hcju/isabelleps huggingface dataset:
simps_1: fixes l :: "nat"   and i :: "int" shows "compile l (Val i) = ([Push i], l)"
distinct_4: fixes trm1' :: "trm"   and trm2' :: "trm"   and name :: "name" shows "App trm1' trm2' \<noteq> Var name"
Sup_mono: fixes x :: "'b"   and y :: "'b" assumes "x \<sqsubseteq> y"   and "x \<in> Y"   and "y \<in> Y" shows "\<Squnion> (f x ` Y) \<le> \<Squnion> (f y ` Y)"
rsum_Nil: fixes f :: "real \<Rightarrow> real" shows "rsum [] f = 0"
prod_defs_3: fixes f :: "nat \<Rightarrow> 'b" shows "convergent_prod f \<equiv> \<exists>M p. raw_has_prod f M p"
assms_3: assumes "{#a, b#} \<subseteq># B" shows "P"
comp_assoc: fixes h :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" shows "(h \<bullet> g) \<bullet> f = h \<bullet> g \<bullet> f"
sdrop_smap: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and s :: "'c stream" shows "sdrop n (smap f s) = smap f (sdrop n s)"
that: fixes p :: "nat \<Rightarrow> nat" assumes "p permutes {..<length ys}"   and "permute_list p ys = xs" shows "thesis"
open_Int_closure_subset: fixes S :: "'a set"   and T :: "'a set" assumes "open S" shows "S \<inter> closure T \<subseteq> closure (S \<inter> T)"
nn_integral_count_space: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite {a \<in> A. 0 < f a}" shows "integral\<^sup>N (count_space A) f = (\<Sum>a | a \<in> A \<and> 0 < f a. f a)"
select_false: fixes n :: "i32"   and v1 :: "v"   and v2 :: "v" assumes "int_eq n 0" shows "\<lparr>[$C v1, $C v2, $C ConstInt32 n, $Select]\<rparr> \<leadsto> \<lparr>[$C v2]\<rparr>"
G: shows "set xs \<subseteq> range p \<union> range q"
cka_1: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
beta_reduces_app_right: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B \<rightarrow>\<beta>\<^sup>* B'" shows "App A B \<rightarrow>\<beta>\<^sup>* App A B'"
fB_def: shows "f \<restriction>\<^sup>l\<^sub>\<circ> B = (f \<restriction>\<^sup>l\<^sub>\<circ> A) \<restriction>\<^sup>l\<^sub>\<circ> B"
reducedAllocation: fixes a :: "('a \<times> 'b set) set"   and b :: "('a \<times> 'b set) set" assumes "a \<in> allocationsUniverse" shows "a - b \<in> allocationsUniverse"
max_comm: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.non_strict_order less_eq" shows "ord.max less_eq x y = ord.max less_eq y x"
assms: shows "x \<in> carrier (residue_ring (p ^ 0))"
obj_simps_2: fixes a :: "'a" assumes "B.obj a" shows "src\<^sub>B a = a"
MOST_inj: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "inj f" shows "\<forall>\<^sub>\<infinity>x. P (f x)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "tarski_absolute_space C B"   and "tarski_absolute_axioms C B" shows "tarski_absolute C B"
learn_fin_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_fin \<psi> U s" shows "\<exists>t. learn_fin \<chi> U t"
oth_class_taut_3_d: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<or> \<phi> \<^bold>\<equiv> \<phi> in v]"
invert_side_of_triangle_2: fixes h :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<phi>.B.arr h"   and "f \<cdot>\<^sub>B g = h"   and "\<phi>.B.iso g" shows "\<phi>.B.seq h (\<phi>.B.inv g) \<and> f = h \<cdot>\<^sub>B \<phi>.B.inv g"
infinite: fixes A :: "'a set" assumes "infinite A" shows "F A = \<^bold>\<bottom>"
a'b': shows "a' \<cdot> b - a' \<cdot> b' \<in> I"
simps_2: fixes uw :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and v :: "'b" shows "lift2_option uw (Some v) None = None"
edom_esing_subset: fixes x :: "'a"   and n :: "'b" shows "edom (esing x\<cdot>n) \<subseteq> {x}"
