hcju/isabelleps huggingface dataset:
map_rev_Suc: fixes f :: "nat \<Rightarrow> 'a"   and j :: "nat" shows "map f (rev [0..<Suc j]) = f j # map f (rev [0..<j])"
remove_vertex_V: fixes x :: "'a" shows "V\<^bsub>H.remove_vertex x\<^esub> = V\<^bsub>remove_vertex x\<^esub> - {x}"
degree_one_roots: fixes a :: "'a"   and a' :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "a' \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes> a' = \<one>" shows "roots [a, b] = {#\<ominus> (a' \<otimes> b)#}"
pd': shows "Postdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
integrable_scaleR_right: fixes c :: "real"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> 0 \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c *\<^sub>R f x)"
get_ACS_union_get_IFS: fixes M :: "'a SecurityInvariant_configured list" shows "set (get_ACS M) \<union> set (get_IFS M) = set M"
fmap_of_list_app: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "fmap_of_list (xs @ ys) = fmap_of_list ys ++\<^sub>f fmap_of_list xs"
assms: shows "inverse (f $$ fls_subdegree f) \<noteq> (0::'a)"
quot_Succ: fixes x :: "tm" shows "\<guillemotleft>SUCC x\<guillemotright> = Q_Succ \<guillemotleft>x\<guillemotright>"
if_stm_def: fixes b :: "'b Assertion"   and x :: "'b"   and y :: "'b" shows "(If b then x else y) = [\<cdot> b ] * x \<sqinter> ([\<cdot> - b ] * y)"
replace_basis_completely: fixes vs :: "'v list"   and us :: "'v list" assumes "set vs \<subseteq> V"   and "R_lin_independent vs"   and "length us = length vs"   and "set us \<subseteq> RSpan vs"   and "R_lin_independent us" shows "set us \<subseteq> RSpan vs \<and> RSpan vs = RSpan us \<and> R_lin_independent us"
prems_1: fixes S :: "'a set" assumes "S \<in> insert S__ \<F>" shows "S \<in> lmeasurable"
normalized_nnf_match_normalize_match: fixes m :: "'b match_expr" shows "\<forall>m'\<in>set (normalize_match m). normalized_nnf_match m'"
sij_def: fixes i :: "nat"   and j :: "nat" shows "local.sij i j = ((zeroes m)[i := local.dij i j], (zeroes n)[j := local.eij i j])"
PrioMap_axioms: shows "PrioMap Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin"
gt_wt: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt s < wt t" shows "t >\<^sub>t s"
restrict_all: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "enum_class.enum \<langle>f\<rangle> enum_class.enum = f"
maxsimp_in_opp_half_apartment: fixes C :: "'a set" assumes "SimplicialComplex.maxsimp folding_g.Y C" shows "C \<in> folding_g.\<C> - g \<turnstile> folding_g.\<C>"
scaleR: fixes r :: "real"   and b :: "'b" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
fmrestrict_fset_fmmap: fixes A :: "'a fset"   and f :: "'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmrestrict_fset A (fmmap f m) = fmmap f (fmrestrict_fset A m)"
ccSup_empty: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
derivation_ge_shift: fixes F :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat" shows "derivation_ge (derivation_shift F 0 j) j"
pdevs_scaleR_def: fixes r :: "real pdevs"   and x :: "'b" shows "pdevs_scaleR r x = unop_pdevs (\<lambda>r. r *\<^sub>R x) r"
insert_before_def: fixes ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and child :: "(_) node_ptr option" shows "insert_before ptr node child = Heap_Error_Monad.bind (local.a_ensure_pre_insertion_validity node ptr child) (\<lambda>_. Heap_Error_Monad.bind (if Some node = child then local.a_next_sibling node else return child) (\<lambda>reference_child. Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>owner_document. Heap_Error_Monad.bind (adopt_node owner_document node) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (remove1 node disc_nodes)) (\<lambda>_. local.a_insert_node ptr node reference_child))))))"
simps_2: fixes t :: "(val \<times> val) list" shows "val_size (VFun t) = 1 + fun_size t"
GB_imp_zero_reducibility: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "i.punit.is_Groebner_basis G"   and "f \<in> punit.pmdl G" shows "(i.punit.red G)\<^sup>*\<^sup>* f 0"
integrable_mult_right: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c * f x)"
