hcju/isabelleps huggingface dataset:
sum_root_unity: fixes x :: "'a"   and n :: "nat" assumes "x ^ n = (1::'a)" shows "sum ((^) x) {..<n} = (if x = (1::'a) then of_nat n else (0::'a))"
distinct_Ex1: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
typeof_addr_sys_xcp: fixes empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and C :: "String.literal" assumes "heap_base.preallocated empty_heap allocate typeof_addr h"   and "C \<in> sys_xcpts" shows "typeof_addr h (heap_base.addr_of_sys_xcpt empty_heap allocate C) = \<lfloor>Class_type C\<rfloor>"
bind_gpv_cong: fixes gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv"   and f :: "'d \<Rightarrow> ('g, 'e, 'f) gpv"   and g :: "'d \<Rightarrow> ('g, 'e, 'f) gpv" assumes "gpv = gpv'"   and "\<And>x. x \<in> results'_gpv gpv' \<Longrightarrow> f x = g x" shows "gpv \<bind> f = gpv' \<bind> g"
nb\<^sub>7: shows "set (delete_effects_of op') = (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
simps_1: fixes uu :: "'b" shows "delete uu \<langle>\<rangle> = \<langle>\<rangle>"
alphaAbs_sym: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs"   and A' :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "A $= A'" shows "A' $= A"
dres_order_simps_3: fixes r :: "'b" shows "dRETURN r \<noteq> dFAIL"
flowD_outside: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "countable_network \<Delta>"   and "flow \<Delta> g"   and "e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub>" shows "g e = 0"
WTrt_env_mono: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and E' :: "String.literal \<Rightarrow> ty option" assumes "P,E,h \<turnstile> e : T"   and "E \<subseteq>\<^sub>m E'" shows "P,E',h \<turnstile> e : T"
IFC_def: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "IFC v t e \<equiv> if t = e then t else IF v t e"
poincare_circle_in_disc_2: fixes r :: "real"   and z :: "complex_homo"   and ze :: "complex"   and re :: "real" assumes "0 < r"   and "z \<in> unit_disc"   and "(ze, re) = poincare_circle_euclidean z r" shows "0 < re"
Collect_conj_eq: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
card_pick: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "Suc (card (A - {pick A})) = card A"
rev_update: fixes k :: "nat"   and xs :: "'b list"   and y :: "'b" assumes "k < length xs" shows "rev (xs[k := y]) = (rev xs)[length xs - k - 1 := y]"
arrI: fixes f :: "'a \<times> 'a \<times> 'a" assumes "arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f))" shows "VVV.arr f"
simps_2: fixes \<mu> :: "'c" shows "Cod \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>C.cod \<mu>\<^bold>\<rangle>"
rsa_map_eq: shows "x ^ e mod (P * Q) = y ^ e mod (P * Q)"
strong_induct: fixes P :: "'a \<Rightarrow> fmla \<Rightarrow> bool"   and c :: "'a"   and fmla :: "fmla" assumes "\<And>x1 x2 c. P c (x1 EQ x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (x1 OR x2)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (neg x)"   and "\<And>x1 x2 c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (exi x1 x2)" shows "P c fmla"
simps_5: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_nat (Comp_op t1 t2) = c_pair 4 (c_pair (sch_to_nat t1) (sch_to_nat t2))"
union_mulex_on_mono1: fixes A :: "'b multiset"   and F :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b multiset"   and B :: "'b multiset"   and D :: "'b multiset" assumes "A \<in> multisets F"   and "(mulex_on P F)\<^sup>=\<^sup>= A C"   and "mulex_on P F B D" shows "mulex_on P F (A + B) (C + D)"
D0'_def: shows "D0' = D0 \<cdot> \<rho>"
