hcju/isabelleps huggingface dataset:
group_add_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a" shows "class.group_add minus plus zero uminus \<equiv> class.monoid_add plus zero \<and> class.group_add_axioms minus plus zero uminus"
f: shows "f \<in> M \<rightarrow>\<^sub>M N"
obj_simps_2: fixes a :: "'a" assumes "B.obj a" shows "trg a = a"
simps_1: fixes uu :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and uv :: "String.literal \<times> value list" shows "ltl_step uu None r uv = (None, [], r)"
\<theta>_def: shows "primes_theta = prime_sum_upto (\<lambda>p. ln (real p))"
normal_form_diff_in_ideal: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "F \<subseteq> P[X]" shows "p - normal_form F p \<in> ideal F"
least_cong: fixes x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "x .= x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed A" shows "least L x A = least L x' A"
wf_bounded_supset: fixes S :: "'g set" assumes "finite S" shows "wf {(Q', Q). Q \<subset> Q' \<and> Q' \<subseteq> S}"
simps_1: fixes uu :: "value" shows "is_lit (L uu) = True"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> ns_public"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> ns_public; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> ns_public; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) # evs1)"   and "\<And>evs2 NB A' B NA A. \<lbrakk>evs2 \<in> ns_public; P evs2; Nonce NB \<notin> used evs2; Says A' B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) # evs2)"   and "\<And>evs3 A B NA B' NB. \<lbrakk>evs3 \<in> ns_public; P evs3; Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3; Says B' A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) (Nonce NB)) # evs3)" shows "P x"
prems_2: fixes i :: "nat" shows "i < length (x # xs) \<longrightarrow> k \<le> i \<longrightarrow> (x # xs) ! i = (y # ys) ! i"
assms_2: shows "clauseFalse (removeAll (opposite literal) clause2) valuation"
meetI: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> A\<^sup>*"   and "(a, c) \<in> A\<^sup>*" shows "(b, c) \<in> A\<^sup>\<up>"
Program: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'p"   and is' :: "instr list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')" shows "direct_computation.concurrent_step (ts, m, \<S>) (ts[i := (p', is @ is', \<theta>, sb, \<D>, \<O>, \<R>)], m, \<S>)"
fl_subdigraph_axioms: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
7: shows "lselect {i. Suc (least s) + i \<in> s - {least s}} (ldropn (Suc (least s)) w) = lselect (s - {least s}) w"
finite_powerset: fixes M :: "'a set" assumes "finite M" shows "finite {S. S \<subseteq> M}"
inf_p: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"
lappfin_finT: fixes s :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "s \<in> A\<^sup>\<star>"   and "t \<in> A\<^sup>\<star>" shows "s @@ t \<in> A\<^sup>\<star>"
all0_but1: fixes i :: "nat" assumes "i \<noteq> is ! 0"   and "i < r0" shows "lookup (tensors_from_net (witness' r0 [M]) $ i) is = 0"
sorted_splay: fixes t :: "('c \<times> 'd) tree"   and x :: "'c"   and l :: "('c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and r :: "('c \<times> 'd) tree" assumes "sorted1 (inorder t)"   and "Splay_Map.splay x t = \<langle>l, a, r\<rangle>" shows "Sorted_Less.sorted (map fst (inorder l) @ x # map fst (inorder r))"
euler_mascheroni_def: shows "euler_mascheroni = of_real (lim (\<lambda>n. harm n - ln (real n)))"
deutsch_transform_coeff_3: shows "U\<^sub>f $$ (2, 3) = of_nat (f 1)"
h_idem: fixes x :: "'a" shows "h (h x) = h x"
powr_float_interval: fixes prec :: "nat"   and X :: "float interval"   and Y :: "float interval"   and R :: "float interval" assumes "powr_float_interval prec X Y = Some R" shows "{x powr y |x y. x \<in>\<^sub>r X \<and> y \<in>\<^sub>r Y} \<subseteq> set_of (real_interval R)"
normalized_sort_subset: fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set" assumes "wf_subclass_loc cs" shows "normalize_sort cs s \<subseteq> s"
prems_2: shows "wf_envs \<sigma> j P P' V db"
