) H a"
map_poly_order_of_real: fixes p :: "real poly"   and t :: "real" assumes "p \<noteq> 0" shows "order (of_real t) (map_poly of_real p) = order t p"
Quotient_ivl: shows "Quotient eq_ivl abs_ivl rep_ivl cr_ivl"
Wbis_trace: fixes c :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and tr :: "(('test, 'atom) com \<times> 'state) list \<times> 'state" assumes "\<And>s. mustT c s"   and "c \<approx>w c"   and "s \<approx> t"   and "finTrace tr"   and "iconfig tr = (c, s)" shows "\<exists>tr'. finTrace tr' \<and> iconfig tr' = (c, t) \<and> fstate tr \<approx> fstate tr'"
st_part_Ex1: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<exists>!t. t \<in> \<real> \<and> x \<approx> t"
less_numeral_extra_4: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
intact_axioms_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and I :: "'b set" shows "intact_axioms slices W I \<equiv> I \<subseteq> W \<and> stellar.quorum slices W I \<and> (\<forall>Q Q'. stellar.quorum (project slices I) W Q \<longrightarrow> stellar.quorum (project slices I) W Q' \<longrightarrow> Q \<inter> I \<noteq> {} \<longrightarrow> Q' \<inter> I \<noteq> {} \<longrightarrow> Q \<inter> Q' \<inter> I \<noteq> {})"
inv_comp: fixes f :: "'a"   and g :: "'a" assumes "local.iso f"   and "local.iso g"   and "seq g f" shows "local.inv (g \<cdot> f) = local.inv f \<cdot> local.inv g"
sum_part_character: fixes w :: "i"   and x :: "i"   and y :: "i" shows "P w (x \<oplus> y) = (\<forall>v. O v w \<longrightarrow> O v x \<or> O v y)"
realrel_def: shows "realrel = (\<lambda>X Y. Real.cauchy X \<and> Real.cauchy Y \<and> Real.vanishes (\<lambda>n. X n - Y n))"
fun_add_eval_car: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F"   and "g \<in> carrier F" shows "(f \<oplus>\<^bsub>F\<^esub> g) a = f a \<oplus> g a"
tp_len_assoc: shows "tp - len + 1 = tp - (len - 1)"
assms_3: fixes n :: "nat" shows "norm (s n) \<le> k / b ^ n"
ex_min_counterex: fixes I :: "'a set"   and N :: "'a literal multiset set" assumes "\<not> I \<Turnstile>s N" shows "\<exists>C\<in>N. \<not> I \<Turnstile> C \<and> (\<forall>D\<in>N. D < C \<longrightarrow> I \<Turnstile> D)"
preserves_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (R f)"
WT_resource_bot: fixes res :: "('c, 'd) resource" shows "\<bottom> \<turnstile>res res \<surd>"
ord_le_eq_trans: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a \<le>s b"   and "b = c" shows "a \<le>s c"
P_Union: fixes M :: "'a set" assumes "\<forall>S\<in>M. P S" shows "P (Inf M)"
simps: shows "prime_nat_main n jj is = (case is of [] \<Rightarrow> case next_candidates jj of (j, is) \<Rightarrow> prime_nat_main n j is | i # is \<Rightarrow> if i dvd n then n \<le> i else if i * i \<le> n then prime_nat_main n jj is else True)"
simps: fixes i :: "nat"   and v :: "rat vec" shows "from_index_geq0_vector i v = map (\<lambda>j. GEQ (lp_monom 1 (i + j)) (v $ j)) [0..<dim_vec v]"
upper_semilattice_def: fixes L :: "('e, 'f) gorder_scheme" shows "upper_semilattice L \<equiv> partial_order L \<and> upper_semilattice_axioms L"
non_trivial_ker_imp_algebraic: fixes K :: "'a set"   and x :: "'a" assumes "a_kernel (K [X]) R (\<lambda>p. local.eval p x) \<noteq> {[]}" shows "(algebraic over K) x"
get_attribute_locs_def: shows "get_attribute_locs \<equiv> l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute_locs"
tuple_in_hom': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "arr f"   and "local.dom f = a"   and "cod f = b"   and "arr g"   and "local.dom g = a"   and "cod g = c" shows "\<guillemotleft>\<langle>f, g\<rangle> : a \<rightarrow> b \<otimes> c\<guillemotright>"
indRelRTPO_impl_TRel_reflects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelRTPO TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
hext_heap_write: fixes h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and v :: "'addr val"   and h' :: "'heap" assumes "heap_write h a al v h'" shows "h \<unlhd> h'"
s_top: fixes n :: "nat" shows "s n \<noteq> \<top>"
is_scc_def: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" shows "is_scc E U = (U \<times> U \<subseteq> E\<^sup>* \<and> (\<forall>V. U \<subset> V \<longrightarrow> \<not> V \<times> V \<subseteq> E\<^sup>*))"
