hcju/isabelleps huggingface dataset:
assign_doesnt_read: fixes y :: "'Var"   and x :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list" assumes "y \<noteq> x"   and "y \<notin> aexp_vars e"   and "x \<notin> \<C>_vars y"   and "\<C>_vars y \<inter> aexp_vars e = {}" shows "doesnt_read_or_modify ((x \<leftarrow> e) \<otimes> annos) y"
rep_eq: fixes x :: "'c monom" shows "monom_list x = Rep_monom x"
stableI: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> A co A" shows "F \<in> stable A"
init: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d" assumes "(\<sigma>, p) \<in> init A"   and "l \<in> labels \<Gamma> p" shows "P (\<sigma>, l)"
a: fixes P :: "atomUni \<Rightarrow> bool" shows "(\<forall>l\<in>set (map EqUni a) \<union> (set (map LessUni b) \<union> (set (map LeqUni c) \<union> set (map NeqUni d))). P l) = ((\<forall>(d, e, f)\<in>set a. P (EqUni (d, e, f))) \<and> (\<forall>(d, e, f)\<in>set b. P (LessUni (d, e, f))) \<and> (\<forall>(d, e, f)\<in>set c. P (LeqUni (d, e, f))) \<and> (\<forall>(d, e, f)\<in>set d. P (NeqUni (d, e, f))))"
lens_equiv_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" shows "X \<approx>\<^sub>L Y = (X \<subseteq>\<^sub>L Y \<and> Y \<subseteq>\<^sub>L X)"
join_def: fixes x :: "nat"   and y :: "nat" shows "nat_dvd.join x y = (THE sup. nat_dvd.is_sup x y sup)"
Max_ge: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "x \<in> A" shows "x \<preceq> ordered_powerprod_lin.Max A"
module_hom_zero: shows "clinear (\<lambda>x. 0::'b)"
WTrts_same_length: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:] Ts" shows "length es = length Ts"
iterates_lmap: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "iterates f x = LCons x (lmap f (iterates f x))"
respects_unit: fixes a :: "'c" assumes "C.obj a" shows "(\<sigma>\<^sub>0 a \<star>\<^sub>D F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<sigma>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<sigma>\<^sub>0 a] = \<sigma>\<^sub>1 a \<cdot>\<^sub>D (G.unit a \<star>\<^sub>D \<sigma>\<^sub>0 a)"
strict_trans: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<prec> a"   and "c \<prec> b" shows "c \<prec> a"
ord_of_nat_add: fixes m :: "nat"   and n :: "nat" shows "(m + n)\<^sub>\<nat> = m\<^sub>\<nat> + n\<^sub>\<nat>"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "comm_monoid f z"   and "monoid_list f z" shows "comm_monoid_list f z"
trancl_to_fmrel: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'c \<rightharpoonup>\<^sub>f 'd"   and ym :: "'c \<rightharpoonup>\<^sub>f 'd" assumes "(fmrel f)\<^sup>+\<^sup>+ xm ym" shows "fmrel f\<^sup>+\<^sup>+ xm ym"
q2_def: shows "q2 \<equiv> [:\<i>, 1:] ^ degree p"
Upair_inject: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(Upair a b = Upair c d) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
induced_arrowI_2: fixes \<chi>' :: "'j \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (induced_arrow a' \<chi>') \<pi> = \<chi>'"
max_top: fixes x :: "'a" shows "max \<top> x = \<top>"
d_def: shows "d \<equiv> \<lambda>x. ball x w \<inter> d1 x"
numeral_code_3: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = (let m = numeral.numeral one plus n in plus (plus m m) one)"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "lex_pp x xa = lex_pm (pp.mapping_of x) (pp.mapping_of xa)"
lformula_nFAnd: fixes \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" assumes "lformula (FAnd \<phi> \<psi>)" shows "lformula (nFAnd \<phi> \<psi>)"
Brouwer_degree2_iff: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and d :: "int" assumes "continuous_map (nsphere p) (nsphere p) f"   and "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "(hom_induced (int p) (nsphere p) {} (nsphere p) {} f x = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d) = (x = \<one>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> \<or> Brouwer_degree2 p f = d)"
