hcju/isabelleps huggingface dataset:
ex_norm1: assumes "UNIV \<noteq> {0::'c}" shows "\<exists>x. norm x = 1"
res: fixes \<nu>3 :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "(\<nu>, \<nu>3) \<in> (prog_sem I ($\<alpha> vid1))\<^sup>*" shows "\<nu>3 \<in> fml_sem I (Pc pid1)"
less_add_same_cancel1: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
pairwise_orthogonal_imp_finite: fixes S :: "'b set" assumes "pairwise orthogonal S" shows "finite S"
right_not_zero: shows "0 < k \<and> \<not> coprime k n \<and> gauss_sum k \<noteq> 0"
R_def: shows "R = {(h, w). fst h = \<^bold>g [^] w \<and> snd h = g' [^] w}"
IdD: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Id" shows "a = b"
semi_mojmir: fixes \<psi> :: "'a ltl" shows "semi_mojmir \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w"
less_option_Some: fixes x :: "'a"   and y :: "'a" shows "(Some x < Some y) = (x < y)"
Rep_eqD: fixes x :: "'a word"   and y :: "'a word" assumes "to_bl x = to_bl y" shows "x = y"
basis_def: fixes K :: "('a, 'b) ring_scheme"   and V :: "('a, 'c, 'd) module_scheme"   and A :: "'c set" assumes "vectorspace K V" shows "vectorspace.basis K V A = (\<not> module.lin_dep K V A \<and> LinearCombinations.module.span K V A = carrier V \<and> A \<subseteq> carrier V)"
legal_wf_execD: fixes is_justification :: "'a prog \<Rightarrow> ('b \<times> ('c, 'b) obs_event action) llist \<times> (nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> ('c, 'b) justifying_execution) \<Rightarrow> bool"   and P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set"   and Ews :: "('b \<times> ('c, 'b) obs_event action) llist \<times> (nat \<Rightarrow> nat)" assumes "gen_legal_execution is_justification P \<E> Ews" shows "P \<turnstile> Ews \<surd>"
normalize_\<pi>: fixes \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < \<A>.max_rank_of \<Sigma> \<chi>"   and "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>, UNIV) w"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (\<A>.Acc\<^sub>\<A> \<Sigma> \<pi> \<chi>) w" obtains \<pi>\<^sub>\<A> :: "'a ltl \<Rightarrow> nat option" where "dom \<pi> = dom \<pi>\<^sub>\<A>" and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> \<pi>\<^sub>\<A> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w {q. dom \<pi>\<^sub>\<A> \<up>\<Turnstile>\<^sub>P q}" and "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>\<^sub>\<A>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (\<A>.Acc\<^sub>\<A> \<Sigma> \<pi>\<^sub>\<A> \<chi>) w"
Zero_not_eq_j0: shows "J.Zero \<noteq> J.j0"
assms: shows "s' = snd (fst (jmpl_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
rev_H: shows "bidel_arc.H G M rev.a' = rev.H"
simps_4: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.mem \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = mem"
quasi_inverse_simps_1: fixes f :: "'a" assumes "equivalence_map f" shows "equivalence_map f\<^sup>~"
HEndPhase0_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
poly_deg_monom_mult: fixes c :: "'b"   and p :: "('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'b"   and t :: "'c \<Rightarrow>\<^sub>0 'd" assumes "c \<noteq> (0::'b)"   and "p \<noteq> 0" shows "poly_deg (MPoly_Type_Class.punit.monom_mult c t p) = deg_pm t + poly_deg p"
fps_lr_inverse_nth_0_2: fixes f :: "'a fps"   and x :: "'a" shows "fps_nth (fps_right_inverse f x) 0 = x"
one_zeroD: assumes "(1::'b) = (0::'b)" shows "UNIV = {0::'b}"
powr_one_eq_one: fixes a :: "'b" shows "(1::'b) powr a = (1::'b)"
discover: fixes s :: "('v, 'es) state_scheme"   and u :: "'v"   and v :: "'v"   and s' :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "I s"   and "cond s"   and "\<not> is_break param s"   and "stack s \<noteq> []"   and "(u, v) \<in> pending s"   and "u = hd (stack s)"   and "v \<notin> dom (discovered s)"   and "s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)" shows "on_discover param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
dim_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C" shows "card (f ` C) = card C"