length_list_of: fixes xs :: "'c llist" assumes "lfinite xs" shows "enat (length (list_of xs)) = llength xs"
jump_jumpF: fixes a :: "real"   and f :: "real \<Rightarrow> real"   and l :: "real"   and r :: "real" assumes "isCont (inverse \<circ> f) a"   and "(f has_sgnx l) (at_left a)"   and "(f has_sgnx r) (at_right a)"   and "l \<noteq> 0"   and "r \<noteq> 0" shows "real_of_int (jump f a) = jumpF f (at_right a) - jumpF f (at_left a)"
kauff_mat_non_empty: fixes ws :: "wall" assumes "is_tangle_diagram ws" shows "kauff_mat ws \<noteq> []"
join_simps_2: fixes x :: "'c option" shows "join_option (Some x) = x"
set_val_locs_impl: shows "set_val_locs = local.a_set_val_locs"
req: shows "rot (l (- dr) - l dr) = (2 * dr / norm (f x)) *\<^sub>R f x"
cases: fixes a1 :: "'item list"   and a2 :: "'item list" assumes "a1 =\<^sub>S a2" obtains a :: "'item" and b :: "'item" and u :: "'item list" and v :: "'item list" where "a1 = u @ [a] @ [b] @ v" and "a2 = u @ [b] @ [a] @ v" and "ind a b"
set_less_eq_def2: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq> B) = (if finite UNIV then A \<sqsubseteq>''' B else if A \<in> infinite_complement_partition then A \<sqsubseteq>''' B \<or> B \<notin> infinite_complement_partition else B \<notin> infinite_complement_partition \<and> - B \<sqsubseteq>''' - A)"
vector_derivative_at: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real" assumes "(f has_vector_derivative f') (at x)" shows "vector_derivative f (at x) = f'"
liftE_def: fixes f :: "'d \<Rightarrow> ('f \<times> 'd) \<times> bool" shows "liftE f \<equiv> do r \<leftarrow> f; return (Inr r) od"
finite_completion_lemma: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "finite I" shows "(\<forall>i\<in>I. F \<in> A i \<longmapsto> A' i \<union> C) \<longrightarrow> (\<forall>i\<in>I. F \<in> A' i co A' i \<union> C) \<longrightarrow> F \<in> \<Inter> (A ` I) \<longmapsto> \<Inter> (A' ` I) \<union> C"
Solution_axioms: shows "Solution task.inPorts task.outPorts nodeOf task.hyps vertices task.labelsIn task.labelsOut (curry to_nat) map_lc lc Predicate_Formulas.closed subst lc_subst map_lc_subst (Var 0 []) id inst task_edges"
is_idirprod_def: fixes A :: "('c \<Rightarrow> 'a) set"   and S :: "'d \<Rightarrow> ('c \<Rightarrow> 'a) set"   and I :: "'d set" shows "DP.is_idirprod A S I = ((\<forall>i\<in>I. S i \<lhd> DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) \<and> A = IDirProds (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) S I \<and> DP.compl_fam S I)"
inf_sup_aci_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
LatticeTriangle2: fixes A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and D :: "'b \<Rightarrow> bool" shows "\<turnstile> (A \<leadsto> B \<or> D) \<and> (B \<leadsto> D) \<longrightarrow> (A \<leadsto> D)"
matrix_vector_mul_3: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "bounded_linear f" shows "(*v) (matrix f) = f"
borel_measurable_vimage: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and x :: "'b" assumes "f \<in> borel_measurable M" shows "f -` {x} \<inter> space M \<in> sets M"
assign_unique: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and a :: "'b"   and i :: "'b" assumes "Hintikka A H"   and "a \<in> assign A H i" shows "assign A H a = assign A H i"
associativitySym: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" shows "(\<Psi> \<otimes> \<Psi>') \<otimes> \<Psi>'' \<simeq> (\<Psi> \<otimes> \<Psi>'') \<otimes> \<Psi>'"
zmult_zless_mono2_lemma: fixes i :: "Int.int"   and j :: "Int.int"   and k :: "nat" assumes "i < j"   and "0 < k" shows "int k * i < int k * j"
get_env_def: fixes f :: "'s \<Rightarrow> ('c, 'm) envT" shows "local.get_env f = EnvT (\<lambda>r. get (\<lambda>s. run_env (f s) r))"
f': fixes x :: "real" assumes "a < x"   and "x < b" shows "(f has_derivative f' x) (at x)"
FinGroupRepresentation_FG: assumes "of_nat (card G) \<noteq> (0::'d)" shows "FinGroupRepresentation G (*) ActingGroup.group_ring"
closure_of_subtopology_subset: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" shows "subtopology X U closure_of S \<subseteq> X closure_of S"