viterbi_ix_def: fixes os :: "'t list" shows "viterbi_ix os = viterbi_ix_rec (IArray os)"
mat_index_conjugate: fixes i :: "nat"   and m :: "'b mat"   and j :: "nat" assumes "i < dim_row m"   and "j < dim_col m" shows "conjugate m $$ (i, j) = conjugate (m $$ (i, j))"
app_invT: fixes r :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "r \<in> A\<^sup>\<star>"   and "r @@ s \<in> A\<^sup>\<omega>" shows "s \<in> A\<^sup>\<omega>"
removing_attractor_induces_no_deadends: fixes v :: "'a"   and S :: "'a set"   and p :: "Player"   and W :: "'a set"   and w :: "'a" assumes "v \<in> S - G'.attractor p W"   and "v \<rightarrow>\<^bsub>G'\<^esub> w"   and "w \<in> S"   and "\<And>w. \<lbrakk>v \<in> G'.VV p**; v \<rightarrow>\<^bsub>G'\<^esub> w\<rbrakk> \<Longrightarrow> w \<in> S" shows "\<exists>w\<in>S - G'.attractor p W. v \<rightarrow>\<^bsub>G'\<^esub> w"
index_order: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and X :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat"   and c :: "nat" assumes "[[.. x .. y .. z ..]X]"   and "ch_by_ord f X"   and "f a = x"   and "f b = y"   and "f c = z"   and "finite X \<longrightarrow> a < card X"   and "finite X \<longrightarrow> b < card X"   and "finite X \<longrightarrow> c < card X" shows "a < b \<and> b < c \<or> c < b \<and> b < a"
cond_helper1: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "test p"   and "pres x p" shows "p \<cdot> (p \<cdot> x + !p \<cdot> y)\<^sup>\<dagger> \<cdot> (p \<cdot> z + !p \<cdot> w) = p \<cdot> x\<^sup>\<dagger> \<cdot> z"
weakCongruenceI: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<leadsto><weakBisimulation> Q"   and "Q \<leadsto><weakBisimulation> P" shows "P \<cong> Q"
cong: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "f = g"   and "x = y" shows "f x = g y"
Set_filter_image: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "Set.filter P (f ` A) = f ` Set.filter (P \<circ> f) A"
msed_map_invR: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and b :: "'b"   and N :: "'b multiset" assumes "image_mset f M = b, N" shows "\<exists>M1 a. M = a, M1 \<and> f a = b \<and> image_mset f M1 = N"
eq: shows "A * matrix_sum m g l = matrix_sum m (\<lambda>k. A * g k) l"
dims_product: shows "st1.d0 = st1.d1 * st1.d2"
simps_1: fixes r :: "real"   and x :: "quat" shows "quat.Re (r *\<^sub>R x) = r * quat.Re x"
pos_less_neg: fixes A :: "'a" shows "Pos A < Neg A"
simps_2: fixes y :: "'c"   and ys :: "'c list"   and x :: "'c" shows "enum_pos' (y # ys) x = (if x = y then 0 else 1 + enum_pos' ys x)"
inject_2: fixes x31 :: "'b val"   and x32 :: "'b val"   and y31 :: "'b val"   and y32 :: "'b val" shows "(Seq x31 x32 = Seq y31 y32) = (x31 = y31 \<and> x32 = y32)"
powhr_less_mono: fixes a :: "real star"   and b :: "real star"   and x :: "real star" assumes "a < b"   and "1 < x" shows "x powhr a < x powhr b"
redT_hext_incr: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr expr \<times> (String.literal \<Rightarrow> 'addr val option)) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) J_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr expr \<times> (String.literal \<Rightarrow> 'addr val option)) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" assumes "P \<turnstile> s -t\<triangleright>ta\<rightarrow> s'" shows "shr s \<unlhd> shr s'"
op_set_ndelete_def: fixes x :: "'a"   and X :: "'a set" shows "op_set_ndelete x X = RES {X - {x}, X}"
assms_4: fixes a :: "'a" assumes "P a" shows "f a \<in> \<real>"
floor_zero: shows "\<lfloor>0::'a\<rfloor> = 0"