S2P_def: fixes A :: "'c set" shows "S2P A \<equiv> \<lambda>x. x \<in> A"
converse_rtranclp_into_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
Ln'_of_real_pos: fixes x :: "real" assumes "0 < x" shows "Ln' (complex_of_real x) = complex_of_real (ln x)"
interfree_aux_def: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set" shows "interfree_aux \<Gamma> \<Theta> F \<equiv> \<lambda>(c\<^sub>1, (P\<^sub>1, Q\<^sub>1, A\<^sub>1), c\<^sub>2, P\<^sub>2). \<forall>p c. atomicsR \<Gamma> \<Theta> P\<^sub>2 c\<^sub>2 (p, c) \<longrightarrow> \<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (Q\<^sub>1 \<inter> p) c Q\<^sub>1, Q\<^sub>1 \<and> \<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (A\<^sub>1 \<inter> p) c A\<^sub>1, A\<^sub>1 \<and> (\<forall>a. assertionsR \<Gamma> \<Theta> Q\<^sub>1 A\<^sub>1 P\<^sub>1 c\<^sub>1 a \<longrightarrow> \<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (a \<inter> p) c a, a)"
OPT2_is_ub: fixes qs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set qs \<subseteq> {x, y}"   and "x \<noteq> y" shows "T\<^sub>p_opt [x, y] qs \<le> T\<^sub>p [x, y] qs (OPT2 qs [x, y])"
subsequence_length: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" shows "length (w [i \<rightarrow> j]) = j - i"
fill_count_item: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat"   and x :: "'c" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "ns \<noteq> []"   and "offs_pred ns ub xs index key mi ma"   and "length xs \<le> ub" shows "count (mset (map the (fill xs ns index key ub mi ma))) x = count (mset xs) x + (if the None = x then ub - length xs else 0)"
map_eq_OkD: fixes f :: "'n \<Rightarrow> 'l"   and g :: "'o \<Rightarrow> 'm"   and x :: "('n, 'o) result"   and y :: "'m" assumes "map_result f g x = Ok y" shows "\<exists>x'. x = Ok x' \<and> y = g x'"
irreducible_linear_field_poly: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "irreducible [:a, b:]"
assms: fixes \<ss>\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "P \<ss>\<^sub>0" shows "wlp \<pi> (c \<ss>\<^sub>0) (Q \<ss>\<^sub>0) \<ss>\<^sub>0"
finite_elements: fixes A :: "'b set"   and P :: "'b set set" assumes "finite A"   and "partition_on A P" shows "finite P"
preserves_hhom: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "\<guillemotleft>F \<mu> : map\<^sub>0 (src\<^sub>C \<mu>) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
iso_is_arr: fixes f :: "'a" assumes "L'.C.iso f" shows "L'.C.arr f"
assms: shows "\<forall>w\<in>X. k \<le> |w|"
callee_invariant_on_fuse: fixes \<I>_adv_core :: "('a, 'b) \<I>"   and \<I>_usr_core :: "('c, 'd) \<I>"   and I_core :: "'e \<Rightarrow> bool"   and core :: "('e, 'f, 'a, 'c, 'b, 'd) core"   and \<I>_adv_rest :: "('g, 'h) \<I>"   and \<I>_usr_rest :: "('i, 'j) \<I>"   and I_rest :: "'k \<Rightarrow> bool"   and rest :: "('k, 'f, 'g, 'i, 'h, 'j, 'k) rest_scheme" assumes "WT_core \<I>_adv_core \<I>_usr_core I_core core"   and "WT_rest \<I>_adv_rest \<I>_usr_rest I_rest rest" shows "callee_invariant_on (fused_resource.fuse core rest) (pred_prod I_core I_rest) ((\<I>_adv_core \<oplus>\<^sub>\<I> \<I>_adv_rest) \<oplus>\<^sub>\<I> (\<I>_usr_core \<oplus>\<^sub>\<I> \<I>_usr_rest))"
vs1: shows "vs1 = getFrN (map snd txs) (map fst txs) [\<phi>1] (length txs)"
vrestr_comp: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "g \<in> M \<rightarrow>\<^sub>M M" shows "qmpt.vimage_restr M (f \<circ> g) A = qmpt.vimage_restr M g (qmpt.vimage_restr M f A)"
add_step_iff: fixes E :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and n :: "nat"   and P :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>xs y. Graph_Defs.steps E (x # xs @ [y]) \<and> length xs = Suc n \<longrightarrow> P xs y) = (\<forall>z xs y. Graph_Defs.steps E (x # z # xs @ [y]) \<and> length xs = n \<longrightarrow> P (z # xs) y)"
Abs_BinomialHeap_inject: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and y :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "x \<in> {q. BinomialHeapStruc.invar q}"   and "y \<in> {q. BinomialHeapStruc.invar q}" shows "(Abs_BinomialHeap x = Abs_BinomialHeap y) = (x = y)"