nats_of_boolss_div2: fixes bs :: "bool list"   and n :: "nat"   and bss :: "bool list list" assumes "length bs = n"   and "list_all (is_alph n) bss" shows "map (\<lambda>x. x div 2) (nats_of_boolss n (bs # bss)) = nats_of_boolss n bss"
invariant_succs: fixes I :: "'c \<Rightarrow> bool"   and succs :: "'c \<Rightarrow> 'c list"   and S :: "'c set" assumes "invariant I succs"   and "\<forall>x\<in>S. I x" shows "\<forall>x\<in>(Rel succs)\<^sup>* `` S. I x"
assms: shows "h \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r children"
induct: fixes x1 :: "'var"   and x2 :: "'node SDG_node"   and P :: "'var \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 \<in> Def\<^bsub>SDG\<^esub> x2"   and "\<And>m V n. \<lbrakk>valid_node m; V \<in> Def m; n = CFG_node m\<rbrakk> \<Longrightarrow> P V n"   and "\<And>n m x p ins outs V. \<lbrakk>valid_SDG_node n; n = Formal_in (m, x); get_proc m = p; (p, ins, outs) \<in> set procs; V = ins ! x\<rbrakk> \<Longrightarrow> P V n"   and "\<And>n m x V. \<lbrakk>valid_SDG_node n; n = Actual_out (m, x); V = ParamDefs m ! x\<rbrakk> \<Longrightarrow> P V n" shows "P x1 x2"
chamber_image_and_complement_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and D :: "'b set" assumes "ThinChamberComplexFolding X f"   and "D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X" shows "f \<turnstile> ChamberComplex.chamber_system X = {B \<in> ChamberComplex.chamber_system X. order.greater (ChamberComplex.chamber_distance X B D) (ChamberComplex.chamber_distance X B (f ` D))}"
Basis_list_nth_nonzero: fixes i :: "nat" assumes "i < length Basis_list" shows "Basis_list ! i \<noteq> (0::'b)"
Lt_lt_conv: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Lt) = lt_of_comp comp x y"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) ((\<otimes>\<^sub>C) (C.cod \<iota>\<^sub>C))"
strict_trans2: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "strict_prefix b a"   and "prefix c b" shows "strict_prefix c a"
normalizeI: fixes x :: "trm"   and xa :: "trm" assumes "x \<Down> xa" shows "pred.eval (Execute.normalize x) xa"
H: shows "\<Delta> @ \<Gamma> \<turnstile> t : T"
return_wp_rule: fixes Q :: "'c \<Rightarrow> assn"   and x :: "'c" shows "<Q x> return x <Q>"
cong: fixes ord :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool" shows "pm_powerprod.is_GB_cofactor_bound ord \<equiv> pm_powerprod.is_GB_cofactor_bound ord"
nonpos_Ints_def: shows "\<int>\<^sub>\<le>\<^sub>0 = {of_int n |n. n \<le> 0}"
selectlike_Sup: fixes M :: "(variable \<Rightarrow> real) set set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "selectlike (\<Union> M) \<nu> V = \<Union> {selectlike X \<nu> V |X. X \<in> M}"
drop_eq: shows "drop (length A_pos) (L_pos @ L_neg) = L_neg"
err_semilat_UnionI: fixes AS :: "'b set set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err" assumes "\<forall>A\<in>AS. err_semilat (A, r, f)"   and "AS \<noteq> {}"   and "\<forall>A\<in>AS. \<forall>B\<in>AS. A \<noteq> B \<longrightarrow> (\<forall>a\<in>A. \<forall>b\<in>B. \<not> a \<sqsubseteq>\<^bsub>r\<^esub> b \<and> a \<squnion>\<^bsub>f\<^esub> b = Err)" shows "err_semilat (\<Union> AS, r, f)"
assms_1: shows "dom (C (AllowPortFromTo a b po)) \<noteq> {}"
assms_4: shows "Domain a \<inter> X \<noteq> {}"
lipschitz_on_nonneg: fixes L :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "L-lipschitz_on X f" shows "0 \<le> L"
vimage_eq: fixes a :: "'e"   and f :: "'e \<Rightarrow> 'f"   and B :: "'f set" shows "(a \<in> f -` B) = (f a \<in> B)"