iwf_edge_match: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and "is" :: "nat list"   and i :: "nat" assumes "iwf fc t ent"   and "is @ [i] \<in> it_paths t" shows "subst (iSubst (tree_at t (is @ [i]))) (freshen (iAnnot (tree_at t (is @ [i]))) (iOutPort (tree_at t (is @ [i])))) = subst (iSubst (tree_at t is)) (freshen (iAnnot (tree_at t is)) (a_conc (inPorts' (iNodeOf (tree_at t is)) ! i)))"
RG_sum_list_rddg_closed: fixes rgs :: "('a \<times> 'g) list" assumes "set (map snd rgs) \<subseteq> G" shows "(\<Sum>(r, g)\<leftarrow>rgs. r \<delta>\<delta> g) \<in> group_ring"
gL_rel_mult: fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "y \<preceq>\<^sub>L x \<oplus> y"
aligned\<^sub>d_mono_exp_Suc: fixes e :: "nat"   and M :: "hmultiset" assumes "local.aligned\<^sub>d (Suc e) M" shows "local.aligned\<^sub>d e M"
4: shows "\<forall>z. z \<in> Filters.filter.Rep_filter (stone_phi x \<sqinter> y) \<longrightarrow> - Rep_regular x \<le> Rep_dense z"
approx_tse_form': fixes prec :: "nat"   and t :: "nat"   and f :: "floatarith"   and s :: "nat"   and ivl :: "float interval"   and cmp :: "float interval \<Rightarrow> bool"   and x :: "real" assumes "approx_tse_form' prec t f s ivl cmp"   and "x \<in>\<^sub>r ivl" shows "\<exists>ivl' ivly. x \<in>\<^sub>r ivl' \<and> ivl' \<le> ivl \<and> cmp ivly \<and> approx_tse prec 0 t (mid ivl') 1 f [Some ivl'] = Some ivly"
algebraic_self: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and k :: "'c" assumes "domain R"   and "subring K R"   and "k \<in> K" shows "((\<lambda>K x. \<not> ring.transcendental R K x) over K) k"
C_def: shows "B.C \<equiv> \<Inter> B.AA"
cod_eqI: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.ide b"   and "VV.seq b f" shows "VV.cod f = b"
R7_4: shows "pmf (sds R7) d = 1 / 2"
\<psi>_minus_\<theta>: fixes x :: "real" assumes "2 \<le> x" shows "\<psi> x - \<theta> x = (\<Sum>i | 2 \<le> i \<and> real i \<le> log 2 x. \<theta> (root i x))"
multi_member_last: fixes x :: "'d" shows "x \<in># {#x#}"
po_subset_po: fixes S :: "'a set" assumes "S \<subseteq> A" shows "\<lparr>pset = S, order = induced S r\<rparr> \<in> PartialOrder"
is_ub_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below" shows "po.is_ub below S x = (\<forall>y\<in>S. below y x)"
le: fixes x :: "'n"   and k :: "nat" assumes "x \<in> cbox a b" shows "f k x \<le> f (Suc k) x"
cmp_simps_4: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi>\<^sub>F (f, g)) = F f \<star>\<^sub>D F g"
span_breakdown_eq: fixes x :: "'b"   and a :: "'b"   and S :: "'b set" shows "(x \<in> local.span (insert a S)) = (\<exists>k. x - k *s a \<in> local.span S)"
assms_2: shows "n \<noteq> 0 \<and> n \<noteq> p"
ordering_ow_axioms: shows "ordering_ow U (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
tangle_compose_matrix: fixes ws1 :: "wall"   and ws2 :: "wall" assumes "is_tangle_diagram ws1 \<and> is_tangle_diagram ws2 \<and> domain_wall ws2 = codomain_wall ws1" shows "kauff_mat (ws1 \<circ> ws2) = rat_poly.matrix_mult (kauff_mat ws1) (kauff_mat ws2)"
step_preserves_inst: fixes g :: "'g"   and u :: "'node \<Rightarrow> 'val set"   and p :: "'node \<times> 'val \<Rightarrow> 'val list option" assumes "inst' g (u, p)"   and "CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs (uses(g := u)) (phis(g := p)) g" shows "inst' g (step g (u, p))"
perm_length: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs <~~> ys" shows "length xs = length ys"
matrix_inv_invertible: fixes M :: "(('a, 'b) vec, 'c) vec" assumes "invertible M" shows "invertible (matrix_inv M)"