Match_1: fixes P :: "pi"   and Rs :: "residual"   and a :: "name" assumes "P \<Longrightarrow>\<^sub>l Rs" shows "[a\<frown>a]P \<Longrightarrow>\<^sub>l Rs"
word_bitwise_m1_simps_1: shows "~~ - 1 = 0"
distinct_remdups_wrt_rev: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and vs :: "'a list" shows "distinct (map f (remdups_wrt_rev f xs vs))"
simps_1: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "send\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = send\<langle>t \<cdot> \<theta>\<rangle>"
IF: fixes \<sigma> :: "'c" assumes "I (dom m) \<sigma>" shows "P \<sigma>"
trg_def: fixes \<mu> :: "'a arrow_of_spans_data" shows "trg \<mu> \<equiv> if V.arr \<mu> then mkObj (C.cod (Leg1 (Dom \<mu>))) else V.null"
igAbsRen_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igAbsRen MOD \<equiv> \<forall>xs y x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
boundOutputChainEqLength: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q" shows "length xvec = length yvec"
simps: shows "\<Delta>''.wf_residual_network = ((\<forall>x xa. (x, xa) \<in> \<^bold>E\<^bsub>\<Delta>''\<^esub> \<longrightarrow> (xa, x) \<notin> \<^bold>E\<^bsub>\<Delta>''\<^esub>) \<and> (source \<Delta>'', sink \<Delta>'') \<notin> \<^bold>E\<^bsub>\<Delta>''\<^esub>)"
set_inorder: fixes t :: "('c \<times> 'd) tree" shows "set (Tree2.inorder t) = Tree2.set_tree t"
inext_le_mono_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I \<le> inext b I"   and "a \<in> I"   and "b \<in> I"   and "\<exists>x\<in>I. inext a I < x" shows "a \<le> b"
minf_8: fixes t :: "'b" shows "\<exists>z. \<forall>x<z. (t \<le> x) = False"
augment_edge_impl_refine: fixes e :: "nat \<times> nat"   and cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and cap :: "'capacity" assumes "case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V"   and "\<forall>u. e \<noteq> (u, u)" shows "augment_edge_impl cf e cap \<le> (spec r. r = Graph.augment_edge cf e cap)"
language: fixes p :: "'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and r :: "'c stream" assumes "p \<in> initial A"   and "run A (w ||| r) p"   and "infs (accepting A) (p ## r)" shows "w \<in> language A"
map_simp_1: fixes f :: "'a" assumes "A.arr f" shows "local.map f = \<tau> (A.cod f) \<cdot>\<^sub>B \<sigma> f"
fold_eq_0_to_bool: fixes v :: "'b word" shows "(v = 0) = (\<not> to_bool v)"
delayedR_def: fixes R :: "('c \<Rightarrow> real) set"   and u :: "'c \<Rightarrow> real" shows "delayedR R u \<equiv> u \<oplus> (let I = SOME I. \<exists>r. Regions.valid_region \<X> k I r \<and> R = Regions.region \<X> I r; m = 1 - Max ({frac (u c) |c. c \<in> \<X> \<and> Regions.isIntv (I c)} \<union> {0}) in SOME t. u \<oplus> t \<in> R \<and> m / 2 \<le> t)"
distinct_1: shows "val \<noteq> Aodv_Basic.inv"
has_vderiv_on_domain: fixes s :: "'a" assumes "s \<in> S" shows " D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on T"
simps: fixes w :: "'a list" shows "LynFac w = (if w \<noteq> \<epsilon> then LynFac (w\<^sup><\<inverse>LynSuf w ) \<cdot> [LynSuf w] else \<epsilon>)"
lim: shows "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
lower_triangular_index: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "lower_triangular A"   and "i < j"   and "i < dim_row A"   and "j < dim_col A" shows "A $$ (i, j) = (0::'b)"
Exit_Def_empty: fixes V :: "'d" assumes "V \<in> lDef NewExit" shows "False"
simps_4: fixes uu :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uw :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom"   and s :: "char list \<Rightarrow> int" shows "postQ ({uu/uv/uw} CONSEQ C) s = postQ C s"
cases: fixes a :: "(nat \<times> nat) convnet" assumes "valid_net a" obtains M :: "nat" where "a = Input M"   | m :: "(nat \<times> nat) convnet" and r1 :: "nat" and r0 :: "nat" where "a = Conv (r0, r1) m" and "output_size m = r1" and "valid_net m"   | m1 :: "(nat \<times> nat) convnet" and m2 :: "(nat \<times> nat) convnet" where "a = Pool m1 m2" and "output_size m1 = output_size m2" and "valid_net m1" and "valid_net m2"