i_take_add: fixes i :: "nat"   and j :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "f \<Down> (i + j) = f \<Down> i @ f \<Up> i \<Down> j"
slice_eq_bounds_empty: fixes i :: "nat"   and xs :: "'a list" shows "Misc.slice i i xs = []"
extends_subst_refl: fixes \<sigma> :: "'a \<Rightarrow> 'b option" shows "extends_subst \<sigma> \<sigma>"
sel_rews_3: shows "shead\<cdot>[::] = \<bottom>"
primekeyrewrite: fixes p :: "nat"   and a :: "nat"   and b :: "nat" assumes "prime p"   and "p dvd a * b"   and "\<not> p dvd a" shows "p dvd b"
prems: fixes x :: "nat" assumes "x \<in> free_vars (Random dst e)" shows "\<sigma>1 x = \<sigma>2 x"
list_diff_closed: fixes gs :: "'r list"   and hs :: "'r list" assumes "set gs \<subseteq> R"   and "set hs \<subseteq> R" shows "set (map2 (-) gs hs) \<subseteq> R"
a_offending: shows "f \<in> set_offending_flows G nP"
rep_list_def: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list r = ideal.rep (pm_of_idx_pm fs (vectorize_poly r))"
cotangent_space_def: fixes p :: "'a" shows "cotangent_space p = dual_space (tangent_space p)"
no_loops: fixes v :: "nat" shows "(v, v) \<notin> E"
subst_compose_assoc: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'e) Term.term"   and \<tau> :: "'e \<Rightarrow> ('b, 'd) Term.term"   and \<mu> :: "'d \<Rightarrow> ('b, 'c) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> \<circ>\<^sub>s \<mu> = \<sigma> \<circ>\<^sub>s (\<tau> \<circ>\<^sub>s \<mu>)"
differenceSumVsCardinality: fixes Z :: "'c set"   and X :: "'c set"   and Y :: "'c set" assumes "Z \<subseteq> X \<union> Y"   and "finite Z" shows "int (sum (X <| Y) X) - int (sum (X <| Y) Z) = int (card X) - int (card (Z \<inter> X))"
cs_single_pd_intermed: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.cs prob \<pi> n = [\<pi> n]"   and "k \<le> n" shows "IFC_def.is_pd prob (\<pi> n) (\<pi> k)"
srev_induct: fixes P :: "[:'r:] \<Rightarrow> bool"   and xs :: "[:'r:]" assumes "P \<bottom>"   and "P [::]"   and "\<And>x xs. \<lbrakk>x \<noteq> \<bottom>; xs \<noteq> \<bottom>; P xs\<rbrakk> \<Longrightarrow> P (xs :@ [:x:])" shows "P xs"
axioms_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "prebicategory_with_homs V H \<a> src trg" shows "prebicategory V H \<a>"
seqE: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f" obtains "J.arr f" and "J.arr g" and "J.dom g = J.cod f"
assms_4: fixes S :: "'x set"   and \<sigma> :: "'\<sigma>" assumes "S \<subseteq> S0"   and "S \<noteq> S0"   and "\<not> c \<sigma>"   and "I S \<sigma>"   and "\<forall>x\<in>S0 - S. \<forall>y\<in>S. R y x" shows "P \<sigma>"
segment_a'_a_neD: assumes "segment G.face_cycle_succ a' a \<noteq> {}" shows "segment G.face_cycle_succ a' a \<in> H.face_cycle_sets"
transitive_subset: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "transitive A less_eq"   and "B \<subseteq> A" shows "transitive B less_eq"
var_not_fn: fixes x :: "'b"   and y :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm" shows "Var x \<noteq> Fn y T A"
coprime_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a b = (\<forall>c. dvd.dvd times c a \<longrightarrow> dvd.dvd times c b \<longrightarrow> dvd.dvd times c one)"
sub_conjD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "sub_conj t"   and "sound P"   and "sound Q" shows "t P && t Q \<tturnstile> t (P && Q)"
two_div_sqrt_two: shows "2 * complex_of_real (sqrt (1 / 2)) = complex_of_real (sqrt 2)"
create_gba__fin: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. finite (g_V \<A>))"