halt_lemma: fixes LE :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" assumes "wf LE"   and "\<forall>n. \<not> P (f n) \<longrightarrow> (f (Suc n), f n) \<in> LE" shows "\<exists>n. P (f n)"
rangeE: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "s2p (range f) b" obtains x :: "'b" where "b = f x"
magQ_scaleQ: fixes x :: "'e"   and y :: "'e['f, 'g]" shows "\<lbrakk>x *\<^sub>Q y\<rbrakk>\<^sub>Q = x \<cdot> \<lbrakk>y\<rbrakk>\<^sub>Q"
VC: shows "s = v \<or> t = v \<or> excess f v \<noteq> (0::'a)"
continuous_on_def: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
simps_2: shows "eval_inl OpMulNum = eval_MulNum"
Star: shows "finite {\<guillemotleft>derivs xs r\<guillemotright> |xs. True}"
order_refl: fixes x :: "'d \<times> 'e" shows "less_eq_prod x x"
replace1: fixes f :: "'c"   and f' :: "'c"   and fs :: "'c list"   and ls :: "'c list" assumes "f \<in> set (replace f' fs ls)"   and "f \<notin> set ls" shows "f \<in> set fs"
hom_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
scope_valid_inport: fixes v' :: "'form \<times> nat list"   and v :: "'form \<times> nat list"   and ip :: "('form, 'var) antecedent" assumes "v' |\<in>| vertices" shows "scope' v ip v' = (\<exists>i. fst v = fst v' \<and> prefix (snd v @ [i]) (snd v') \<and> ip = in_port_at v i)"
reduction_derivable: fixes D :: "'a literal multiset"   and L' :: "'a literal"   and CC :: "'a literal multiset set"   and L :: "'a literal"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "D + {#L'#} \<in> CC"   and "- L = L' \<cdot>l \<sigma>"   and "D \<cdot> \<sigma> \<subseteq># C" shows "sr_ext.derive (grounding_of_clss (CC \<union> {C + {#L#}})) (grounding_of_clss (CC \<union> {C}))"
sel_3: fixes x21 :: "'e"   and x22 :: "'d \<Rightarrow> ('e, 'd, 'f) resumption" shows "resume (Pause x21 x22) = x22"
psubset_card_mono: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
compactin_sing: fixes X :: "'a topology"   and a :: "'a" shows "compactin X {a} = (a \<in> topspace X)"
no_hide_write0: fixes a :: "'a"   and B :: "'a set"   and P :: "'a process" assumes "a \<notin> B" shows "(a \<rightarrow> P) \ B = (a \<rightarrow> P \ B)"
min_gen_is_li: fixes A :: "'c set" assumes "minimal A (\<lambda>S. S \<subseteq> carrier V \<and> gen_set S)" shows "lin_indpt A"
left_gpv_Inl_transfer: fixes gpv :: "('h, 'i, 'k) gpv" shows "rel_gpv'' (=) (\<lambda>l r. l = Inl r) (\<lambda>l r. l = Inl r) (left_gpv gpv) gpv"
IH_2: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> G, \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
BackwardSlice_axioms: shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WWeakControlDependence.PDG_BS_w prog)"
bidirected_digraph_rev_conv_pair: fixes G :: "'b pair_pre_digraph"   and rev_G :: "'b \<times> 'b \<Rightarrow> 'b \<times> 'b" assumes "bidirected_digraph (with_proj G) rev_G" shows "rev_G = swap_in (parcs G)"
abc_step_red: fixes as :: "nat"   and am :: "nat list"   and ap :: "abc_inst list"   and stp :: "nat"   and bs :: "nat"   and bm :: "nat list" assumes "abc_steps_l (as, am) ap stp = (bs, bm)" shows "abc_steps_l (as, am) ap (Suc stp) = abc_step_l (bs, bm) (abc_fetch bs ap)"
Ass: fixes S :: "pi"   and T :: "pi"   and U :: "pi" shows "((S \<parallel> T) \<parallel> U, S \<parallel> T \<parallel> U) \<in> Rel"
interchange: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<tau> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<sigma> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>"   and "Span.seq \<tau> \<sigma>" shows "\<nu> \<bullet> \<mu> \<circ> \<tau> \<bullet> \<sigma> = (\<nu> \<circ> \<tau>) \<bullet> (\<mu> \<circ> \<sigma>)"