axioms_1: fixes \<eta> :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and G' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'" shows "monoid_homomorphism \<eta> G composition unit G' composition' unit'"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>tvar. P (TVar tvar)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"
imirror_iMin_Max: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<in> imirror I" shows "iMin I \<le> n \<and> n \<le> Max I"
iso_is_arr: fixes f :: "'a" assumes "local.iso f" shows "arr f"
full_functors_compose: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "full_functor A B F"   and "full_functor B C G" shows "full_functor A C (G \<circ> F)"
diff_diff_add_mset: fixes M :: "'a multiset"   and N :: "'a multiset"   and P :: "'a multiset" shows "M - N - P = M - (N + P)"
hom_0_iff: fixes x :: "'c"   and d :: "nat" shows "(monom x d = 0) = (x = (0::'c))"
only_non_strict: shows "lec_rel ` c ` Is \<subseteq> {Leq_Rel}"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
DIVERGENCES_def: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "DIVERGENCES P = snd P"
aprimedivisor_gt_0_nat: fixes d :: "nat" assumes "Suc 0 < d" shows "0 < aprimedivisor d"
combine_infinity_invariant_5: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_weak_until (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and d :: "bool"   and r :: "'b tree" shows "set_of (Node l x d r) = (if d then {} else {x}) \<union> set_of l \<union> set_of r"
him_of_real: fixes a :: "real" shows "HIm (quat_of_real a) = 0"
bernoulli_stream_component_probability_compl: fixes N :: "bool stream measure"   and p :: "real" assumes "N = bernoulli_stream p"   and "0 \<le> p"   and "p \<le> 1" shows "\<forall>n. emeasure N {w \<in> space N. \<not> w !! n} = ennreal (1 - p)"
exhaust: fixes y :: "pnat" obtains x1 :: "nat" where "y = pnat x1"   | "y = pnat.PInfty"
monoAux: fixes A :: "(pi \<times> pi) set"   and B :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "A \<subseteq> B" shows "P \<leadsto>[A] Q \<longrightarrow> P \<leadsto>[B] Q"
i_take_Suc_conv_app_nth: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> Suc n = f \<Down> n @ [f n]"
open_Times: fixes S :: "'b set"   and T :: "'c set" assumes "open S"   and "open T" shows "open (S \<times> T)"
match_rews_3: fixes k :: "'a match" shows "match_snil\<cdot>[::]\<cdot>k = k"
rkp'_def: shows "rkp' \<equiv> interpret_floatarith rkp vs"
Node_notin_subtrees_if: fixes a :: "'b"   and t :: "'b tree"   and l :: "'b tree"   and r :: "'b tree" assumes "a \<notin> set_tree t" shows "\<langle>l, a, r\<rangle> \<notin> subtrees t"
a_next_sibling_def: fixes node_ptr :: "(_) node_ptr" shows "a_next_sibling node_ptr = Heap_Error_Monad.bind (get_parent node_ptr) (\<lambda>parent_opt. case parent_opt of None \<Rightarrow> return None | Some parent \<Rightarrow> Heap_Error_Monad.bind (get_child_nodes parent) (\<lambda>children. case dropWhile (\<lambda>ptr. ptr = node_ptr) (dropWhile (\<lambda>ptr. ptr \<noteq> node_ptr) children) of [] \<Rightarrow> return None | x # xa \<Rightarrow> return (Some x)))"
sel_1: fixes x1 :: "le_rel"   and x2 :: "linear_poly"   and x3 :: "'a" shows "lec_rel (Le_Constraint x1 x2 x3) = x1"
split_elimvar: fixes L :: "atomUni list"   and eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list"   and At :: "atomUni"   and x :: "real" assumes "separateAtoms L = (eq, les, leq, neq)" shows "evalUni (elimVarUni_atom L At) x = evalUni (elimVarUni_atom (map EqUni eq @ map LessUni les @ map LeqUni leq @ map NeqUni neq) At) x"
chord_property: fixes C :: "'b"   and S :: "'b"   and T :: "'b"   and X :: "'b" assumes "dist C S = dist C T"   and "between (S, T) X" shows "dist S X * dist X T = (dist C S)\<^sup>2 - (dist C X)\<^sup>2"
sep_add_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj" shows "plus x zero = x"
iff_def: fixes p :: "fm"   and q :: "fm" shows "Ferrack.iff p q = (if p = q then T else if p = NOT q \<or> NOT p = q then F else if p = F then not q else if q = F then not p else if p = T then q else if q = T then p else Iff p q)"