collinear_3_affine_hull: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> b" shows "collinear {a, b, c} = (c \<in> affine hull {a, b})"
dotprod_le: fixes x :: "nat list"   and u :: "nat list"   and y :: "nat list"   and v :: "nat list" assumes "x \<le>\<^sub>v u"   and "y \<le>\<^sub>v v"   and "length y = length x"   and "length v = length u" shows "x \<bullet> y \<le> u \<bullet> v"
dense_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and z :: "'a" assumes "class.dense_linorder less_eq less"   and "\<And>x. less x y \<Longrightarrow> less_eq x z" shows "less_eq y z"
awalk_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" shows "pre_digraph.awalk G u p v \<equiv> u \<in> verts G \<and> set p \<subseteq> arcs G \<and> pre_digraph.cas G u p v"
presCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'g, 'opSym, 'var, 'e, 'f, 'h) Recursion.model_scheme" shows "presCons h hA MOD \<equiv> presVar h MOD \<and> presAbs h hA MOD \<and> presOp h hA MOD"
prems_2: shows "\<forall>i. pointwise P (fst (Y i)) (snd (Y i))"
vector_derivative_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real"   and x :: "real" shows "vector_derivative (part_circlepath z r s t) (at x) = \<i> * complex_of_real r * (complex_of_real t - complex_of_real s) * exp (\<i> * complex_of_real (linepath s t x))"
singleE: fixes x :: "'a"   and y :: "'a" assumes "eval (single x) y" obtains "y = x"
locally_full: fixes f :: "'b"   and f' :: "'b"   and \<nu> :: "'c" assumes "C.ide f"   and "C.ide f'"   and "src\<^sub>C f = src\<^sub>C f'"   and "trg\<^sub>C f = trg\<^sub>C f'"   and "\<guillemotleft>\<nu> : G f \<Rightarrow>\<^sub>D G f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> G \<mu> = \<nu>"
HMA_M_1: shows "HMA_Connect.HMA_M (1\<^sub>m CARD('e)) (mat (1::'d))"
N_fact_le: fixes q :: "real"   and m :: "nat" assumes "fbnorm q" shows "fbnorm.N q (fact m) \<le> 1 / (q - 1) * 1 / q ^ m"
horner_sum_append: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and xs :: "'c list"   and ys :: "'c list" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_0.horner_sum plus zero times f a (xs @ ys) = plus (comm_semiring_0.horner_sum plus zero times f a xs) (times (power.power one times a (length xs)) (comm_semiring_0.horner_sum plus zero times f a ys))"
emeasure_cball_aux: fixes A :: "'b set"   and r :: "real" assumes "finite A"   and "0 < r" shows "emeasure (Pi\<^sub>M A (\<lambda>_. lborel)) ({f. sqrt (\<Sum>i\<in>A. (f i)\<^sup>2) \<le> r} \<inter> space (Pi\<^sub>M A (\<lambda>_. lborel))) = ennreal (unit_ball_vol (real (card A)) * r ^ card A)"
assms_2: shows "subgroup_generated (reduced_homology_group (int p) (nsphere p)) {a} = reduced_homology_group (int p) (nsphere p)"
rep_eq: fixes x :: "'b \<Rightarrow> 'd \<Rightarrow> 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'd" shows "lookup (mapp x xa) = (\<lambda>k. if k \<in> keys xa then x k (lookup xa k) else (0::'c))"
S4EnvUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [ENext p]_(e p) \<and> $S4 rmhist p \<longrightarrow> unchanged e p"
simps: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "min_list_param rel (x # xs) = (case xs of [] \<Rightarrow> x | a # list \<Rightarrow> let m = min_list_param rel xs in if rel x m then x else m)"
step: shows "(is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is', \<theta>', x', m', \<D>', \<O>', \<R>', \<S>')"
get_child_nodes_reads: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_child_nodes_locs ptr) (get_child_nodes ptr) h h'"
scc_root_of_node_exists: fixes v :: "'v" assumes "v \<in> dom (discovered s')" shows "\<exists>r. scc_root s' r (scc_of E v)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f" obtains a :: "'a" where "a \<in> A \<Gamma>" and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" and "d_OUT f a < weight \<Gamma> a"