deleted_ids_last_ins: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c option"   and val :: "'d" shows "deleted_ids (xs @ [(oid, Insert ref val)]) = deleted_ids xs"
ccw'_subst_collinear: fixes t :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real"   and p :: "real \<times> real" assumes "coll t r s"   and "s \<noteq> t"   and "ccw' t r p" shows "ccw' t s p \<or> ccw' t p s"
mirror_pair_equal: fixes p :: "'b \<times> 'b" shows "equal_pair (mirror_pair p) = equal_pair p"
primitive_extractor_correct_6: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr"   and neg :: "bool" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "\<forall>disc2. \<not> has_disc_negated disc2 neg m \<longrightarrow> \<not> has_disc_negated disc2 neg ms"
antisym: fixes b :: "'c list"   and a :: "'c list" assumes "sublist b a"   and "sublist a b" shows "a = b"
telescopic_base: fixes K :: "'a set"   and F :: "'a set"   and n :: "nat"   and m :: "nat"   and E :: "'a set" assumes "subfield K R"   and "subfield F R"   and "dimension n K F"   and "dimension m F E" shows "dimension (n * m) K E"
continuous_map_product_coordinates: fixes i :: "'c"   and I :: "'c set"   and T :: "'c \<Rightarrow> 'd topology" assumes "i \<in> I" shows "continuous_map (product_topology T I) (T i) (\<lambda>x. x i)"
assms: shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F \<^bold>\<or> G)"
HARoot_def: fixes HA :: "('a, 'b, 'c) hierauto" shows "HARoot HA \<equiv> Root (SAs HA) (CompFun HA)"
that: fixes y' :: "'c" assumes "(x, y') \<in> set (zip xs ys)"   and "f y' = y" shows "thesis"
cofactorM_eq: shows "cofactorM = to_vec \<circ> Square_Matrix.cofactor \<circ> from_vec"
asym: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "asymmetric A less"   and "less x y"   and "less y x"   and "x \<in> A"   and "y \<in> A" shows "False"
right: fixes N :: "trm"   and x :: "name"   and a :: "coname"   and M :: "trm" assumes "\<not> fin N x"   and "a \<sharp> N"   and "x \<sharp> M" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>c N{x:=<a>.M}"
cycle_leave_ex_enter: fixes G :: "('a, 'b) graph_scheme"   and p :: "'a list"   and x :: "'a"   and y :: "'a" assumes "cycle G p"   and "(x, y) \<in> set (cycle_edges p)" shows "\<exists>z. (z, x) \<in> set (cycle_edges p)"
elements_mat: fixes A :: "'a mat" shows "elements_mat A = {A $$ (i, j) |i j. i < dim_row A \<and> j < dim_col A}"
widest_spread_axis_Un_2: fixes k :: "'a"   and K :: "'a set"   and P :: "(real, 'a) vec set"   and k' :: "'a" assumes "widest_spread_axis k K P"   and "spread k P \<le> spread k' P" shows "widest_spread_axis k' (K \<union> {k'}) P"
elem_set: fixes x :: "'c"   and xo :: "'c option" shows "(x \<in> set_option xo) = (xo = Some x)"
mat_multrow_def: fixes mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c"   and A :: "'c mat" shows "mat_multrow_gen mul k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
space_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "space (P J) = space (Pi\<^sub>M J (\<lambda>_. borel))"
pair_map_spmf1: fixes f :: "'c \<Rightarrow> 'a"   and p :: "'c spmf"   and q :: "'b spmf" shows "pair_spmf (map_spmf f p) q = map_spmf (apfst f) (pair_spmf p q)"
assms_3: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<paragraph>''invar-post''"   and "modifies modset s s\<^sub>0"   and "I s"   and "\<not> bval b s" shows "Q s"
module_iota_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('e, 'f, 'h) Module_scheme" shows "m\<iota>\<^bsub>R\<^esub> M = (\<lambda>x\<in>carrier M. x)"