simps_2: fixes n :: "nat"   and x :: "dB"   and i :: "nat" shows "unlift' n (Pure x) i = liftn n x 0"
representant_exists: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and A :: "'c set" assumes "equivalence S E"   and "A \<in> S / E" shows "\<exists>a\<in>S. a \<in> A \<and> A = equivalence.Class S E a"
all_fin_gen_ideals_are_principal_imp_bezout: assumes "\<forall>I. finitely_generated_ideal I \<longrightarrow> principal_ideal I" shows "OFCLASS('b, bezout_ring_class)"
nonzero_mod_p: fixes x :: "int" assumes "0 < x"   and "x < int p" shows "[x \<noteq> 0] (mod int p)"
wp_eval_24: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wlp (Bind g f) = (\<lambda>P s. wlp (f (g s)) P s)"
upRules_not_left_principal_for_modal: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and M :: "'b"   and Ms :: "('a, 'b) form list"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "r \<in> upRules" shows "\<not> leftPrincipal r (Modal M Ms) R"
assms_3: shows "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"
mat_diag_smult: fixes n :: "nat"   and k :: "'b" shows "mat_diag n (\<lambda>x. k) = k \<cdot>\<^sub>m 1\<^sub>m n"
supp_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'d" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
zero_min: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "ordered_powerprod ord ord_strict" shows "ord (0::'b) t"
affine_alt: fixes s :: "'b set" shows "affine s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
essentially_equalE: fixes G :: "('c, 'd) monoid_scheme"   and fs1 :: "'c list"   and fs2 :: "'c list" assumes "Group.monoid G"   and "essentially_equal G fs1 fs2" obtains fs1' :: "'c list" where "fs1 <~~> fs1'" and "fs1' [\<sim>]\<^bsub>G\<^esub> fs2"
simps_40: fixes f1 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f3 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f4 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f5 :: "form \<Rightarrow> 'a"   and f6 :: "form \<Rightarrow> 'a"   and x41 :: "form"   and x42 :: "form" shows "(case FDisj x41 x42 of PAtom x xa \<Rightarrow> f1 x xa | NAtom x xa \<Rightarrow> f2 x xa | FConj x xa \<Rightarrow> f3 x xa | FDisj x xa \<Rightarrow> f4 x xa | FAll x \<Rightarrow> f5 x | FEx x \<Rightarrow> f6 x) = f4 x41 x42"
prod_add: fixes A :: "'a set"   and f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(\<Prod>x\<in>A. f1 x + f2 x) = (\<Sum>X\<in>Pow A. prod f1 X * prod f2 (A - X))"
lookup_higher: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and u :: "'c"   and v :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "lookup (ordered_term.higher ord_term p u) v = (if ord_term_strict u v then lookup p v else (0::'f))"
not_el: shows "\<nexists>e. e \<in> set noteq \<and> m < e"
continuous_map_o_Pair: fixes X :: "'b topology"   and Y :: "'c topology"   and Z :: "'d topology"   and h :: "'b \<times> 'c \<Rightarrow> 'd"   and t :: "'b" assumes "continuous_map (prod_topology X Y) Z h"   and "t \<in> topspace X" shows "continuous_map Y Z (h \<circ> Pair t)"
simps_5: fixes r :: "'b mrexp2"   and s :: "'b mrexp2"   and f :: "bool"   and n :: "bool" shows "ok2 (Times2 r s f n) = (ok2 r \<and> ok2 s \<and> (let rs = Times (strip2 r) (strip2 s) in f = final rs \<and> n = nullable rs))"
mod_exp_aux_correct: fixes m :: "'a"   and y :: "'a"   and x :: "'a"   and n :: "nat" shows "mod_exp_aux m y x n mod m = x ^ n * y mod m"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "monomials (MPoly x) = keys x"
assms: shows "low_equal s1 s2 \<and> t1 = s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr> \<and> t2 = s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f"   and b :: "bool" shows "thread_ok ts (ThreadExists t b) = (b \<noteq> free_thread_id ts t)"
assms_1: shows "qReachable cfg1 (Proc - Q) cfg2"
fgs_submodule: fixes a :: "'a set" assumes "a \<subseteq> carrier M" shows "submodule R M (fgs R M a)"
cring_simprules_16: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