permute_fun_app_eq: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "p \<bullet> f x = (p \<bullet> f) (p \<bullet> x)"
ideal_prod_sub_Int: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "I \<diamondsuit>\<^sub>r J \<subseteq> I \<inter> J"
pNode_2: fixes bdt1 :: "bdt" assumes "Node lst stp rst \<le> rt"   and "bdt rt var = Some bdt1" shows "\<exists>bdt2. bdt (Node lst stp rst) var = Some bdt2"
borel_measurable_uminus: fixes g :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "g \<in> borel_measurable M" shows "(\<lambda>x. - g x) \<in> borel_measurable M"
resample_le_sample: shows "ord_spmf (=) (resample A B) (spmf_of_set B)"
Un_left_absorb: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> (A \<union> B) = A \<union> B"
heap_wellformed_induct_rev: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "\<And>child. (\<And>parent child_node. \<lbrakk>cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child; h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent) \<Longrightarrow> P child" shows "P ptr"
subsumeseq_term_trans: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" assumes "s \<le>\<cdot> t"   and "t \<le>\<cdot> u" shows "s \<le>\<cdot> u"
span_base: fixes Sa :: "('a \<Rightarrow> real) set"   and a :: "'a \<Rightarrow> real" assumes "Sa \<subseteq> diff_fun_space"   and "a \<in> Sa" shows "a \<in> diff_fun_space.m1.span Sa"
graph_union_idemp_1: fixes A :: "('e, 'f) labeled_graph" shows "graph_union A A = A"
bheight_rbtreeify_f: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "bheight (fst (rbtreeify_f n kvs)) = RBT_Impl.plog2 n"
rev_exhaust: fixes xs :: "'g list" obtains "xs = []"   | ys :: "'g list" and y :: "'g" where "xs = ys @ [y]"
psimps_1: fixes c :: "'b"   and M :: "'b trm"   and N :: "'b trm" assumes "unify_dom (Const c, M \<cdot> N)" shows "unify (Const c) (M \<cdot> N) = None"
epath_split_distinct: fixes G :: "'b uprod set"   and v :: "'b"   and p :: "'b uprod list"   and v' :: "'b"   and a :: "'b"   and b :: "'b" assumes "epath G v p v'"   and "Upair a b \<in> set p" shows "\<exists>p' p'' u u'. epath G v p' u \<and> epath G u' p'' v' \<and> length p' < length p \<and> length p'' < length p \<and> (u \<in> {a, b} \<and> u' \<in> {a, b}) \<and> Upair a b \<notin> set p' \<and> Upair a b \<notin> set p''"
mult_neg_neg: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "b < (0::'a)" shows "(0::'a) < a * b"
simps_1: fixes css :: "nat \<Rightarrow> 'd list" shows "seq css 0 = []"
empty_tokens_idempotent: fixes T :: "(symbol \<times> character list) set" shows "empty_tokens (empty_tokens T) = empty_tokens T"
that: fixes g :: "real \<Rightarrow> 'a" assumes "\<forall>u\<in>{a..b}. \<forall>v\<in>{a..b}. u \<le> v \<longrightarrow> (f has_integral g v - g u) {u..v}" shows "thesis"
divide_pos_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less zero y" shows "less zero (divide x y)"
ardens_lemma_var: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x\<^sup>\<omega> = (0::'a)"   and "z + x \<cdot> y = y" shows "x\<^sup>\<star> \<cdot> z = y"
positions_in_row_unique: fixes s :: "('b \<times> bool list) stream"   and i :: "nat" assumes "\<exists>!p. snd (s !! p) ! i" shows "the_elem (positions_in_row s i) = (THE p. snd (s !! p) ! i)"
bezout_coefficients: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus"   and "euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b = (x, y)" shows "plus (times x a) (times y b) = gcd a b"
le_conv_radius_iff: fixes r :: "real"   and a :: "nat \<Rightarrow> 'a"   and \<xi> :: "'a" shows "(ereal r \<le> conv_radius a) = (\<forall>x. norm (x - \<xi>) < r \<longrightarrow> summable (\<lambda>i. a i * (x - \<xi>) ^ i))"
subt: fixes s :: "('d, 'e) Term.term"   and ss :: "('d, 'e) Term.term list"   and t :: "('d, 'e) Term.term"   and f :: "'d" assumes "s \<in> set ss"   and "s \<rhd> t" shows "Fun f ss \<rhd> t"
ttop_def: fixes A :: "'b set" shows "ttop A = topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s})"