order_iff_strict: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
simps_400: fixes x201 :: "char list"   and x202 :: "'a exp"   and x203 :: "char list list"   and x204 :: "'a exp"   and x191 :: "char list"   and x192 :: "char list list"   and x193 :: "bool"   and x194 :: "'a exp" shows "RI (x201,x202) ; x203 \<leftarrow> x204 \<noteq> INIT x191 (x192,x193) \<leftarrow> x194"
all_tran_formulas_validated: fixes fresh :: "'a set \<Rightarrow> 'a"   and F :: "'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "freshstuff_sema fresh" shows "\<forall>J\<in>set (subformulae F). freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> J = freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> freshstuff.tseytin_toatom fresh F J"
mset_subset_eq_exists_conv: fixes A :: "'c multiset"   and B :: "'c multiset" shows "(A \<subseteq># B) = (\<exists>C. B = A + C)"
bij_betw_id_iff: fixes A :: "'c set"   and B :: "'c set" shows "bij_betw id A B = (A = B)"
ipoly_uminus_inner: fixes as :: "int list"   and x :: "'a" shows "ipoly (poly_uminus_inner as) x = ipoly (Poly as) (- x)"
o_iso_def: fixes hom2 :: "('i, 'j, 'g, 'h) digraph_isomorphism"   and hom1 :: "('e, 'f, 'i, 'j) digraph_isomorphism" shows "o_iso hom2 hom1 = \<lparr>iso_verts = iso_verts hom2 \<circ> iso_verts hom1, iso_arcs = iso_arcs hom2 \<circ> iso_arcs hom1, iso_head = iso_head hom2, iso_tail = iso_tail hom2\<rparr>"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "P F"   and "P T"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
NextPropagate'_preserves_prop_invs: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('d, 'c, 'a) Combined.configuration"   and q :: "'d"   and c1 :: "('d, 'c, 'a) Combined.configuration"   and p :: "'d" assumes "dataflow_topology summary results_in"   and "inv_implications_nonneg (prop_config c0 q) \<and> dataflow_topology.inv_imps_work_sum summary results_in (prop_config c0 q)"   and "dataflow_topology.NextPropagate' summary results_in c0 c1 p" shows "inv_implications_nonneg (prop_config c1 q) \<and> dataflow_topology.inv_imps_work_sum summary results_in (prop_config c1 q)"
tree2_induct: fixes P :: "('a \<times> 'b) tree \<Rightarrow> bool"   and treea :: "('a \<times> 'b) tree" assumes "P \<langle>\<rangle>"   and "\<And>x1 a b x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P \<langle>x1, (a, b), x3\<rangle>" shows "P treea"
action_loc_aux_compP: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'd \<Rightarrow> 'c"   and P :: "'d prog" shows "action_loc_aux (compP f P) = action_loc_aux P"
sym_join: fixes r :: "('b \<times> 'b) set" shows "sym (r\<^sup>\<down>)"
sign_simps_36: fixes am :: "'c"   and bm :: "'c"   and ck :: "'c" shows "am * (bm + ck) = am * bm + am * ck"
ocondition_K_true: fixes T :: "'d \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" shows "ocondition (\<lambda>_. True) T F = T"
jumpF_not_infinity: fixes F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "continuous F g"   and "F \<noteq> bot" shows "jumpF g F = 0"
mult_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc m * n = n + m * n"
IntegerLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "int"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (IntegerLiteral c) : \<tau>" obtains "\<tau> = Integer[1]"
conj_disj_not_abs: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<and> (\<not> P \<or> Q)) = (P \<and> Q)"
distinct_7: shows "ccode.Lt \<noteq> ccode.Eq"
output_size_shallow_model': fixes Z :: "nat"   and M :: "nat"   and N :: "nat" shows "output_size (shallow_model' Z M N) = Z"