C_eq_s: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "p \<noteq> []" shows "C (list2FWpolicy (separate p)) = C (list2FWpolicy p)"
dimQ_def: fixes x :: "'a['b, 'c]" shows "dimQ x = Quantity.dim (fromQ x)"
leg1_commutes: shows "\<epsilon>.cod.leg1 \<cdot> \<epsilon>.chine = \<epsilon>.dom.leg1"
i_set_mult_0_eq_i_set_cont: shows "i_set_mult 0 = i_set_cont"
globs_upd_gobj_upd: fixes s :: "st"   and r :: "oref"   and obj :: "obj"   and n :: "vn"   and v :: "val" assumes "globs s r = Some obj" shows "globs (upd_gobj r n v s) = globs s(r \<mapsto> upd_obj n v obj)"
isNotDSourceX_def: fixes i :: "AbstrLevelsID"   and S :: "CSet"   and C :: "CSet" shows "isNotDSourceX i S C \<equiv> \<forall>x\<in>OUT S. C \<notin> AbstrLevel i \<or> x \<notin> IN C"
comp_in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
supr_greater: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "local.Above B \<noteq> {}"   and "b \<in> B" shows "(b, local.supr B) \<in> r"
simps_4: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "ltl (ltakeWhile P xs) = ltakeWhile P (ltl xs)"
real_interval_sup: fixes a :: "float interval"   and b :: "float interval" shows "real_interval (sup a b) = sup (real_interval a) (real_interval b)"
prems_2: shows "insert x X \<subseteq> natUnion C"
EF_lem00: fixes f :: "'a set" shows "EF f = f \<union> EX' (lfp (\<lambda>Z. f \<union> EX' Z))"
integral_less_AE: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "integrable (density M (\<lambda>x. ennreal (D x))) X"   and "integrable (density M (\<lambda>x. ennreal (D x))) Y"   and "emeasure (density M (\<lambda>x. ennreal (D x))) A \<noteq> 0"   and "A \<in> N.events"   and "AE x\<in>A in density M (\<lambda>x. ennreal (D x)). X x \<noteq> Y x"   and "AE x in density M (\<lambda>x. ennreal (D x)). X x \<le> Y x" shows "N.expectation X < N.expectation Y"
Int_Diff_Un: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B \<union> (A - B) = A"
TC_map_subc: fixes G :: "('a, 'b) form list"   and c :: "'a"   and s :: "'a FOL_Fitting.term" assumes "infinite UNIV"   and "\<stileturn> G" shows "\<stileturn> map (subc c s) G"
intro: assumes "\<And>x y. PP x y = (P x y \<and> x \<noteq> y)"   and "\<And>x y. PP x y = (P x y \<and> \<not> P y x)" shows "M2_axioms"
derivations_cross_compare: fixes X :: "'a set"   and Y :: "'a set"   and x :: "'a"   and y :: "'a" assumes "derivation X"   and "derivation Y"   and "x \<in> X"   and "y \<in> Y" shows "x \<sqsubset> y \<and> x \<in> Y \<or> x = y \<or> y \<sqsubset> x \<and> y \<in> X"
S_irrefl: fixes t :: "('a, 'b) Term.term" shows "\<not> fst (kbo t t)"
get_prism_lens: fixes X :: "'d \<Longrightarrow>\<^sub>\<triangle> 'c" shows "get\<^bsub>prism_lens X\<^esub> = the \<circ> match\<^bsub>X\<^esub>"
implc_get_offending_flows_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G" shows "set ` set (implc_get_offending_flows (get_impl M) G) = get_offending_flows (get_spec M) (list_graph_to_graph G)"
ide_is_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso a"
assms_2: shows "f x \<in> {- exp (- 1)<..<0}"
confluent_quotient_cyclic1: shows "confluent_quotient cyclic1 cyclic cyclic cyclic cyclic cyclic (map fst) (map snd) (map fst) (map snd) list_all2 list_all2 list_all2 set set"
pmf_pair: fixes M :: "'a pmf"   and N :: "'b pmf"   and a :: "'a"   and b :: "'b" shows "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
simps_2: fixes n :: "'c"   and xs :: "('c \<times> 'd \<times> bool) list" shows "\<langle>Delete n\<rangle> xs = delete xs n"
cr_uint32_def: shows "cr_uint32 \<equiv> \<lambda>x y. x = Rep_uint32 y"
conv_complement_sub_leq: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>T * - (x * y) \<le> - y"
aux6: fixes a1 :: "'a set set"   and a2 :: "'a set set"   and a :: "'a set set"   and b :: "'a set set" assumes "twoNetsDistinct a1 a2 a b" shows "dom (deny_all_from_to a1 a2) \<inter> dom (deny_all_from_to a b) = {}"
lrev_work_def: shows "lrev_work \<equiv> fix\<cdot>(unwrapH oo lrev_body oo wrapH)"