IH: assumes "x \<le> n"   and "y \<in> older_seniors x n"   and "older_seniors x n = older_seniors x (n + m)" shows "older_seniors y n = older_seniors y (n + m)"
subst_img_update: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b"   and t :: "('a, 'b) Term.term" assumes "\<sigma> v = Var v"   and "t \<noteq> Var v" shows "range_vars (\<sigma>(v := t)) = range_vars \<sigma> \<union> fv t"
imageI: fixes x :: "'a::type"   and A :: "'a::type set"   and f :: "'a::type \<Rightarrow> 'b::type" assumes "(x::'a::type) \<in> (A::'a::type set)" shows "(f::'a::type \<Rightarrow> 'b::type) (x::'a::type) \<in> f ` (A::'a::type set)"
ft: fixes n :: "nat" assumes "0 < n" shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (replicate n 0) = fold (\<lambda>i P. P|||FORK i) [1..<n] (FORK 0)"
mult_cancel_left: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](g1)"   and "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'b)" shows "((\<lambda>x. f1 x * f2 x) \<in> L F (\<lambda>x. g1 x * g2 x)) = (f2 \<in> L F g2)"
cross7_simps_34: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x + y) \<bullet> z = x \<bullet> z + y \<bullet> z"
simple_distributed_joint: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e"   and Px :: "'d \<times> 'e \<Rightarrow> real" assumes "simple_distributed M (\<lambda>x. (X x, Y x)) Px" shows "distributed M (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M)) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Px x else 0))"
cut_greater_ge_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t = I \<down>\<ge> t - {t}"
prems_3: shows "P,shp s1 \<turnstile>\<^sub>b (e1,b1) \<surd>"
set_shadow_root_get_disconnected_nodes_is_l_set_shadow_root_get_disconnected_nodes: shows "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs"
HC_mono: fixes S :: "'b formula set"   and F :: "'b formula"   and T :: "'b formula set" assumes "(S::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"   and "(S::'b formula set) \<subseteq> (T::'b formula set)" shows "(T::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"
contract_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div> \<phi> \<subseteq> K"
sum_image_mset_mono_mem: fixes n :: "'a"   and M :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "n \<in># M" shows "f n \<le> \<Sum>\<^sub># (image_mset f M)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Rval x1 = Rval y1) = (x1 = y1)"
one_index: fixes i :: "'e" shows "1 $ i = (1::'d)"
degree_pderiv_le: fixes f :: "'c poly" shows "degree (pderiv f) \<le> degree f - 1"
type_definition_lifted_distrib_lattice_top: shows "type_definition Rep_lifted_distrib_lattice_top Abs_lifted_distrib_lattice_top {xf. True}"
induct: fixes P :: "msg \<Rightarrow> bool"   and msg :: "msg" assumes "\<And>x. P (Num x)"   and "\<And>x. P (Agent x)"   and "\<And>x. P (Pwd x)"   and "\<And>x. P (Key x)"   and "\<And>x1a x2a. P (x1a \<otimes> x2a)"   and "\<And>x. P x \<Longrightarrow> P (Hash x)"   and "\<And>x1a x2a. P x2a \<Longrightarrow> P (Crypt x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P \<lbrace>x1a, x2a\<rbrace>"   and "\<And>x1a x2a. P x2a \<Longrightarrow> P \<langle>x1a, x2a\<rangle>"   and "\<And>x. P x \<Longrightarrow> P (Log x)" shows "P msg"
clinear_def: fixes f :: "'a \<Rightarrow> 'b" shows "clinear f \<equiv> Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) f"
L\<^sub>3_def: fixes X :: "'b ltln set"   and Y :: "'b ltln set" shows "L\<^sub>3 X Y = {w. \<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)}"
periodic: fixes a :: "nat" shows "\<chi> (a + n) = \<chi> a"
bprv_\<phi>L_eqv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (eqv (\<phi>L \<phi>) (imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>))"
intensional_simps_25: fixes y :: "'b \<Rightarrow> bool" shows "y \<or> #False \<equiv> y"
Vset_def: fixes x :: "hf" shows "Vset x = ord_rec 0 HPow (\<lambda>z. 0) x"
cblinfun_extension_def: fixes S :: "'a set"   and \<phi> :: "'a \<Rightarrow> 'b" shows "cblinfun_extension S \<phi> = (SOME B. \<forall>x\<in>S. B *\<^sub>V x = \<phi> x)"