UINF_is_exists: fixes P :: "'b \<Rightarrow> 'a upred" shows "(\<Sqinter> x \<bullet> P x) = ushEx P"
exec_step_ind_Invoke_NoMethod: fixes r :: "val"   and stk :: "val list"   and n :: "nat"   and C :: "char list"   and h :: "heap"   and P :: "jvm_prog"   and M :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "r = stk ! n"   and "C = cname_of h (the_Addr r)"   and "r \<noteq> Null"   and "\<nexists>Ts T m D b. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "exec_step_ind (StepI (Invoke M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchMethodError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
nsumTail: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>Suc n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f Suc n = \<Sigma>\<^sub>e A (\<lambda>i. f (Suc i)) n \<plusminus> f 0"
less_add_same_cancel1: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (power_integer x xa) = Optimize_Integer.power_int (int_of_integer x) (int_of_integer xa)"
E_ss_VxV: shows "cf.E \<subseteq> cf.V \<times> cf.V"
ffb_inj_iff: fixes f :: "'d \<Rightarrow> 'c set"   and g :: "'d \<Rightarrow> 'c set" shows "(fb\<^sub>\<F> f = fb\<^sub>\<F> g) = (f = g)"
quotients_length: shows "length comp\<HH>.quotients + 1 = length \<HH>"
in_KeysE: fixes s :: "'a"   and F :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "s \<in> Keys F" obtains f :: "'a \<Rightarrow>\<^sub>0 'b" where "s \<in> keys f" and "f \<in> F"
tickFree_append: fixes s :: "'c event list"   and t :: "'c event list" shows "tickFree (s @ t) = (tickFree s \<and> tickFree t)"
fsub_def: fixes m :: "roundmode"   and a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fsub m a b = (if is_nan a \<or> is_nan b \<or> is_infinity a \<and> is_infinity b \<and> sign a = sign b then some_nan else if is_infinity a then a else if is_infinity b then - b else zerosign (if is_zero a \<and> is_zero b \<and> sign a \<noteq> sign b then sign a else if m = To_ninfinity then 1 else 0) (IEEE.round m (valof a - valof b)))"
deg_pp_def: shows "deg_pp \<equiv> map_fun pp.mapping_of id deg_pm"
Sup_set_def: fixes A :: "'a set set" shows "\<Union> A = {x. \<Squnion>((\<in>) x ` A)}"
mkKripke_def: fixes ws :: "'c set"   and rels :: "'a \<Rightarrow> ('c \<times> 'c) set"   and val :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "mkKripke ws rels val \<equiv> \<lparr>worlds = ws, relations = \<lambda>a. Restr (rels a) ws, valuation = val\<rparr>"
renamings_apart_length: fixes Cs :: "'a literal multiset list" shows "length (renamings_apart Cs) = length Cs"
sams123231: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A \<noteq> C"   and "B \<noteq> C" shows "SAMS A B C B C A"
deg_fun_plus: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" assumes "finite (supp_fun s)"   and "finite (supp_fun t)" shows "deg_fun (s + t) = deg_fun s + deg_fun t"
assms_2: shows "bijective (c * top) \<and> bijective (c[[top]])"
disj_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<or>. q) = (A \<union> {.\<not> p} \<turnstile> q)"
simps_5: fixes p :: "real \<times> real \<times> real"   and L :: "atomUni list" shows "separateAtoms (NeqUni p # L) = (let (a, b, c, d) = separateAtoms L in (a, b, c, p # d))"
prefix_def: fixes xs :: "'c list"   and ys :: "'c list" shows "xs \<le> ys = (\<exists>zs. ys = xs @ zs)"
dropWhile_idem_iff: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(dropWhile P xs = xs) = no_leading P xs"