cexpr_typing_branch_prob_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and vs :: "nat list"   and vs' :: "nat list" assumes "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL" shows "\<Gamma> \<turnstile>\<^sub>c branch_prob_cexpr (vs, vs', \<Gamma>, \<delta>) : REAL"
epsilon0_def: shows "\<epsilon>\<^sub>0 = oFix ((**) \<omega>) 0"
Ri_mono: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "Ri N \<subseteq> Ri N'"
simps_1: fixes ah :: "'c"   and as :: "'c list"   and i :: "nat"   and a :: "'c" shows "list_insert (ah # as) i a = (case i of 0 \<Rightarrow> a # ah # as | Suc j \<Rightarrow> ah # list_insert as j a)"
MinLVarNotInBounds_axioms: shows "MinLVarNotInBounds min_lvar_not_in_bounds"
simps_2: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "state_list s (a # as) = s # state_list (state_succ s a) as"
cl_ivl_Suc: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "prod g {m..Suc n} = (if Suc n < m then 1::'b else prod g {m..n} * g (Suc n))"
simps_2: fixes As :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "(\<turnstile>As \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>) = (case As of [] \<Rightarrow> fst s \<subseteq> snd s | [a] \<Rightarrow> \<turnstile>a \<and> attack a = s | a # ab # list \<Rightarrow> \<turnstile>a \<and> fst (attack a) = fst s \<and> \<turnstile>ab # list \<oplus>\<^sub>\<and>\<^bsup>(snd (attack a), snd s)\<^esup>)"
cexpr_sem_cexpr_comp: fixes \<sigma> :: "nat \<Rightarrow> val"   and f :: "cexpr"   and g :: "cexpr" shows "cexpr_sem \<sigma> (f \<circ>\<^sub>c g) = cexpr_sem (\<sigma>(0 := cexpr_sem \<sigma> g)) f"
update_conv_update_with_aux: fixes k :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" shows "AList.update k v xs = update_with_aux v k (\<lambda>_. v) xs"
Sup_less_def: fixes x :: "'c \<Rightarrow> 'a"   and w :: "'c" shows "Sup_less x w = \<Squnion>{y. \<exists>v<w. y = x v}"
eq_refl: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C \<phi>"
elims: fixes x :: "'b list"   and y :: "('b \<times> 'b option) set" assumes "succ_rel x = y" obtains "xb__ = []" and "y = {}"   | head :: "'b" where "xb__ = [head__]" and "y = {(head__, None)}"   | head :: "'b" and x :: "'b" and xs :: "'b list" where "xb__ = head # x # xs" and "y = {(head, Some x)} \<union> succ_rel (x # xs)"
interval_integral_sum: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f" shows "interval_lebesgue_integral lborel a b f + interval_lebesgue_integral lborel b c f = interval_lebesgue_integral lborel a c f"
arr_inv: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.arr (VV.inv f)"
retraction_imp_surjective_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "retraction_map X Y f" shows "f ` topspace X = topspace Y"
long_div_theorem: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "g \<noteq> \<zero>\<^bsub>P\<^esub>" shows "\<exists>q r k. q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g [^] k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r \<and> (r = \<zero>\<^bsub>P\<^esub> \<or> deg R r < deg R g)"
fin_iter_step_mono: fixes c :: "'a::refinement_lattice" shows "mono (\<lambda>x::'a::refinement_lattice. nil \<sqinter> (c::'a::refinement_lattice) ; x)"
linear_sub_deriv: fixes f :: "nat \<Rightarrow> 'a"   and d :: "'a"   and g :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "f \<in> carrier (UP R)"   and "d \<in> carrier R"   and "g = Cring_Poly.compose R f (monom (UP R) d 1)"   and "c \<in> carrier R" shows "pderiv g = d \<odot>\<^bsub>UP R\<^esub> Cring_Poly.compose R (pderiv f) (monom (UP R) d 1)"
line_in: shows "(\<lambda>s. (t, x t) + s *\<^sub>R (h * p) *\<^sub>R (1, f (t, x t))) ` {0..1} \<subseteq> T \<times> X"
differing_vars_initially_empty: fixes i :: "nat"   and n :: "nat"   and x :: "'Var"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "i < n" shows "x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 (zip (replicate n mem\<^sub>1) (replicate n mem\<^sub>2)) i"
chain_def: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" shows "Complete_Partial_Order.chain ord S = (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