notin_fset: fixes x :: "'x"   and S :: "'x fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
dvd_defD2: fixes a :: "nat"   and b :: "nat" assumes "a dvd b" shows "\<exists>k. b = k * a"
coeff_nth: fixes i :: "nat"   and p :: "'a list" assumes "i < length p" shows "local.coeff p i = p ! (degree p - i)"
RAT_F1_i: shows "map of_int_hom.vec_hom fs' ! i = map of_int_hom.vec_hom fs ! i - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j"
cod_simp: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) f" shows "partial_magma.cod (product_category.comp C1 C2) f = (partial_magma.cod C1 (fst f), partial_magma.cod C2 (snd f))"
start_addrs_def: shows "start_addrs = fst (snd start_heap_data)"
ntadj_sub_assign: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'b) trm"   and e :: "('e + 'c, 'f) trm"   and x :: "'f" shows "(\<Union>y\<in>{y. Inr y \<in> SIGT e}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGP (x := e)}. FVT (\<sigma> y))"
assms_5: fixes R :: "ccs"   and S :: "ccs" assumes "(R \<parallel> !R, S) \<in> Rel'" shows "(!R, S) \<in> Rel'"
select_convs_2: fixes nodes :: "'c set"   and edges :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
rel_FGco_pos_distr_cond_def: fixes Co1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Co2 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Co2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Co3 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Co3' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Co4 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Co4' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Co5 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Co5' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Co6 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and Co6' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra1 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and Contra1' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra2 :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra2' :: "'ar \<Rightarrow> 'as \<Rightarrow> bool"   and Contra3 :: "'at \<Rightarrow> 'au \<Rightarrow> bool"   and Contra3' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Contra4 :: "'aw \<Rightarrow> 'ax \<Rightarrow> bool"   and Contra4' :: "'ax \<Rightarrow> 'ay \<Rightarrow> bool"   and uu :: "('az \<times> 'ba \<times> 'bb \<times> 'bc \<times> 'bd) itself" shows "rel_FGco_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Co6 Co6' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' uu = (\<forall>L1 L1'. rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 OO rel_FGco L1' Co1' Co2' Co3' Co4' Co5' Co6' Contra1' Contra2' Contra3' Contra4' \<le> rel_FGco (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Co5 OO Co5') (Co6 OO Co6') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4'))"
acyclic_2a_def: fixes x :: "'a" shows "acyclic_2a x \<equiv> \<forall>y. y \<squnion> y\<^sup>T \<le> x \<and> asymmetric y \<longrightarrow> pd_kleene_allegory_class.acyclic y"
DDH0_eq_ddh_0: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" assumes "ddh_ext \<G>" shows "ddh.DDH0 \<G> \<A> = ddh.ddh_0 \<G> \<A>"
ProblemIkPol_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool" shows "ProblemIkPol wtFsym wtPsym arOf resOf parOf \<Phi> infTp \<equiv> ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp"
list_slice_append_mult: fixes xs :: "'b list"   and m :: "nat"   and k :: "nat"   and ys :: "'b list" assumes "length xs = m * k" shows "list_slice (xs @ ys) k = list_slice xs k @ list_slice ys k"
elims_2: fixes x :: "'a option"   and xa :: "'a list"   and xb :: "'a list" assumes "proper_interval_Compl_set_aux x xa xb" obtains ao :: "'a option" and xc :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = ao" and "xa = xc # xs" and "xb = y # ys" and "if xc < y then \<not> proper_interval ao (Some xc) \<and> proper_interval_Compl_set_aux (Some xc) xs (y # ys) else if y < xc then \<not> proper_interval ao (Some y) \<and> proper_interval_Compl_set_aux (Some y) (xc # xs) ys else \<not> proper_interval ao (Some xc) \<and> (ys = [] \<longrightarrow> xs \<noteq> [])"