read: shows "action_obs E a = NormalAction (ReadMem ad al v)"
OclAnd_idem: fixes X :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and X) = X"
unique_connecting_path_properties_3: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "hd (v \<leadsto>\<^bsub>G\<^esub> w) = v"
rel_interior_polyhedron_explicit: fixes F :: "'b set set"   and S :: "'b set"   and a :: "'b set \<Rightarrow> 'b"   and b :: "'b set \<Rightarrow> real" assumes "finite F"   and "S = affine hull S \<inter> \<Inter> F"   and "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> (0::'b) \<and> h = {x. a h \<bullet> x \<le> b h}"   and "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter> F'" shows "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
SUP_eq_const: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<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 I :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Sup (f ` I) = x"
summable_const_iff: fixes c :: "'b" shows "summable (\<lambda>_. c) = (c = (0::'b))"
S_transform_def: fixes x :: "('a, 'c, 'b) formula" shows "S_transform x = Abs_formula (S_transform_formula x)"
poly_roots_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" shows "poly_roots (prod f A) = (\<Sum>x\<in>A. poly_roots (f x))"
subset_eq_atLeast0_lessThan_card: fixes N :: "nat set"   and n :: "nat" assumes "N \<subseteq> {0..<n}" shows "card N \<le> n"
prv_cnjEL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (cnj \<phi> \<chi>)" shows "prv \<phi>"
tendsto: fixes f :: "'e \<Rightarrow> 'a"   and a :: "'a"   and F :: "'e filter"   and g :: "'e \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
Skolem_list_nth: fixes k :: "nat"   and P :: "nat \<Rightarrow> 'c \<Rightarrow> bool" shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. length xs = k \<and> (\<forall>i<k. P i (xs ! i)))"
case_rews_2: fixes f1 :: "'b"   and f2 :: "'c \<rightarrow> ['c] \<rightarrow> 'b" shows "list_case\<cdot>f1\<cdot>f2\<cdot>[] = f1"
assms: shows "HT_mods \<pi> mods P (PCall p) Q"
assms: shows "x\<^sup>2 + y\<^sup>2 = 1"
norm_cauchy_schwarz: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y \<le> norm x * norm y"
strict_implies_order: fixes a :: "'t"   and b :: "'t" assumes "lt_of_nat_term_order cmp_term a b" shows "le_of_nat_term_order cmp_term a b"
worst_chain_pred: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and i :: "nat" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "p (worst_chain p gt i) (worst_chain p gt (Suc i))"
endpoint_in_S_2: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S" shows "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt (endpoint_in_S a b)"
fSpan_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and ms :: "'c list" assumes "FGModule G smult V"   and "set ms \<subseteq> V" shows "R_scalar_mult.RSpan UNIV (aezfun_scalar_mult.fsmult smult) ms \<subseteq> V"
finite_set_and_nat_induct: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> nat \<Rightarrow> bool"   and k :: "nat" assumes "finite A"   and "\<And>A. finite A \<Longrightarrow> P A 0"   and "\<And>k. P {} k"   and "\<And>A k x. \<lbrakk>finite A; x \<notin> A; P A (Suc k); P (insert x A) k\<rbrakk> \<Longrightarrow> P (insert x A) (Suc k)" shows "P A k"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(extC.Val x2 = extC.Val y2) = (x2 = y2)"
monad_altc_stateT': assumes "monad_altc Monad_Overloading.return Monad_Overloading.bind altc" shows "monad_altc Monad_Overloading.return Monad_Overloading.bind altc"
mult_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma I \<FF>1"   and "(V, y) \<in> Sigma I \<FF>1"   and "W \<in> I"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stk1.mult_rel (stk1.class_of U x) (stk1.class_of V y) = stk1.class_of W (mult_str1 W (\<rho>1 U W x) (\<rho>1 V W y))"
A: shows "\<forall>\<^sub>F n in sequentially. ereal (Busemann_function_at (to_Gromov_completion (t n)) x y) < ereal (Busemann_function_at xi x y) + ereal e"
fp_cop_F_unfold: fixes ds :: "'b set" shows "fp_cop_F ds = cop_F ds (fp_cop_F ds)"