\<Delta>_naturality: shows "(h \<star> tab\<^sub>0 h) \<cdot> h.tab = (h \<star> h.the_\<theta>) \<cdot> \<a>[h, tab\<^sub>0 h, h.chine] \<cdot> (h.tab \<star> h.chine) \<cdot> h.the_\<nu>"
exhaust: fixes y :: "('c, 'd) blindable\<^sub>m" obtains x1 :: "'c" where "y = Unblinded x1"   | x2 :: "'d blindable\<^sub>h" where "y = Blinded x2"
piecewise_continuous_onE: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "piecewise_continuous_on a b I f" obtains l :: "'a \<Rightarrow> 'b" and u :: "'a \<Rightarrow> 'b" where "finite I" and "continuous_on ({a..b} - I) f" and "\<And>i. \<lbrakk>i \<in> I; a < i; i \<le> b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> l i) (at_left i)" and "\<And>i. \<lbrakk>i \<in> I; a \<le> i; i < b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> u i) (at_right i)"
invariantStateSpace_thm_9: fixes ss :: "('d, 'e) fmap set"   and vs1 :: "'d set"   and vs2 :: "'d set" shows "ss_proj ss (vs1 \<inter> vs2) = ss_proj (ss_proj ss vs2) vs1"
Nil: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "[] \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (update\<^sub>s\<^sub>t \<S> [], \<A>)"
unrestrictn_edges: fixes g :: "'a ugraph"   and V :: "'a set" shows "edges (restrict_nodes g V) \<subseteq> edges g"
arbitrary_union_of_relative_to: fixes P :: "'d set \<Rightarrow> bool"   and U :: "'d set" shows "arbitrary union_of P relative_to U = arbitrary union_of (P relative_to U)"
hyps_4: fixes b :: "name"   and bb :: "(name \<times> trm) list"   and ba :: "trm" assumes "b \<sharp> bb" shows "bb<t[b::=ba]> = bb<t>[b::=bb<ba>]"
ComplD: fixes c :: "'c"   and A :: "'c set" assumes "s2p (- A) c" shows "c \<notin> A"
mult_implies_one_step: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mult r" shows "\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r)"
leg1_simps_4: shows "trg r\<^sub>0s\<^sub>1.p\<^sub>1 = trg (r\<^sub>0\<^sup>* \<star> s\<^sub>1)"
simps_1: fixes \<phi> :: "formula" shows "convert_multiway (Neg \<phi>) = Neg (convert_multiway \<phi>)"
trg_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "trg\<^sub>D (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
thetaParW_def: shows "thetaParW \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>w d1 \<and> c2 \<approx>w d2}"
normalize_gauss_i: shows "normalize \<i>\<^sub>\<int> = 1"
rel_spmf_return_spmf2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and x :: "'b" shows "rel_spmf R p (return_spmf x) = (lossless_spmf p \<and> (\<forall>a\<in>set_spmf p. R a x))"
nths_prod_list_split: fixes B :: "nat set"   and A :: "nat set"   and xs :: "nat list" assumes "B \<subseteq> A" shows "prod_list (nths xs A) = prod_list (nths xs B) * prod_list (nths xs (A - B))"
simps_1: fixes tbq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.insertList [] tbq = tbq"
rlex_Cons: fixes x :: "nat"   and xs :: "nat list"   and y :: "nat"   and ys :: "nat list" shows "(x # xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x y # ys) = (xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x ys \<or> ys = xs \<and> x < y)"
uminus_bounded_comp: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "bounded ((\<lambda>x. - f x) ` S) = bounded (f ` S)"
11: shows "length B2 = length (B1 \<circ> B2)"
phiNodes_of_def: fixes g :: "'g" shows "phiNodes_of g = fold (phis_addN g) (sorted_list_of_set (dom (phis g))) Mapping.empty"
Units: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "carrier G \<subseteq> Units G"
append_child_preserves_type_wf: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h'" shows "type_wf h'"
5: shows "inconsist i (f \<triangleright> (n + m)) \<down>= (LEAST x. x < Suc (n + m) \<and> \<phi> i x \<noteq> f x)"
tauLaw1SimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "eqvt Rel"   and "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi> P Q R. \<lbrakk>(\<Psi>, P, Q) \<in> Rel; \<Psi> \<rhd> Q \<sim> R\<rbrakk> \<Longrightarrow> (\<Psi>, P, R) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> \<tau>.Q"