topological_basis_def: fixes B :: "'a set set" shows "local.topological_basis B = ((\<forall>b\<in>B. p b) \<and> (\<forall>x. p x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
assms: shows "extend S ( \<LM> At i  \<RM> \<Rightarrow>* At i) = ( \<Gamma> \<Rightarrow>* \<Delta>)"
Cons: fixes x :: "'a"   and xs :: "'a list" shows "sum_list (x # xs) = x + sum_list xs"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<tau>"
variants_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.variants subst_atm C D = (substitution_ops.generalizes subst_atm C D \<and> substitution_ops.generalizes subst_atm D C)"
finite_Collect: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "finite S"   and "inj f" shows "finite {a. f a \<in> S}"
HAInitState_InitConf: fixes A :: "('d, 'e, 'f) hierauto" shows "HAInitState A \<in> InitConf A"
simptm_ci: fixes vs :: "'b list"   and bs :: "'b list"   and t :: "tm" shows "Itm vs bs (simptm t) = Itm vs bs t"
domains_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.domains (g \<cdot>\<^sub>C f) = C.domains f"
document_ptr_kinds_eq_h2: shows "document_ptr_kinds h2 = document_ptr_kinds h' |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr|}"
Hermite_of_row_i_def: fixes ass :: "'d \<Rightarrow> 'd"   and res :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f" shows "Hermite_of_row_i ass res A i = (if is_zero_row i A then A else let j = LEAST n. A $ i $ n \<noteq> (0::'d); Aij = A $ i $ j; A' = mult_row A i (ass Aij div Aij) in Hermite_reduce_above A' (mod_type_class.to_nat i) i j res)"
condition_split: fixes P :: "('c \<times> 'd) set \<times> bool \<Rightarrow> bool"   and C :: "'d \<Rightarrow> bool"   and a :: "'d \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and b :: "'d \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and s :: "'d" shows "P (condition C   a   b s) = ((C s \<longrightarrow> P (a s)) \<and> (\<not> C s \<longrightarrow> P (b s)))"
transpose_def: fixes A :: "(('e, 'g) vec, 'f) vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $ j $ i)"
qfree_subst_peps: fixes \<phi> :: "atom fm"   and rcs :: "real \<times> real list" assumes "nqfree \<phi>" shows "qfree (subst\<^sub>+ \<phi> rcs)"
1: shows "fv (\<Gamma>, Terms.Let as body) \<subseteq> set L \<union> domA \<Gamma>"
induct: fixes P :: "Core.term \<Rightarrow> bool"   and a0 :: "Core.term" assumes "\<And>T t. \<lbrakk>P t; \<And>x51 x52 x3. \<lbrakk>eta_norm t = x51 $ x52; x52 = Bv x3; x3 = 0; \<not> is_dependent x51\<rbrakk> \<Longrightarrow> P x51\<rbrakk> \<Longrightarrow> P (Abs T t)"   and "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (t $ u)"   and "\<And>v va. P (Ct v va)"   and "\<And>v va. P (Fv v va)"   and "\<And>v. P (Bv v)" shows "P a0"
assms_1: shows "matcher_agree_on_exact_matches \<gamma> \<beta>"
unrestricted_resolution_is_deductive_complete: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "saturated_binary_rule resolvent S"   and "all_fulfill finite S"   and "C \<in> implicates S" shows "redundant C S"
create_element_preserves_wellformedness_2: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "type_wf h'"
mult_n1n: fixes n :: "nat"   and m :: "nat" assumes "n = m * n"   and "0 < n" shows "m = 1"
cgenideal_eq_genideal: fixes i :: "'a list" assumes "i \<in> carrier (poly_ring R)" shows "PIdl\<^bsub>poly_ring R\<^esub> i = Idl\<^bsub>poly_ring R\<^esub> {i}"
inf_llist_lnth: fixes xs :: "'a llist" assumes "\<not> lfinite xs" shows "inf_llist (lnth xs) = xs"
size_mset_tree: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.tree_invar t" shows "size (BinomialHeapStruc.tree_to_multiset t) = 2 ^ BinomialHeapStruc.rank t"
VarP_sf: fixes x :: "tm" shows "Sigma_fm (VarP x)"
assms_1: fixes t :: "('fun, 'a) Term.term" assumes "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
H0_2: shows "\<forall>n. u n \<in> to_Gromov_completion ` X \<union> Gromov_boundary"
assms_1: shows "mat (row_length M1) (length M1) M1"