P_accD_2: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b"   and K1 :: "'a \<Rightarrow> 'a pmf"   and K2 :: "'b \<Rightarrow> 'b pmf" assumes "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (MC_pair.Kp K1 K2 x))\<^sup>*" shows "(x2, y2) \<in> (SIGMA x:UNIV. set_pmf (K2 x))\<^sup>*"
split_by_inter_def: fixes X :: "'d set" shows "split_by_inter X = SPEC (\<lambda>(Y, YS). X = Y \<union> YS)"
weak_transition_weakI: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act"   and Q' :: "'state"   and P' :: "'state" assumes "P \<Rightarrow> Q"   and "Q \<Rightarrow>\<langle>\<alpha>\<rangle> Q'"   and "Q' \<Rightarrow> P'" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle> P'"
plus_enat_eq_0_conv: fixes m :: "enat"   and n :: "enat" shows "(m + n = 0) = (m = 0 \<and> n = 0)"
poly_deg_zero: shows "poly_deg 0 = (0::'b)"
leg1_uniquely_isomorphic_1: shows "tab\<^sub>1 (local.dom \<mu>) \<cong> tab\<^sub>1 (cod \<mu>) \<star> \<mu>.chine"
cases: fixes a1 :: "'a list"   and a2 :: "'a literal multiset" assumes "eligible a1 a2" obtains DA :: "'a literal multiset" and As :: "'a list" where "a1 = As" and "a2 = DA" and "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0) DA"
det_snd_Gauss_Jordan_det_P: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det (snd (Gauss_Jordan_det_P A)) = (\<Prod>i\<in>UNIV. snd (Gauss_Jordan_det_P A) $ i $ i)"
af_interchange: fixes f :: "('e \<Rightarrow> 'd) af"   and x :: "'e" shows "Applicative.ap f (pure x) = Applicative.ap (pure (\<lambda>g. g x)) f"
rep_eq: fixes x :: "((real, 'c) vec, 'd) vec" shows "blinfun_apply (blinfun_of_vmatrix x) = (*v) x"
extract_Nil_code: fixes P :: "'a \<Rightarrow> bool" shows "List.extract P [] = None"
IH: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>a {wp\<^sub>3\<^sub>' c Q} c { Q}"
un_consec_seq: fixes m :: "nat"   and n :: "nat"   and l :: "nat" shows "m \<le> n \<and> n + 1 \<le> l \<longrightarrow> {m..n} \<union> {n + 1..l} = {m..l}"
subset_inext_in_imp_in: fixes n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "inext n A \<in> B"   and "A \<subseteq> B" shows "n \<in> B"
simps_6: fixes x :: "octo"   and y :: "octo" shows "Im5 (x - y) = Im5 x - Im5 y"
loc1_a_s_1: shows "s \<in>\<^sub>A path_summary loc1 loc2"
head_minus_poly_rev_list: fixes d :: "'b list"   and r :: "'b list" assumes "length d \<le> length r"   and "d \<noteq> []" shows "hd (minus_poly_rev_list (map ((*) (last d)) r) (map ((*) (hd r)) (rev d))) = (0::'b)"
Cons_3: shows "ys \<noteq> [] \<longrightarrow> last (a # xs) = hd ys"
monom_is_polynomial: fixes K :: "'a list set"   and a :: "'a list"   and n :: "nat" assumes "subring K (K [X])"   and "a \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}" shows "polynomial\<^bsub>K [X]\<^esub> K (UP.monom a n)"
FSwImorph_def: fixes h :: "'e \<Rightarrow> 'f"   and hA :: "'g \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'e, 'g, 'j) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'f, 'h, 'k) model_scheme" shows "FSwImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"
sup_inf_distrib_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<sqinter> z) = x \<squnion> y \<sqinter> x \<squnion> z"
NEA: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>" shows "\<^bold>\<not> \<AA> # \<Gamma> \<Rightarrow> \<Theta>"
f_def: shows "f \<equiv> \<lambda>x. 1 - (1 + c * inverse b * ln x powr - (1 + e)) powr p"
Derivation_swap_single_end_to_front: fixes i :: "nat"   and j :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "i < j"   and "derivation_ge D j"   and "Derivation a (D @ [(i, r)]) b" shows "Derivation a ((i, r) # derivation_shift D 1 (length (snd r))) b"
filter_shuffles: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "filter P ` shuffles xs ys = shuffles (filter P xs) (filter P ys)"
genPrefix_antisym: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> genPrefix r"   and "antisym r"   and "(ys, xs) \<in> genPrefix r" shows "xs = ys"
in_set_repl: fixes f :: "'a"   and fs :: "'a list"   and f' :: "'a"   and fs' :: "'a list" assumes "f \<in> set fs"   and "f \<noteq> f'" shows "f \<in> set (replace f' fs' fs)"
euler_char_def: shows "rev.H.euler_char \<equiv> int (card (verts rev.H)) - int (card (arcs rev.H) div 2) + int (card rev.H.face_cycle_sets)"