MakeCatCompDef: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" shows "f \<approx>>\<^bsub>MakeCat C\<^esub> g = f \<approx>>\<^bsub>C\<^esub> g"
set_fold_insert: fixes xs1 :: "'c list"   and xs2 :: "'c list" shows "set (List.fold List.insert xs1 xs2) = set xs1 \<union> set xs2"
sup2E: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "sup A B x y" obtains "A x y"   | "B x y"
open_map_restrict: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T' :: "'d set"   and U :: "'c set"   and T :: "'d set" assumes "openin (top_of_set (S \<inter> f -` T')) U"   and "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)"   and "T' \<subseteq> T" shows "openin (top_of_set T') (f ` U)"
leg1_simps_6: shows "cod fg\<^sub>0h\<^sub>1.p\<^sub>1 = fg\<^sub>0h\<^sub>1.p\<^sub>1"
ipresIGSubstAbs_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'k, 'l, 'a, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGSubstAbs Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresSubstAbs Delta h hA MOD"
tm2: fixes x :: "'a" shows "n x \<cdot> t x = n (1::'a)"
dom_char: fixes f :: "'c \<times> 'c" shows "C'.VV.dom f = (if C'.VV.arr f then C'.VxV.dom f else C'.VxV.null)"
parent_child_rel_node_ptr: fixes parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "(parent, child_ptr) \<in> parent_child_rel h" shows "is_node_ptr_kind child_ptr"
map_sel_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b inference" shows "concl_of (map_inference f a) = f (concl_of a)"
trancl_reflcl: fixes r :: "('b \<times> 'b) set" shows "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
2: shows "b \<bullet> y \<in> {b \<bullet> y |y. y \<in> sat_dual A c}"
2: fixes s :: "'a"   and vl :: "'b"   and s1 :: "'state"   and vl1 :: "'value list" assumes "\<Delta> s vl s1 vl1" shows "\<Delta>' s vl s1 vl1"
rel_dist_to_stop_expr_def: fixes v :: "nat"   and a :: "nat" shows "rel_dist_to_stop_expr v a = Mult (Minus (Power (Var v) 2)) (Inverse (Mult (Num 2) (Var a)))"
iT_Minus_Minus_cut_eq: fixes k :: "nat"   and I :: "nat set" shows "k \<ominus> (k \<ominus> I \<down>\<le> k) = I \<down>\<le> k"
assms: shows "\<Gamma> \<turnstile> SimplyTyped.Pair A B : \<theta>"
assms: fixes i :: "'a"   and j :: "'b" assumes "i \<in> Basis"   and "j \<in> Basis" shows "((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
cases: fixes x :: "'e \<times> ('a, 'f, 'g, 'h) SLin_state_scheme" obtains s1 :: "'e" and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" where "x = (s1, s2)"
mk_conn_orderA: fixes H1 :: "'a hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>1 :: "'b hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>2 :: "'a hrel \<Rightarrow> 'b hrel"   and H2 :: "'b hrel \<Rightarrow> 'b hrel" shows "\<X>\<^bsub>H1 \<Leftarrow>\<langle>\<H>\<^sub>1,\<H>\<^sub>2\<rangle>\<Rightarrow> H2\<^esub> = utp_order H1"
of_circline_of_ocircline: fixes H' :: "ocircline" shows "of_circline (of_ocircline H') = H' \<or> of_circline (of_ocircline H') = opposite_ocircline H'"
Bad_def: shows "Bad = (\<lambda>U n. {x \<in> (T ^^ n) --` U. local_time B n x < k})"
disc_2: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done x \<or> \<not> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x))))" shows "\<not> Resumption.resumption.is_Done (x \<bind> f)"
poly_squarefree_decomp_order: fixes p :: "'b poly"   and q :: "'b poly"   and d :: "'b poly"   and e :: "'b poly"   and r :: "'b poly"   and s :: "'b poly"   and a :: "'b" assumes "pderiv p \<noteq> 0"   and "p = q * d"   and "pderiv p = e * d"   and "d = r * p + s * pderiv p" shows "order a q = (if order a p = 0 then 0 else 1)"
q1: fixes x :: "'a" assumes "x \<in> space_as_set A" shows "x \<in> closure {\<psi> + \<phi> |\<psi> \<phi>. \<psi> \<in> space_as_set B \<and> \<phi> \<in> space_as_set C}"
units_of_inv: fixes x :: "'a" assumes "x \<in> Units G" shows "inv\<^bsub>units_of G\<^esub> x = inv x"
sdrop_simps_2: fixes n :: "nat"   and s :: "'d stream" shows "stl (sdrop n s) = sdrop (Suc n) s"
last_conv_nth: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - 1)"