msubstlt_def: fixes c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm"   and a :: "poly"   and r :: "tm" shows "msubstlt c t d s a r = evaldjf (\<lambda>(x, y). Parametric_Ferrante_Rackoff.conj x y) [let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Lt (tm.Add (tm.Mul (~\<^sub>p a) (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), let cd = c *\<^sub>p d in (lt (CP cd), Lt (tm.Sub (tm.Mul a (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Lt (tm.Add (tm.Mul (~\<^sub>p a) t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP c)) (Eq (CP d)), Lt (tm.Sub (tm.Mul a t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Lt (tm.Add (tm.Mul (~\<^sub>p a) s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP d)) (Eq (CP c)), Lt (tm.Sub (tm.Mul a s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (Eq (CP d)), Lt r)]"
tauLaw2SimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<tau>.P \<leadsto><Rel> P \<oplus>\<^sub>\<top> \<tau>.P"
hom: shows "homeomorphism (cbox w z) (cbox w z) f g"
Pbij_compose_Rng: fixes \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set" shows "Pbij_Rng (Pbij_compose \<beta> \<gamma>) \<subseteq> Pbij_Rng \<gamma>"
Tau_def: shows "\<tau> \<equiv> Abs_action Tau_Rep"
sigma_sets_subseteq: fixes A :: "'d set set"   and B :: "'d set set"   and X :: "'d set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
respects_label_fixes_chamber_imp_fixespointwise: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'c set"   and \<phi> :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "ChamberComplex.label_wrt X B \<phi>"   and "SimplicialComplex.maxsimp X C"   and "f ` C = C"   and "\<forall>v\<in>C. \<phi> (f v) = \<phi> v" shows "fixespointwise f C"
table_auxlist': shows "\<forall>(t, X)\<in>set auxlist'. table n R X"
mod_mult_self1_is_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times b a) b = zero"
ge_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.non_strict_order less_eq"   and "less_eq y x"   and "less_eq z y" shows "less_eq z x"
ext: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. f x = g x" shows "f = g"
length_rotate: fixes n :: "nat"   and xs :: "'b list" shows "\<^bold>|rotate n xs\<^bold>| = \<^bold>|xs\<^bold>|"
assms_1: fixes i :: "nat" shows "emeasure M (N i) = 0"
facetrel_subset: fixes y :: "'b set"   and x :: "'b set" assumes "y \<lhd> x" shows "order.greater_eq x y"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and f :: "nat \<Rightarrow> 'a" shows "is_prefix (x # xs) f = (x = f (length xs) \<and> is_prefix xs f)"
l8_18_existence: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<exists>X. Col A B X \<and> A B Perp C X"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) node_ptr"   and setter :: "('e \<Rightarrow> 'e) \<Rightarrow> 'c RNode_scheme \<Rightarrow> 'c RNode_scheme"   and v :: "'e" assumes "h \<turnstile> ok put_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e ptr setter v" shows "ptr |\<in>| node_ptr_kinds h"
cons_perm_eq: fixes z :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "(z # xs <~~> z # ys) = (xs <~~> ys)"
power3_eq_cube: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "power.power one times a 3 = times (times a a) a"
IH: shows "card {i. n < i \<and> i \<le> n' \<and> p i} \<le> card {i. n < i \<and> i \<le> n'' \<and> p i}"
sub_sup_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y" shows "- x \<squnion> - z \<le> - y \<squnion> - z"
induceES_def: fixes SES :: "('d, 'c) SES_rec" shows "induceES SES \<equiv> \<lparr>E_ES = E\<^bsub>SES\<^esub>, I_ES = I\<^bsub>SES\<^esub>, O_ES = O\<^bsub>SES\<^esub>, Tr_ES = possible_traces SES\<rparr>"
Posix_elims_4: fixes s :: "'b list"   and r1 :: "'b rexp"   and r2 :: "'b rexp"   and v :: "'b val" assumes "s \<in> Plus r1 r2 \<rightarrow> v" obtains v :: "'b val" where "va__ = val.Left vb__" and "s \<in> r1 \<rightarrow> vb__"   | v :: "'b val" where "va__ = val.Right v" and "s \<in> r2 \<rightarrow> v" and "s \<notin> lang r1"
KIR: fixes x :: "'v" assumes "x \<in> RF\<^sup>\<circ> (TER f)"   and "x \<notin> A \<Gamma>" shows "KIR h x"