of_real_sum: fixes f :: "'b \<Rightarrow> real"   and s :: "'b set" shows "of_real (sum f s) = (\<Sum>x\<in>s. of_real (f x))"
simps_5: fixes v1 :: "val"   and v2 :: "val" shows "val_type <|v1, v2|> = PRODUCT (val_type v1) (val_type v2)"
fold_cong: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and A :: "'k set"   and s :: "'l"   and t :: "'l"   and B :: "'k set" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "s = t"   and "A = B" shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
ccomp_exec_n: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and n :: "nat"   and t :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "ccomp c \<turnstile> (0, s, stk) \<rightarrow>^n (size (ccomp c), t, stk')" shows "(c, s) \<Rightarrow> t \<and> stk' = stk"
mult_scalar_minus_mult_left: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair (- p) q = - term_powerprod.mult_scalar pair_of_term term_of_pair p q"
one_smult: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and m :: "'b" assumes "VectorSpace smult V"   and "m \<in> V" shows "smult (1::'a) m = m"
well_order_on: fixes A :: "'a set" shows "\<exists>r. well_order_on A r"
f_def: shows "f \<equiv> fds_newman + fds_deriv fds_zeta"
pp_point_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_point f = (\<exists>i. \<forall>j. f (i, j) \<noteq> bot \<and> (\<forall>k. f (i, j) = f (i, k)) \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
sum_encode: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and dims :: "nat list"   and vars :: "nat set" shows "(\<Sum>i = 0..<partial_state.d1 dims vars. sum (f i) {0..<partial_state.d2 dims vars}) = (\<Sum>k = 0..<state_sig.d dims. f (partial_state.encode1 dims vars k) (partial_state.encode2 dims vars k))"
nt_abbrevE: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and G :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" assumes "NT f : F \<Longrightarrow> G" obtains "NatTrans f" and "NTDom f = F" and "NTCod f = G"
approx_hrabs_add_Infinitesimal: fixes e :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal" shows "\<bar>x\<bar> \<approx> \<bar>x + e\<bar>"
allT: fixes F :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "\<turnstile> (\<forall>x. \<box>F x) = (\<box>(\<forall>x. F x))"
components_are_iso: fixes a :: "'c" assumes "Cop.ide a" shows "S.iso (\<Phi>' a)"
min_in_range: fixes k :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "0 < k" shows "min\<^sub>k T k \<in> {1..k}"
combine_block_groups_pairwise: shows "pairwise_balance \<V> (\<B> + mset_set \<G>) 1"
sums_def: fixes f :: "nat \<Rightarrow> 'd"   and s :: "'d" shows "f sums s = (\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> s"
assms_1: shows "ipresIGWlsAll h hA MOD MOD'"
path_edges: fixes g :: "'a ugraph"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "path g u p v" shows "set p \<subseteq> edges g"
assms_6: shows "qf2 x1 y1 = 0"
R_sub_r: shows "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r} \<subseteq> r"
sym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(x \<sim> y) = (y \<sim> x)"
le_div_geq: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "n \<le> m" shows "m div n = Suc ((m - n) div n)"
nrows_eq_card_rows: fixes A :: "(('c, 'd) vec, 'e) vec" shows "nrows_iarray (matrix_to_iarray A) = CARD('e)"
p_is_fourier: shows "poly_of_list (map g [0..<k]) = finite_fourier_poly' f k"
is_run_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep" shows "df_system.is_run s\<^sub>0 lstep s \<equiv> \<exists>l. df_system.is_lrun s\<^sub>0 lstep l s"