assms_3: shows "\<forall>i<n. FlexRayController (nReturn i) rcv (nC i) (nStore i) (nSend i) (nGet i)"
sets_PiF: fixes I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" shows "sets (Pi\<^sub>F I M) = sigma_sets (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
simps_2: fixes i :: "'b memref"   and "is" :: "'b memref list" shows "all_shared (i # is) = (case i of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then R \<union> all_shared is else all_shared is | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> R \<union> all_shared is | _ \<Rightarrow> all_shared is)"
FunctorId: fixes X :: "'a" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "\<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y"
Deliver_added_ids: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "i \<in> set (added_ids xs e)" shows "Deliver (i, Add i e) \<in> set xs"
fake_mono: fixes bad :: "agent set"   and bad' :: "agent set"   and IK :: "msg set"   and IK' :: "msg set"   and chan :: "chan set"   and chan' :: "chan set" assumes "bad \<subseteq> bad'"   and "IK \<subseteq> IK'"   and "chan \<subseteq> chan'" shows "fake bad IK chan \<subseteq> fake bad' IK' chan'"
Src_Dom_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Src (Dom t) = Src t \<and> Src (Cod t) = Src t"
check_no_overlap_def: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and ft :: "('a, 'c) flow_entry_match list" shows "check_no_overlap \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. \<forall>p\<in>UNIV. ofe_prio a = ofe_prio b \<and> \<gamma> (ofe_fields a) p \<and> a \<noteq> b \<longrightarrow> \<not> \<gamma> (ofe_fields b) p)"
tail_suffix_suc: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "s |\<^sub>s Suc n = tail (s |\<^sub>s n)"
nat_of_ipv4addr_def: fixes a :: "32 word" shows "nat_of_ipv4addr a = unat a"
simps_2: fixes A :: "'at" shows "negative (Neg A) = True"
A81_Acc: shows "Acc level1 sA81 = {sA81, sA91, sA92}"
MMD_def: fixes Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set" shows "MMD Is P \<equiv> {(x, y). card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x} \<le> card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y}}"
sink_out: fixes x :: "'v" shows "\<not> edge \<Delta> (sink \<Delta>) x"
inv_closed: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
map_of_inject_set: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "(map_of xs = map_of ys) = (list.set xs = list.set ys)"
fa_def: shows "fa = (\<lambda>b f Y. if Y = x then b else f Y)"
map_tpOf_getTvars: fixes \<sigma>l :: "'tp list" shows "map GE.tpOf (GE.getTvars \<sigma>l) = \<sigma>l"
path2_app': fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and ms :: "'node list"   and l :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "g \<turnstile> m-ms\<rightarrow>l" shows "g \<turnstile> n-butlast ns @ ms\<rightarrow>l"
rg_succ_def: fixes am :: "'c \<Rightarrow> 'b list"   and cf :: "'c \<times> 'b \<Rightarrow> 'd"   and u :: "'c" shows "rg_succ am cf u \<equiv> filter_rev (\<lambda>v. (0::'d) < cf (u, v)) (am u)"
RPCInit_def: fixes rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCInit rcv rst p \<equiv> rst!p = #rpcA \<and> \<not> Calling rcv p"
re_geq_one: shows "\<forall>c. 1 \<le> |re c|"
vector_space_with_overloaded: shows "vector_space = vector_space_with (+) (0::'e) (-) uminus"
cases: fixes x :: "'b match_expr list" obtains "x = []"   | e :: "'b match_expr" where "x = [e__]"   | e :: "'b match_expr" and v :: "'b match_expr" and va :: "'b match_expr list" where "x = e # v # va"
is_image_is_image_spec: fixes A :: "'p"   and B :: "'p"   and P' :: "'p"   and P :: "'p" assumes "A \<noteq> B" shows "(P' P Reflect A B) = (P' P ReflectL A B)"
cf_pos_poly_0: shows "cf_pos_poly 0 = 0"
assms_4: shows "h2 \<in> o[F](h1)"
sa_iso_right_invertible: fixes p :: "'c stone_phi_pair" shows "sa_iso (sa_iso_inv p) = p"