tinres_steps2: fixes r :: "cell list"   and r' :: "cell list"   and ss :: "nat"   and l :: "cell list"   and t :: "(action \<times> nat) list"   and stp :: "nat"   and sa :: "nat"   and la :: "cell list"   and ra :: "cell list"   and sb :: "nat"   and lb :: "cell list"   and rb :: "cell list" assumes "tinres r r'"   and "steps0 (ss, l, r) t stp = (sa, la, ra)"   and "steps0 (ss, l, r') t stp = (sb, lb, rb)" shows "la = lb \<and> tinres ra rb \<and> sa = sb"
that: fixes A' :: "'a set"   and x' :: "'a" assumes "A = A'"   and "x = x'"   and "x \<in> A" shows "thesis"
hom_uminus: fixes x :: "'b poly"   and a :: "'b" shows "poly (- x) a = - poly x a"
eq_Eq_conv: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Eq) = (x = y)"
bounded_norm_comp: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
even_diff: shows "int 2 dvd x ^ 2 ^ multiplicity 2 n - y ^ 2 ^ multiplicity 2 n"
map2_val_compat_def: fixes f :: "('a \<times> 'd) list \<Rightarrow> ('a \<times> 'e) list" shows "map2_val_compat f = (\<forall>zs. oalist_inv_raw zs \<longrightarrow> oalist_inv_raw (f zs) \<and> fst ` set (f zs) \<subseteq> fst ` set zs)"
ext: fixes g :: "'a"   and f :: "'a" assumes "Left_a.comp g f \<noteq> Left_a.null" shows "Left_a.seq g f"
simps: fixes G :: "('a, 'b) Graph_scheme"   and a :: "'a list" assumes "Digraph G" shows "Digraph.walk G a = (a = [] \<or> (\<exists>v. a = [v] \<and> v \<in> V\<^bsub>G\<^esub>) \<or> (\<exists>v w vs. a = v # w # vs \<and> v \<rightarrow>\<^bsub>G\<^esub> w \<and> Digraph.walk G (w # vs)))"
YFtor'Obj1: fixes X :: "'d"   and C :: "('d, 'e, 'f) LSCategory_scheme" assumes "X \<in> obj\<^bsub>CatDom (YFtor' C)\<^esub>"   and "LSCategory C" shows "YFtor' C ## id\<^bsub>CatDom (YFtor' C)\<^esub> X = id\<^bsub>CatCod (YFtor' C)\<^esub> (Hom\<^bsub>C\<^esub>[\<midarrow>,X])"
algebraic_int_sqrt_iff: fixes x :: "real" shows "algebraic_int (sqrt x) = algebraic_int x"
arbitrary_union_of_inc: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "P S" shows "(arbitrary union_of P) S"
l2: shows "z \<notin> path_image (linepath (Complex (Re ub) (Im lb)) ub)"
eq_onp: shows "eq_onp (\<lambda>x. (\<forall>z\<in>set1_F x. P1 z) \<and> (\<forall>z\<in>set2_F x. P2 z) \<and> (\<forall>z\<in>set3_F x. P3 z)) x y"
lookup_delete: fixes k :: "'a"   and al :: "('a, 'b) alist" shows "DAList.lookup (delete k al) = (DAList.lookup al)(k := None)"
wfactors_cong_r: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c"   and a' :: "'c" assumes "Group.monoid G"   and "wfactors G fs a"   and "a \<sim>\<^bsub>G\<^esub> a'"   and "a \<in> carrier G"   and "a' \<in> carrier G"   and "set fs \<subseteq> carrier G" shows "wfactors G fs a'"
gfp_F_upperbound: shows "(fst XD_XH, dual (snd XD_XH)) \<le> gfp (F ds)"
ch2ch_fun: fixes S :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'a" assumes "chain S" shows "chain (\<lambda>i. S i x)"
f_eq_ma_na: shows "F = set_mset {#y \<in># M. y \<noteq> x#} \<union> set_mset {#y \<in># N. y \<noteq> x#}"
subspace_inter: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "module scale"   and "module.subspace scale A"   and "module.subspace scale B" shows "module.subspace scale (A \<inter> B)"
Inv_in_Hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Inv C t \<in> {ta. monoidal_language.Arr C ta \<and> monoidal_language.Dom C ta = monoidal_language.Cod C t \<and> monoidal_language.Cod C ta = monoidal_language.Dom C t}"
cryptinverts_hilf3: fixes q :: "nat"   and m :: "nat"   and k :: "nat"   and p :: "nat" assumes "prime q" shows "m * m ^ (k * pred p * pred q) mod q = m mod q"