Path_Null_iff: fixes h :: "ref \<Rightarrow> ref"   and y :: "ref"   and xs :: "ref list" shows "Path Null h y xs = (xs = [] \<and> y = Null)"
measurable_bot: fixes M :: "'a measure" shows "\<bottom> \<in> M \<rightarrow>\<^sub>M count_space UNIV"
inner_prod_swap: fixes y :: "complex vec"   and n :: "nat"   and x :: "complex vec" assumes "y \<in> carrier_vec n"   and "x \<in> carrier_vec n" shows "inner_prod y x = conjugate (inner_prod x y)"
b: shows "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. b)"
coprime_m_def: fixes f :: "int poly"   and g :: "int poly" shows "coprime_m f g = (\<forall>h. h dvdm f \<longrightarrow> h dvdm g \<longrightarrow> h dvdm 1)"
axioms_2: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "Group.group H"
strict_iff_order: fixes a :: "'b"   and b :: "'b" shows "codomain.greater b a = (codomain.greater_eq b a \<and> a \<noteq> b)"
inner_Basis_add: fixes b1 :: "'a"   and b2 :: "'a"   and v :: "'a" assumes "b1 \<in> S"   and "b2 \<in> S" shows "local.inner_Basis (b1 + b2) v = local.inner_Basis b1 v + local.inner_Basis b2 v"
IntegralI: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>\<delta>. gauge {a..b} \<delta> \<and> (\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> < e)" shows "Integral (a, b) f k"
add_assoc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semigroup_add plus" shows "plus (plus a b) c = plus a (plus b c)"
simps_1: fixes S :: "'a set" shows "f S 0 = []"
not_gr0: fixes n :: "nat" shows "(\<not> 0 < n) = (n = 0)"
length_pos_if_in_set: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "0 < |xs|"
vec_def: fixes n :: "nat"   and x :: "'c list" shows "vec n x = (length x = n)"
ring_simprules_3: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<in> carrier R"
simps_1: fixes a :: "'a" shows "nqfree (Atom a) = True"
set_list_bind: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'b list" shows "set (List.bind xs f) = (\<Union>x\<in>set xs. set (f x))"
last_polychain_of: fixes xs :: "'a list"   and p :: "'a" assumes "0 < length xs" shows "snd (last (polychain_of p xs)) = p + sum_list xs"
expr_list_inj_term_def: fixes es :: "expr list" shows "\<langle>es\<rangle> \<equiv> In3 es"
assms_4: shows "S\<lparr>ArrDom\<rparr> = T\<lparr>ArrDom\<rparr>"
when_def: fixes zero :: "'c"   and a :: "'c"   and P :: "bool" shows "zero.when zero a P = (if P then a else zero)"
simps_23: fixes e1 :: "('a, 'a, 'b) exp"   and C :: "String.literal"   and V :: "'a"   and e2 :: "('a, 'a, 'b) exp" shows "syncvars (try e1 catch(C V) e2) = (syncvars e1 \<and> syncvars e2)"
has_integralD: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and a :: "'c"   and b :: "'c"   and e :: "real" assumes "(f has_integral y) (cbox a b)"   and "0 < e" obtains \<gamma> :: "'c \<Rightarrow> 'c set" where "gauge \<gamma>" and "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e"
\<phi>: shows "\<guillemotleft>\<phi> : local.map (src\<^sub>C \<mu>) \<rightarrow>\<^sub>D src\<^sub>D (local.map \<mu>)\<guillemotright> \<and> D.iso \<phi>"
less_numeral_extra_3: fixes one :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
residue_1_prop: shows "\<ominus>\<^bsub>residue_ring 1\<^esub> \<zero>\<^bsub>residue_ring 1\<^esub> = \<zero>\<^bsub>residue_ring 1\<^esub>"
ith: shows "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"
length_negGorbit_list_sublist: fixes fs :: "'a list"   and smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and gs :: "'c list"   and T :: "'d \<Rightarrow> 'a"   and as :: "'d list" assumes "fs \<in> set (aezfun_scalar_mult.negGorbit_list smult gs T as)" shows "length fs = length as"
perm_app: fixes pi :: "(name \<times> name) list"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "pi \<bullet> f x = (pi \<bullet> f) (pi \<bullet> x)"
ex1: shows "\<exists>!fs. g = \<Prod>\<^sub># fs \<and> set_mset fs \<subseteq> Irr_Mon"